From ff949389c64108e35ed93c5f08d02e767914dd16 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 17 Nov 2025 17:04:00 +0000
Subject: [PATCH 01/37] Adaptive Multigrid

---
 firedrake/__init__.py                         |   2 +
 firedrake/mg/__init__.py                      |  10 +-
 firedrake/mg/adaptive_hierarchy.py            | 360 ++++++++++++++++++
 firedrake/mg/adaptive_transfer_manager.py     | 147 +++++++
 .../multigrid/test_adaptive_multigrid.py      | 360 ++++++++++++++++++
 5 files changed, 875 insertions(+), 4 deletions(-)
 create mode 100644 firedrake/mg/adaptive_hierarchy.py
 create mode 100644 firedrake/mg/adaptive_transfer_manager.py
 create mode 100644 tests/firedrake/multigrid/test_adaptive_multigrid.py

diff --git a/firedrake/__init__.py b/firedrake/__init__.py
index 70dec1d6ca..fdc9e9e38c 100644
--- a/firedrake/__init__.py
+++ b/firedrake/__init__.py
@@ -73,6 +73,8 @@ def init_petsc():
 from firedrake.mg.interface import *
 from firedrake.mg.embedded import *
 from firedrake.mg.opencascade_mh import *
+from firedrake.mg.adaptive_hierarchy import *
+from firedrake.mg.adaptive_transfer_manager import *
 from firedrake.norms import *
 from firedrake.nullspace import *
 from firedrake.parameters import *
diff --git a/firedrake/mg/__init__.py b/firedrake/mg/__init__.py
index f5e043c3d1..6be5fcb7eb 100644
--- a/firedrake/mg/__init__.py
+++ b/firedrake/mg/__init__.py
@@ -1,4 +1,6 @@
-from .mesh import *              # noqa: F401
-from .interface import *         # noqa: F401
-from .embedded import *          # noqa: F401
-from .opencascade_mh import *    # noqa: F401
+from .mesh import *                       # noqa: F401
+from .interface import *                  # noqa: F401
+from .embedded import *                   # noqa: F401
+from .opencascade_mh import *             # noqa: F401
+from .adaptive_hierarchy import *         # noqa: F401
+from .adaptive_transfer_manager import *  # noqa: F401
diff --git a/firedrake/mg/adaptive_hierarchy.py b/firedrake/mg/adaptive_hierarchy.py
new file mode 100644
index 0000000000..0749d42b5f
--- /dev/null
+++ b/firedrake/mg/adaptive_hierarchy.py
@@ -0,0 +1,360 @@
+"""
+This module contains the class for the AdaptiveMeshHierarchy and
+related helper functions
+"""
+
+from fractions import Fraction
+from collections import defaultdict
+import numpy as np
+
+from firedrake.cofunction import Cofunction
+from firedrake.function import Function
+from firedrake.functionspace import FunctionSpace
+from firedrake.mesh import Mesh, Submesh, RelabeledMesh
+from firedrake.mg import HierarchyBase
+from firedrake.mg.utils import set_level, get_level
+from ufl import conditional, gt
+
+__all__ = ["AdaptiveMeshHierarchy"]
+
+
+class AdaptiveMeshHierarchy(HierarchyBase):
+    """
+    HierarchyBase for hierarchies of adaptively refined meshes
+    """
+
+    def __init__(self, mesh, refinements_per_level=1, nested=True):
+        self.meshes = tuple(mesh)
+        self._meshes = tuple(mesh)
+        self.submesh_hierarchies = []
+        self.coarse_to_fine_cells = {}
+        self.fine_to_coarse_cells = {}
+        self.fine_to_coarse_cells[Fraction(0, 1)] = None
+        self.refinements_per_level = refinements_per_level
+        self.nested = nested
+        set_level(mesh[0], self, 0)
+        self.split_cache = {}
+
+    def add_mesh(self, mesh):
+        """
+        Adds newly refined mesh into hierarchy.
+        Then computes the coarse_to_fine and fine_to_coarse mappings.
+        Constructs intermediate submesh hierarchies with this.
+        """
+        if mesh.topological_dimension <= 2:
+            max_children = 4
+        else:
+            max_children = 16
+        self._meshes += tuple(mesh)
+        self.meshes += tuple(mesh)
+        coarse_mesh = self.meshes[-2]
+        level = len(self.meshes)
+        set_level(self.meshes[-1], self, level - 1)
+        self._shared_data_cache = defaultdict(dict)
+
+        # extract parent child relationships from netgen meshes
+        (c2f, f2c) = get_c2f_f2c_fd(mesh, coarse_mesh)
+        c2f_global_key = Fraction(len(self.meshes) - 2, 1)
+        f2c_global_key = Fraction(len(self.meshes) - 1, 1)
+        self.coarse_to_fine_cells[c2f_global_key] = c2f
+        self.fine_to_coarse_cells[f2c_global_key] = np.array(f2c)
+
+        # split both the fine and coarse meshes into the submeshes
+        (coarse_splits, fine_splits, num_children) = split_to_submesh(
+            mesh, coarse_mesh, c2f, f2c
+        )
+        for i in range(1, max_children + 1):
+            coarse_mesh.mark_entities(coarse_splits[i], i)
+            mesh.mark_entities(fine_splits[i], int(f"10{i}"))
+
+        coarse_indicators = [
+            coarse_splits[i]
+            for i in range(1, max_children + 1)
+        ]
+        coarse_labels = list(range(1, max_children + 1))
+        coarse_mesh = RelabeledMesh(
+            coarse_mesh,
+            coarse_indicators,
+            coarse_labels,
+            name="Relabeled_coarse",
+        )
+        c_subm = {
+            j: Submesh(coarse_mesh, coarse_mesh.topology_dm.getDimension(), j)
+            for j in range(1, max_children + 1)
+            if any(num_children == j)
+        }
+        set_level(coarse_mesh, self, level - 2)
+
+        fine_indicators = [
+            fine_splits[i]
+            for i in range(1, max_children + 1)
+        ]
+        fine_labels = list(range(1, max_children + 1))
+        mesh = RelabeledMesh(
+            mesh,
+            fine_indicators,
+            fine_labels,
+        )
+        f_subm = {
+            int(str(j)[-2:]): Submesh(mesh, mesh.topology_dm.getDimension(), j)
+            for j in [int("10" + str(i)) for i in range(1, max_children + 1)]
+            if any(num_children == int(str(j)[-2:]))
+        }
+        set_level(mesh, self, level - 1)
+
+        # update c2f and f2c for submeshes by mapping numberings
+        # on full mesh to numberings on coarse mesh
+        parents_per_child_count = [
+            len([el for el in c2f if len(el) == j])
+            for j in range(1, max_children + 1)
+        ]  # stores number of parents for each amount of children
+        c2f_adjusted = {
+            j: np.zeros((num_parents, j))
+            for j, num_parents in enumerate(parents_per_child_count, 1)
+            if num_parents != 0
+        }
+        f2c_adjusted = {
+            j: np.zeros((num_parents * j, 1))
+            for j, num_parents in enumerate(parents_per_child_count, 1)
+            if num_parents != 0
+        }
+
+        coarse_full_to_sub_map = {
+            i: full_to_sub(coarse_mesh, c_subm[i])
+            for i in c_subm
+        }
+        fine_full_to_sub_map = {
+            j: full_to_sub(mesh, f_subm[j])
+            for j in f_subm
+        }
+
+        for i, children in enumerate(c2f):
+            n = len(children)
+            if 1 <= n <= max_children:
+                coarse_id_sub = coarse_full_to_sub_map[n][i]
+                fine_id_sub = fine_full_to_sub_map[n][np.array(children)]
+                c2f_adjusted[n][coarse_id_sub] = fine_id_sub
+
+        for j, parent in enumerate(f2c):
+            n = num_children[parent].item()
+            if 1 <= n <= max_children:
+                fine_id_sub = fine_full_to_sub_map[n][j]
+                coarse_id_sub = coarse_full_to_sub_map[n][parent.item()]
+                f2c_adjusted[n][fine_id_sub, 0] = coarse_id_sub
+
+        c2f_subm = {
+            i: {Fraction(0, 1): c2f_adjusted[i].astype(int)}
+            for i in c2f_adjusted
+        }
+        f2c_subm = {i: {Fraction(1, 1): f2c_adjusted[i]} for i in f2c_adjusted}
+
+        hierarchy_dict = {
+            i: HierarchyBase(
+                [c_subm[i], f_subm[i]], c2f_subm[i], f2c_subm[i], nested=True
+            )
+            for i in c_subm
+        }
+        self.submesh_hierarchies.append(hierarchy_dict)
+
+    def refine(self, refinements):
+        """
+        Refines and adds mesh if input a boolean vector corresponding to cells
+        """
+        ngmesh = self.meshes[-1].netgen_mesh
+        for i, el in enumerate(ngmesh.Elements2D()):
+            el.refine = refinements[i]
+
+        ngmesh.Refine(adaptive=True)
+        mesh = Mesh(ngmesh)
+        self.add_mesh(mesh)
+
+    def adapt(self, eta, theta):
+        """
+        Implement Dorfler marking, refines mesh from error estimator
+        """
+        mesh = self.meshes[-1]
+        W = FunctionSpace(mesh, "DG", 0)
+        markers = Function(W)
+
+        with eta.dat.vec_ro as eta_:
+            eta_max = eta_.max()[1]
+
+        should_refine = conditional(gt(eta, theta * eta_max), 1, 0)
+        markers.interpolate(should_refine)
+
+        refined_mesh = mesh.refine_marked_elements(markers)
+        self.add_mesh(refined_mesh)
+        return refined_mesh
+
+    def split_function(self, u, child=True):
+        """
+        Split input function across submeshes
+        """
+        V = u.function_space()
+        full_mesh = V.mesh()
+        _, level = get_level(full_mesh)
+
+        ind = 1 if child else 0
+        hierarchy_dict = self.submesh_hierarchies[int(level) - ind]
+        parent_mesh = hierarchy_dict[[*hierarchy_dict][0]].meshes[ind].submesh_parent
+        parent_space = V.reconstruct(parent_mesh)
+        u_corr_space = Function(parent_space, val=u.dat)
+        key = (u, child)
+        try:
+            split_functions = self.split_cache[key]
+        except KeyError:
+            split_functions = self.split_cache.setdefault(key, {})
+
+        for i in hierarchy_dict:
+            try:
+                f = split_functions[i].zero()
+            except KeyError:
+                V_split = V.reconstruct(mesh=hierarchy_dict[i].meshes[ind])
+                assert (
+                    V_split.mesh().submesh_parent
+                    == u_corr_space.function_space().mesh()
+                )
+                f = split_functions.setdefault(
+                    i,
+                    Function(V_split, name=str(i))
+                )
+
+            f.assign(u_corr_space)
+        return split_functions
+
+    def use_weight(self, V, child):
+        """
+        Counts DoFs across submeshes, computes partition of unity
+        """
+        w = Function(V).assign(1)
+        splits = self.split_function(w, child)
+
+        self.recombine(splits, w, child)
+        with w.dat.vec as wvec:
+            wvec.reciprocal()
+        return w
+
+    def recombine(self, split_funcs, f, child=True):
+        """
+        Recombines functions on submeshes back full mesh
+        """
+        V = f.function_space()
+        f.zero()
+        parent_mesh = (
+            split_funcs[[*split_funcs][0]].function_space().mesh().submesh_parent
+        )
+        V_label = V.reconstruct(mesh=parent_mesh)
+        if isinstance(f, Function):
+            f_label = Function(V_label, val=f.dat)
+        elif isinstance(f, Cofunction):
+            f_label = Cofunction(V_label, val=f.dat)
+
+        for split_label, val in split_funcs.items():
+            assert val.function_space().mesh().submesh_parent == parent_mesh
+            if child:
+                split_label = int("10" + str(split_label))
+            if isinstance(f_label, Function):
+                f_label.assign(val, allow_missing_dofs=True)
+            else:
+                curr = Function(f_label.function_space()).assign(
+                    val, allow_missing_dofs=True
+                )
+                f_label.assign(f_label + curr)  # partition of unity for restriction
+        return f
+
+
+def get_c2f_f2c_fd(mesh, coarse_mesh):
+    """
+    Construct coarse->fine and fine->coarse relations by mapping netgen elements to firedrake ones
+    """
+    ngmesh = mesh.netgen_mesh
+    num_parents = coarse_mesh.num_cells()
+
+    if mesh.topology_dm.getDimension() == 2:
+        parents = ngmesh.parentsurfaceelements.NumPy()
+        elements = ngmesh.Elements2D()
+    elif mesh.topology_dm.getDimension() == 3:
+        parents = ngmesh.parentelements.NumPy()
+        elements = ngmesh.Elements3D()
+    else:
+        raise RuntimeError("Adaptivity not implemented in dimension of mesh")
+
+    c2f = [[] for _ in range(num_parents)]
+    f2c = [[] for _ in range(mesh.num_cells())]
+
+    if parents.shape[0] == 0:
+        raise RuntimeError("Added mesh has not refined any cells from previous mesh")
+    for l, _ in enumerate(elements):
+        if parents[l][0] == -1 or l < num_parents:
+            f2c[mesh._cell_numbering.getOffset(l)].append(
+                coarse_mesh._cell_numbering.getOffset(l)
+            )
+            c2f[coarse_mesh._cell_numbering.getOffset(l)].append(
+                mesh._cell_numbering.getOffset(l)
+            )
+
+        elif parents[l][0] < num_parents:
+            fine_ind = mesh._cell_numbering.getOffset(l)
+            coarse_ind = coarse_mesh._cell_numbering.getOffset(parents[l][0])
+            f2c[fine_ind].append(coarse_ind)
+            c2f[coarse_ind].append(fine_ind)
+
+        else:
+            a = parents[parents[l][0]][0]
+            while a >= num_parents:
+                a = parents[a][0]
+
+            f2c[mesh._cell_numbering.getOffset(l)].append(
+                coarse_mesh._cell_numbering.getOffset(a)
+            )
+            c2f[coarse_mesh._cell_numbering.getOffset(a)].append(
+                mesh._cell_numbering.getOffset(l)
+            )
+
+    return c2f, np.array(f2c).astype(int)
+
+
+def split_to_submesh(mesh, coarse_mesh, c2f, f2c):
+    """
+    Computes submesh split from full mesh.
+    Returns splits which are Functions denoting whether elements
+    belong to the corresponing submesh (bool)
+    """
+    if mesh.topological_dimension <= 2:
+        max_children = 4
+    else:
+        max_children = 16
+    V = FunctionSpace(mesh, "DG", 0)
+    V2 = FunctionSpace(coarse_mesh, "DG", 0)
+    coarse_splits = {
+        i: Function(V2, name=f"{i}_elements") for i in range(1, max_children + 1)
+    }
+    fine_splits = {
+        i: Function(V, name=f"{i}_elements") for i in range(1, max_children + 1)
+    }
+    num_children = np.zeros((len(c2f)))
+
+    for i, children in enumerate(c2f):
+        n = len(children)
+        if 1 <= n <= max_children:
+            coarse_splits[n].dat.data[i] = 1
+            num_children[i] = n
+
+    for i in range(1, max_children + 1):
+        fine_splits[i].dat.data[num_children[f2c.squeeze()] == i] = 1
+
+    return coarse_splits, fine_splits, num_children
+
+
+def full_to_sub(mesh, submesh):
+    """
+    Returns the submesh element id associated with the full mesh element id
+    """
+    V1 = FunctionSpace(mesh, "DG", 0)
+    V2 = FunctionSpace(submesh, "DG", 0)
+    u1 = Function(V1)
+    u2 = Function(V2)
+    u2.dat.data[:] = np.arange(len(u2.dat.data))
+    u1.assign(u2, allow_missing_dofs=True)
+
+    return u1.dat.data.astype(int)
diff --git a/firedrake/mg/adaptive_transfer_manager.py b/firedrake/mg/adaptive_transfer_manager.py
new file mode 100644
index 0000000000..895794711a
--- /dev/null
+++ b/firedrake/mg/adaptive_transfer_manager.py
@@ -0,0 +1,147 @@
+"""
+This module contains the AdaptiveTransferManager used to perform
+transfer operations on AdaptiveMeshHierarchies
+"""
+import numpy as np
+from firedrake.function import Function
+from firedrake.mg.embedded import TransferManager
+from firedrake.mg.utils import get_level
+
+
+__all__ = ("AdaptiveTransferManager",)
+
+
+class AdaptiveTransferManager(TransferManager):
+    """
+    TransferManager for adaptively refined mesh hierarchies
+    """
+    def __init__(self, *, native_transfers=None, use_averaging=True):
+        super().__init__(native_transfers=native_transfers, use_averaging=use_averaging)
+        self.tm = TransferManager()
+        self.weight_cache = {}
+        self.work_function_cache = {}
+        self.perm_cache = {}
+
+    def generic_transfer(self, source, target, transfer_op):
+        """
+        Generalized implementation of transfer operations by wrapping
+        transfer operations from TransferManager()
+        """
+        amh, source_level = get_level(source.function_space().mesh())
+        _, target_level = get_level(target.function_space().mesh())
+
+        # decide order of iteration depending on coarse -> fine or fine -> coarse
+        order = 1
+        if target_level < source_level:
+            order = -1
+
+        curr_source = source
+        if source_level == target_level:
+            target.assign(source)
+            return
+
+        for level in range(source_level, target_level, order):
+            if level + order == target_level:
+                curr_target = target
+            else:
+                target_mesh = amh.meshes[level + order]
+                curr_space = curr_source.function_space()
+                target_space = curr_space.reconstruct(mesh=target_mesh)
+                curr_target = self.get_work_function(target_space)
+
+            if transfer_op == self.tm.restrict:
+                w = self.get_weight(curr_source.function_space())
+                wsource = self.get_work_function(curr_source.function_space())
+                with (
+                    curr_source.dat.vec as svec,
+                    w.dat.vec as wvec,
+                    wsource.dat.vec as wsvec,
+                ):
+                    wsvec.pointwiseMult(svec, wvec)
+                curr_source = wsource
+
+            if order == 1:
+                source_function_splits = amh.split_function(curr_source, child=False)
+                target_function_splits = amh.split_function(curr_target, child=True)
+            else:
+                source_function_splits = amh.split_function(curr_source, child=True)
+                target_function_splits = amh.split_function(curr_target, child=False)
+
+            for split_label, _ in source_function_splits.items():
+                if split_label == 1:
+                    # we don't want to transfer across unsplit parts,
+                    # instead we copy dofs
+                    us_func = source_function_splits[1]
+                    ut_func = target_function_splits[1]
+                    permutations = self.get_perm(
+                        us_func,
+                        ut_func,
+                        transfer_op
+                    )
+                    ut_func.dat.data_wo[permutations] = us_func.dat.data_ro
+                else:
+                    transfer_op(
+                        source_function_splits[split_label],
+                        target_function_splits[split_label],
+                    )
+
+            amh.recombine(target_function_splits, curr_target, child=order + 1)
+            curr_source = curr_target
+
+    def get_work_function(self, func_space):
+        """
+        Cache for function on function space
+        """
+        try:
+            return self.work_function_cache[func_space]
+        except KeyError:
+            return self.work_function_cache.setdefault(func_space, Function(func_space))
+
+    def get_weight(self, V_source):
+        """
+        Cache for weights from partition of unity used during restriction
+        """
+        try:
+            return self.weight_cache[V_source]
+        except KeyError:
+            amh, _ = get_level(V_source.mesh())
+            return self.weight_cache.setdefault(
+                V_source, amh.use_weight(V_source, child=True)
+            )
+
+    def get_perm(self, unsplit_source, unsplit_target, transfer_op):
+        """
+        Cache permutations of DoFs from unsplit source
+        to unsplit target. This is used to skip transfer
+        across unsplit mesh hierarchies
+        """
+        key = (unsplit_source.function_space(),
+               unsplit_target.function_space())
+        try:
+            return self.perm_cache[key]
+        except KeyError:
+            source_nodes = Function(key[0])
+            permutation = Function(key[1])
+            source_nodes.dat.data[:] = np.arange(len(source_nodes.dat.data))
+            transfer_op(source_nodes, permutation)
+            return self.perm_cache.setdefault(
+                key, np.rint(permutation.dat.data_ro).astype(int)
+            )
+
+    def prolong(self, uc, uf):
+        """
+        Prolongation of AdaptiveMeshHierarchy
+        """
+        self.generic_transfer(uc, uf, transfer_op=self.tm.prolong)
+
+    def inject(self, uf, uc):
+        """
+        Injection of AdaptiveMeshHierarchy
+        """
+        self.generic_transfer(uf, uc, transfer_op=self.tm.inject)
+
+    def restrict(self, source, target):
+        """
+        Restriction of AdaptiveMeshHierarchy
+        """
+        self.generic_transfer(source, target, transfer_op=self.tm.restrict)
diff --git a/tests/firedrake/multigrid/test_adaptive_multigrid.py b/tests/firedrake/multigrid/test_adaptive_multigrid.py
new file mode 100644
index 0000000000..45e846b2c3
--- /dev/null
+++ b/tests/firedrake/multigrid/test_adaptive_multigrid.py
@@ -0,0 +1,360 @@
+"""
+Tests for AdaptiveMeshHierarchy
+and AdaptiveTransferManager
+"""
+
+import random
+import pytest
+import numpy as np
+from firedrake import *
+
+
+@pytest.fixture(params=[2, 3])
+def amh(request):
+    """
+    Generate AdaptiveMeshHierarchies
+    """
+    from netgen.occ import WorkPlane, OCCGeometry, Box, Pnt
+    dim = request.param
+    random.seed(1234)
+    if dim == 2:
+        wp = WorkPlane()
+        wp.Rectangle(2, 2)
+        face = wp.Face()
+        geo = OCCGeometry(face, dim=2)
+        maxh = 0.5
+    else:
+        cube = Box(Pnt(0, 0, 0), Pnt(2, 2, 2))
+        geo = OCCGeometry(cube, dim=3)
+        maxh = 0.5
+
+    ngmesh = geo.GenerateMesh(maxh=maxh)
+    base = Mesh(ngmesh)
+    amh_test = AdaptiveMeshHierarchy(base)
+
+    if dim == 2:
+        els = ngmesh.Elements2D()
+    else:
+        els = ngmesh.Elements3D()
+
+    for _ in range(2):
+        for _, el in enumerate(els):
+            el.refine = 0
+            if random.random() < 0.5:
+                el.refine = 1
+        ngmesh.Refine(adaptive=True)
+        mesh = Mesh(ngmesh)
+        amh_test.add_mesh(mesh)
+    return amh_test
+
+
+@pytest.fixture
+def mh_res():
+    """
+    Generate MeshHierarchy for reference
+    """
+    from netgen.occ import WorkPlane, OCCGeometry
+    wp = WorkPlane()
+    wp.Rectangle(2, 2)
+    face = wp.Face()
+    geo = OCCGeometry(face, dim=2)
+    maxh = 0.5
+    ngmesh = geo.GenerateMesh(maxh=maxh)
+    base = Mesh(ngmesh)
+    mesh2 = Mesh(ngmesh)
+    amh_unif = AdaptiveMeshHierarchy([base])
+    for _ in range(2):
+        refs = np.ones(len(ngmesh.Elements2D()))
+        amh_unif.refine(refs)
+    mh = MeshHierarchy(mesh2, 2)
+
+    return amh_unif, mh
+
+
+@pytest.fixture
+def atm():
+    """atm used in tests"""
+    return AdaptiveTransferManager()
+
+
+@pytest.fixture
+def tm():
+    """tm used for restrict consistency"""
+    return TransferManager()
+
+
+@pytest.mark.skipnetgen
+@pytest.mark.parametrize("operator", ["prolong", "inject"])
+def test_DG0(amh, atm, operator):  # pylint: disable=W0621
+    """
+    Prolongation & Injection test for DG0
+    """
+    V_coarse = FunctionSpace(amh[0], "DG", 0)
+    V_fine = FunctionSpace(amh[-1], "DG", 0)
+    u_coarse = Function(V_coarse)
+    u_fine = Function(V_fine)
+    xc, *_ = SpatialCoordinate(V_coarse.mesh())
+    stepc = conditional(ge(xc, 0), 1, 0)
+    xf, *_ = SpatialCoordinate(V_fine.mesh())
+    stepf = conditional(ge(xf, 0), 1, 0)
+
+    if operator == "prolong":
+        u_coarse.interpolate(stepc)
+        assert errornorm(stepc, u_coarse) <= 1e-12
+
+        atm.prolong(u_coarse, u_fine)
+        assert errornorm(stepf, u_fine) <= 1e-12
+    if operator == "inject":
+        u_fine.interpolate(stepf)
+        assert errornorm(stepf, u_fine) <= 1e-12
+
+        atm.inject(u_fine, u_coarse)
+        assert errornorm(stepc, u_coarse) <= 1e-12
+
+
+@pytest.mark.skipnetgen
+@pytest.mark.parametrize("operator", ["prolong", "inject"])
+def test_CG1(amh, atm, operator):  # pylint: disable=W0621
+    """
+    Prolongation & Injection test for CG1
+    """
+    V_coarse = FunctionSpace(amh[0], "CG", 1)
+    V_fine = FunctionSpace(amh[-1], "CG", 1)
+    u_coarse = Function(V_coarse)
+    u_fine = Function(V_fine)
+    xc, *_ = SpatialCoordinate(V_coarse.mesh())
+    xf, *_ = SpatialCoordinate(V_fine.mesh())
+
+    if operator == "prolong":
+        u_coarse.interpolate(xc)
+        assert errornorm(xc, u_coarse) <= 1e-12
+
+        atm.prolong(u_coarse, u_fine)
+        assert errornorm(xf, u_fine) <= 1e-12
+    if operator == "inject":
+        u_fine.interpolate(xf)
+        assert errornorm(xf, u_fine) <= 1e-12
+
+        atm.inject(u_fine, u_coarse)
+        assert errornorm(xc, u_coarse) <= 1e-12
+
+
+@pytest.mark.skipnetgen
+def test_restrict_consistency(mh_res, atm, tm):  # pylint: disable=W0621
+    """
+    Test restriction consistency of amh with uniform refinement vs mh
+    """
+    amh_unif = mh_res[0]
+    mh = mh_res[1]
+
+    V_coarse = FunctionSpace(amh_unif[0], "DG", 0)
+    V_fine = FunctionSpace(amh_unif[-1], "DG", 0)
+    u_coarse = Function(V_coarse)
+    u_fine = Function(V_fine)
+    xc, _ = SpatialCoordinate(V_coarse.mesh())
+
+    u_coarse.interpolate(xc)
+    atm.prolong(u_coarse, u_fine)
+
+    rf = assemble(TestFunction(V_fine) * dx)
+    rc = Cofunction(V_coarse.dual())
+    atm.restrict(rf, rc)
+
+    # compare with mesh_hierarchy
+    xcoarse, _ = SpatialCoordinate(mh[0])
+    Vcoarse = FunctionSpace(mh[0], "DG", 0)
+    Vfine = FunctionSpace(mh[-1], "DG", 0)
+
+    mhuc = Function(Vcoarse)
+    mhuc.interpolate(xcoarse)
+    mhuf = Function(Vfine)
+    tm.prolong(mhuc, mhuf)
+
+    mhrf = assemble(TestFunction(Vfine) * dx)
+    mhrc = Cofunction(Vcoarse.dual())
+
+    tm.restrict(mhrf, mhrc)
+
+    assert (
+        (assemble(action(mhrc, mhuc)) - assemble(action(mhrf, mhuf)))
+        / assemble(action(mhrf, mhuf))
+    ) <= 1e-12
+    assert (
+        (assemble(action(rc, u_coarse)) - assemble(action(mhrc, mhuc)))
+        / assemble(action(mhrc, mhuc))
+    ) <= 1e-12
+
+
+@pytest.mark.skipnetgen
+def test_restrict_CG1(amh, atm):  # pylint: disable=W0621
+    """
+    Test restriction with CG1
+    """
+    V_coarse = FunctionSpace(amh[0], "CG", 1)
+    V_fine = FunctionSpace(amh[-1], "CG", 1)
+    u_coarse = Function(V_coarse)
+    u_fine = Function(V_fine)
+    xc, *_ = SpatialCoordinate(V_coarse.mesh())
+
+    u_coarse.interpolate(xc)
+    atm.prolong(u_coarse, u_fine)
+
+    rf = assemble(TestFunction(V_fine) * dx)
+    rc = Cofunction(V_coarse.dual())
+    atm.restrict(rf, rc)
+
+    assert np.allclose(
+        assemble(action(rc, u_coarse)),
+        assemble(action(rf, u_fine)),
+        rtol=1e-12
+    )
+
+
+@pytest.mark.skipnetgen
+def test_restrict_DG0(amh, atm):  # pylint: disable=W0621
+    """
+    Test restriction with DG0
+    """
+    V_coarse = FunctionSpace(amh[0], "DG", 0)
+    V_fine = FunctionSpace(amh[-1], "DG", 0)
+    u_coarse = Function(V_coarse)
+    u_fine = Function(V_fine)
+    xc, *_ = SpatialCoordinate(V_coarse.mesh())
+
+    u_coarse.interpolate(xc)
+    atm.prolong(u_coarse, u_fine)
+
+    rf = assemble(TestFunction(V_fine) * dx)
+    rc = Cofunction(V_coarse.dual())
+    atm.restrict(rf, rc)
+
+    assert np.allclose(
+        assemble(action(rc, u_coarse)),
+        assemble(action(rf, u_fine)),
+        rtol=1e-12
+    )
+
+
+@pytest.mark.skipnetgen
+def test_mg_jacobi(amh, atm):  # pylint: disable=W0621
+    """
+    Test multigrid with jacobi smoothers
+    """
+    V_J = FunctionSpace(amh[-1], "CG", 1)
+    x = SpatialCoordinate(amh[-1])
+    u_ex = Function(V_J, name="u_fine_real").interpolate(
+        sin(2 * pi * x[0]) * sin(2 * pi * x[1])
+    )
+    u = Function(V_J)
+    v = TestFunction(V_J)
+    bc = DirichletBC(V_J, u_ex, "on_boundary")
+    F = inner(grad(u - u_ex), grad(v)) * dx
+
+    params = {
+        "snes_type": "ksponly",
+        "ksp_max_it": 20,
+        "ksp_type": "cg",
+        "ksp_norm_type": "unpreconditioned",
+        "ksp_rtol": 1e-8,
+        "ksp_atol": 1e-8,
+        "pc_type": "mg",
+        "mg_levels_pc_type": "jacobi",
+        "mg_levels_ksp_type": "chebyshev",
+        "mg_levels_ksp_max_it": 2,
+        "mg_coarse_ksp_type": "preonly",
+        "mg_coarse_pc_type": "lu",
+        "mg_coarse_pc_factor_mat_solver_type": "mumps",
+    }
+
+    problem = NonlinearVariationalProblem(F, u, bc)
+    solver = NonlinearVariationalSolver(problem, solver_parameters=params)
+    solver.set_transfer_manager(atm)
+    solver.solve()
+    assert errornorm(u_ex, u) <= 1e-8
+
+
+@pytest.mark.skipnetgen
+@pytest.mark.parametrize("params", ["jacobi", "asm", "patch"])
+def test_mg_patch(amh, atm, params):  # pylint: disable=W0621
+    """
+    Test multigrid with patch relaxation
+    """
+    if params == "jacobi":
+        solver_params = {
+            "mat_type": "matfree",
+            "ksp_type": "cg",
+            "pc_type": "mg",
+            "mg_levels": {
+                "ksp_type": "chebyshev",
+                "ksp_max_it": 1,
+                "pc_type": "jacobi",
+            },
+            "mg_coarse": {
+                "ksp_type": "preonly",
+                "pc_type": "python",
+                "pc_python_type": "firedrake.AssembledPC",
+                "assembled": {"ksp_type": "preonly", "pc_type": "lu"},
+            },
+        }
+    elif params == "patch":
+        solver_params = {
+            "mat_type": "matfree",
+            "ksp_type": "cg",
+            "pc_type": "mg",
+            "mg_levels": {
+                "ksp_type": "chebyshev",
+                "ksp_max_it": 1,
+                "pc_type": "python",
+                "pc_python_type": "firedrake.PatchPC",
+                "patch": {
+                    "pc_patch": {
+                        "construct_type": "star",
+                        "construct_dim": 0,
+                        "sub_mat_type": "seqdense",
+                        "dense_inverse": True,
+                        "save_operators": True,
+                        "precompute_element_tensors": True,
+                    },
+                    "sub_ksp_type": "preonly",
+                    "sub_pc_type": "lu",
+                },
+            },
+            "mg_coarse": {
+                "ksp_type": "preonly",
+                "pc_type": "python",
+                "pc_python_type": "firedrake.AssembledPC",
+                "assembled": {"ksp_type": "preonly", "pc_type": "lu"},
+            },
+        }
+    else:
+        solver_params = {
+            "mat_type": "aij",
+            "ksp_type": "cg",
+            "pc_type": "mg",
+            "mg_levels": {
+                "ksp_type": "chebyshev",
+                "ksp_max_it": 1,
+                "pc_type": "python",
+                "pc_python_type": "firedrake.ASMStarPC",
+                "pc_star_backend": "tinyasm",
+            },
+            "mg_coarse": {"ksp_type": "preonly", "pc_type": "lu"},
+        }
+
+    V_J = FunctionSpace(amh[-1], "CG", 1)
+    x = SpatialCoordinate(amh[-1])
+    u_ex = Function(V_J, name="u_fine_real").interpolate(
+        sin(2 * pi * x[0]) * sin(2 * pi * x[1])
+    )
+    u = Function(V_J)
+    v = TestFunction(V_J)
+    bc = DirichletBC(V_J, u_ex, "on_boundary")
+    F = inner(grad(u - u_ex), grad(v)) * dx
+
+    problem = NonlinearVariationalProblem(F, u, bc)
+    solver = NonlinearVariationalSolver(problem,
+                                        solver_parameters=solver_params)
+    solver.set_transfer_manager(atm)
+
+    solver.solve()
+    assert errornorm(u_ex, u) <= 1e-8

From cf0b36e9567b1f46732dad27e23cd06a6a5c5888 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 1 Dec 2025 11:41:47 +0000
Subject: [PATCH 02/37] Fix complex tests

---
 .../multigrid/test_adaptive_multigrid.py        | 17 +++++++++++------
 1 file changed, 11 insertions(+), 6 deletions(-)

diff --git a/tests/firedrake/multigrid/test_adaptive_multigrid.py b/tests/firedrake/multigrid/test_adaptive_multigrid.py
index 45e846b2c3..678a3c0efd 100644
--- a/tests/firedrake/multigrid/test_adaptive_multigrid.py
+++ b/tests/firedrake/multigrid/test_adaptive_multigrid.py
@@ -7,6 +7,7 @@
 import pytest
 import numpy as np
 from firedrake import *
+from firedrake.utils import complex_mode
 
 
 @pytest.fixture(params=[2, 3])
@@ -107,8 +108,12 @@ def test_DG0(amh, atm, operator):  # pylint: disable=W0621
     if operator == "inject":
         u_fine.interpolate(stepf)
         assert errornorm(stepf, u_fine) <= 1e-12
-
-        atm.inject(u_fine, u_coarse)
+        if complex_mode:
+            with pytest.raises(NotImplementedError):
+                atm.inject(u_fine, u_coarse)
+            return
+        else:
+            atm.inject(u_fine, u_coarse)
         assert errornorm(stepc, u_coarse) <= 1e-12
 
 
@@ -156,7 +161,7 @@ def test_restrict_consistency(mh_res, atm, tm):  # pylint: disable=W0621
     u_coarse.interpolate(xc)
     atm.prolong(u_coarse, u_fine)
 
-    rf = assemble(TestFunction(V_fine) * dx)
+    rf = assemble(conj(TestFunction(V_fine)) * dx)
     rc = Cofunction(V_coarse.dual())
     atm.restrict(rf, rc)
 
@@ -170,7 +175,7 @@ def test_restrict_consistency(mh_res, atm, tm):  # pylint: disable=W0621
     mhuf = Function(Vfine)
     tm.prolong(mhuc, mhuf)
 
-    mhrf = assemble(TestFunction(Vfine) * dx)
+    mhrf = assemble(conj(TestFunction(Vfine)) * dx)
     mhrc = Cofunction(Vcoarse.dual())
 
     tm.restrict(mhrf, mhrc)
@@ -199,7 +204,7 @@ def test_restrict_CG1(amh, atm):  # pylint: disable=W0621
     u_coarse.interpolate(xc)
     atm.prolong(u_coarse, u_fine)
 
-    rf = assemble(TestFunction(V_fine) * dx)
+    rf = assemble(conj(TestFunction(V_fine)) * dx)
     rc = Cofunction(V_coarse.dual())
     atm.restrict(rf, rc)
 
@@ -224,7 +229,7 @@ def test_restrict_DG0(amh, atm):  # pylint: disable=W0621
     u_coarse.interpolate(xc)
     atm.prolong(u_coarse, u_fine)
 
-    rf = assemble(TestFunction(V_fine) * dx)
+    rf = assemble(conj(TestFunction(V_fine)) * dx)
     rc = Cofunction(V_coarse.dual())
     atm.restrict(rf, rc)
 

From 8312598eac50268f9c58b16ab6752cbf55a6ff0f Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 1 Dec 2025 14:02:45 +0000
Subject: [PATCH 03/37] complex

---
 tests/firedrake/multigrid/test_adaptive_multigrid.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/tests/firedrake/multigrid/test_adaptive_multigrid.py b/tests/firedrake/multigrid/test_adaptive_multigrid.py
index 678a3c0efd..cc9d7fee32 100644
--- a/tests/firedrake/multigrid/test_adaptive_multigrid.py
+++ b/tests/firedrake/multigrid/test_adaptive_multigrid.py
@@ -180,11 +180,11 @@ def test_restrict_consistency(mh_res, atm, tm):  # pylint: disable=W0621
 
     tm.restrict(mhrf, mhrc)
 
-    assert (
+    assert abs(
         (assemble(action(mhrc, mhuc)) - assemble(action(mhrf, mhuf)))
         / assemble(action(mhrf, mhuf))
     ) <= 1e-12
-    assert (
+    assert abs(
         (assemble(action(rc, u_coarse)) - assemble(action(mhrc, mhuc)))
         / assemble(action(mhrc, mhuc))
     ) <= 1e-12

From 6b69bf9af527a87bcc7893139d3ce8acfa87cfd5 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 19 Jan 2026 11:03:12 +0000
Subject: [PATCH 04/37] Apply suggestions from code review

---
 firedrake/mg/adaptive_hierarchy.py | 21 +++++++++++++++++++--
 1 file changed, 19 insertions(+), 2 deletions(-)

diff --git a/firedrake/mg/adaptive_hierarchy.py b/firedrake/mg/adaptive_hierarchy.py
index 0749d42b5f..3913544c8a 100644
--- a/firedrake/mg/adaptive_hierarchy.py
+++ b/firedrake/mg/adaptive_hierarchy.py
@@ -170,12 +170,29 @@ def refine(self, refinements):
 
     def adapt(self, eta, theta):
         """
-        Implement Dorfler marking, refines mesh from error estimator
+        Add the next refinement level to the MeshHierarchy by local refinement
+        with a simplified variant of Dorfler marking.
+
+        Parameters
+        ----------
+        eta : Function
+            A DG0 `Function` with the local error estimator.
+        theta : float
+            The threshold for marking as a fraction of the maximum error.
+
+        Note
+        ----
+        Dorfler marking involves sorting all of the elements by decreasing
+        error estimator and taking the minimal set that exceeds some fixed
+        fraction of the total error. What this code implements is the simpler
+        variant that doesn't have a proof of convergence (as far as I know) 
+        but works as well in practice.
         """
         mesh = self.meshes[-1]
         W = FunctionSpace(mesh, "DG", 0)
         markers = Function(W)
 
+        # Take the maximum over all processes
         with eta.dat.vec_ro as eta_:
             eta_max = eta_.max()[1]
 
@@ -236,7 +253,7 @@ def use_weight(self, V, child):
 
     def recombine(self, split_funcs, f, child=True):
         """
-        Recombines functions on submeshes back full mesh
+        Recombines functions on submeshes back to the parent mesh
         """
         V = f.function_space()
         f.zero()

From 18de9b87b8eb2512ecd0aa871fc37a74a11a48a6 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 19 Jan 2026 18:24:56 +0000
Subject: [PATCH 05/37] WIP

---
 firedrake/mg/adaptive_hierarchy.py            | 181 +++++++++---------
 firedrake/mg/adaptive_transfer_manager.py     |   8 +-
 .../multigrid/test_adaptive_multigrid.py      |  35 ++--
 3 files changed, 111 insertions(+), 113 deletions(-)

diff --git a/firedrake/mg/adaptive_hierarchy.py b/firedrake/mg/adaptive_hierarchy.py
index 3913544c8a..29b0b33dad 100644
--- a/firedrake/mg/adaptive_hierarchy.py
+++ b/firedrake/mg/adaptive_hierarchy.py
@@ -13,7 +13,7 @@
 from firedrake.mesh import Mesh, Submesh, RelabeledMesh
 from firedrake.mg import HierarchyBase
 from firedrake.mg.utils import set_level, get_level
-from ufl import conditional, gt
+from firedrake.petsc import PETSc
 
 __all__ = ["AdaptiveMeshHierarchy"]
 
@@ -22,17 +22,16 @@ class AdaptiveMeshHierarchy(HierarchyBase):
     """
     HierarchyBase for hierarchies of adaptively refined meshes
     """
-
-    def __init__(self, mesh, refinements_per_level=1, nested=True):
-        self.meshes = tuple(mesh)
-        self._meshes = tuple(mesh)
+    def __init__(self, base_mesh, refinements_per_level=1, nested=True):
+        self.meshes = (base_mesh,)
+        self._meshes = (base_mesh,)
         self.submesh_hierarchies = []
         self.coarse_to_fine_cells = {}
         self.fine_to_coarse_cells = {}
         self.fine_to_coarse_cells[Fraction(0, 1)] = None
         self.refinements_per_level = refinements_per_level
         self.nested = nested
-        set_level(mesh[0], self, 0)
+        set_level(base_mesh, self, 0)
         self.split_cache = {}
 
     def add_mesh(self, mesh):
@@ -41,13 +40,10 @@ def add_mesh(self, mesh):
         Then computes the coarse_to_fine and fine_to_coarse mappings.
         Constructs intermediate submesh hierarchies with this.
         """
-        if mesh.topological_dimension <= 2:
-            max_children = 4
-        else:
-            max_children = 16
-        self._meshes += tuple(mesh)
-        self.meshes += tuple(mesh)
-        coarse_mesh = self.meshes[-2]
+        max_children = 2 ** mesh.topological_dimension
+        coarse_mesh = self.meshes[-1]
+        self._meshes += (mesh,)
+        self.meshes += (mesh,)
         level = len(self.meshes)
         set_level(self.meshes[-1], self, level - 1)
         self._shared_data_cache = defaultdict(dict)
@@ -67,11 +63,14 @@ def add_mesh(self, mesh):
             coarse_mesh.mark_entities(coarse_splits[i], i)
             mesh.mark_entities(fine_splits[i], int(f"10{i}"))
 
-        coarse_indicators = [
-            coarse_splits[i]
-            for i in range(1, max_children + 1)
-        ]
         coarse_labels = list(range(1, max_children + 1))
+        coarse_indicators = [coarse_splits[i] for i in coarse_labels]
+
+        has_children = {}
+        for label in coarse_labels:
+            with coarse_splits[label].dat.vec as v:
+                has_children[label] = v.norm() > 1E-12
+
         coarse_mesh = RelabeledMesh(
             coarse_mesh,
             coarse_indicators,
@@ -79,35 +78,35 @@ def add_mesh(self, mesh):
             name="Relabeled_coarse",
         )
         c_subm = {
-            j: Submesh(coarse_mesh, coarse_mesh.topology_dm.getDimension(), j)
+            j: Submesh(coarse_mesh, coarse_mesh.topological_dimension, j)
             for j in range(1, max_children + 1)
-            if any(num_children == j)
+            if has_children[j]
         }
         set_level(coarse_mesh, self, level - 2)
 
-        fine_indicators = [
-            fine_splits[i]
-            for i in range(1, max_children + 1)
-        ]
         fine_labels = list(range(1, max_children + 1))
+        fine_indicators = [fine_splits[i] for i in fine_labels]
         mesh = RelabeledMesh(
             mesh,
             fine_indicators,
             fine_labels,
+            name="Relabeled_fine",
         )
         f_subm = {
-            int(str(j)[-2:]): Submesh(mesh, mesh.topology_dm.getDimension(), j)
+            int(str(j)[-2:]): Submesh(mesh, mesh.topological_dimension, j)
             for j in [int("10" + str(i)) for i in range(1, max_children + 1)]
-            if any(num_children == int(str(j)[-2:]))
+            if has_children[int(str(j)[-2:])]
         }
         set_level(mesh, self, level - 1)
 
-        # update c2f and f2c for submeshes by mapping numberings
-        # on full mesh to numberings on coarse mesh
+        # stores number of parents for each amount of children
         parents_per_child_count = [
             len([el for el in c2f if len(el) == j])
             for j in range(1, max_children + 1)
-        ]  # stores number of parents for each amount of children
+        ]
+
+        # update c2f and f2c for submeshes by mapping numberings
+        # on full mesh to numberings on coarse mesh
         c2f_adjusted = {
             j: np.zeros((num_parents, j))
             for j, num_parents in enumerate(parents_per_child_count, 1)
@@ -120,30 +119,30 @@ def add_mesh(self, mesh):
         }
 
         coarse_full_to_sub_map = {
-            i: full_to_sub(coarse_mesh, c_subm[i])
+            i: get_full_to_sub_numbering(coarse_mesh, c_subm[i])
             for i in c_subm
         }
         fine_full_to_sub_map = {
-            j: full_to_sub(mesh, f_subm[j])
+            j: get_full_to_sub_numbering(mesh, f_subm[j])
             for j in f_subm
         }
 
         for i, children in enumerate(c2f):
             n = len(children)
-            if 1 <= n <= max_children:
+            if n in coarse_full_to_sub_map:
                 coarse_id_sub = coarse_full_to_sub_map[n][i]
                 fine_id_sub = fine_full_to_sub_map[n][np.array(children)]
                 c2f_adjusted[n][coarse_id_sub] = fine_id_sub
 
         for j, parent in enumerate(f2c):
             n = num_children[parent].item()
-            if 1 <= n <= max_children:
-                fine_id_sub = fine_full_to_sub_map[n][j]
+            if n in coarse_full_to_sub_map:
                 coarse_id_sub = coarse_full_to_sub_map[n][parent.item()]
+                fine_id_sub = fine_full_to_sub_map[n][j]
                 f2c_adjusted[n][fine_id_sub, 0] = coarse_id_sub
 
         c2f_subm = {
-            i: {Fraction(0, 1): c2f_adjusted[i].astype(int)}
+            i: {Fraction(0, 1): c2f_adjusted[i].astype(PETSc.IntType)}
             for i in c2f_adjusted
         }
         f2c_subm = {i: {Fraction(1, 1): f2c_adjusted[i]} for i in f2c_adjusted}
@@ -168,16 +167,16 @@ def refine(self, refinements):
         mesh = Mesh(ngmesh)
         self.add_mesh(mesh)
 
-    def adapt(self, eta, theta):
+    def adapt(self, eta: Function | Cofunction, theta: float):
         """
-        Add the next refinement level to the MeshHierarchy by local refinement
+        Add a refinement level to the hierarchy by local refinement
         with a simplified variant of Dorfler marking.
 
         Parameters
         ----------
-        eta : Function
-            A DG0 `Function` with the local error estimator.
-        theta : float
+        eta
+            A DG0 :class:`~firedrake.function.Function` with the local error estimator.
+        theta
             The threshold for marking as a fraction of the maximum error.
 
         Note
@@ -185,19 +184,28 @@ def adapt(self, eta, theta):
         Dorfler marking involves sorting all of the elements by decreasing
         error estimator and taking the minimal set that exceeds some fixed
         fraction of the total error. What this code implements is the simpler
-        variant that doesn't have a proof of convergence (as far as I know) 
+        variant that doesn't have a proof of convergence (as far as I know)
         but works as well in practice.
+
         """
+        if not isinstance(eta, (Function, Cofunction)):
+            raise TypeError(f"eta must be a Function or Cofunction, not a {type(eta).__name__}")
+        M = eta.function_space()
+        if M.finat_element.space_dimension() != 1:
+            raise ValueError("eta must be a Function or Cofunction in DG0")
         mesh = self.meshes[-1]
-        W = FunctionSpace(mesh, "DG", 0)
-        markers = Function(W)
+        if M.mesh() is not mesh:
+            raise ValueError("eta must be defined on the finest mesh of the hierarchy")
 
         # Take the maximum over all processes
         with eta.dat.vec_ro as eta_:
             eta_max = eta_.max()[1]
 
-        should_refine = conditional(gt(eta, theta * eta_max), 1, 0)
-        markers.interpolate(should_refine)
+        threshold = theta * eta_max
+        should_refine = eta.dat.data_ro > threshold
+
+        markers = Function(M)
+        markers.dat.data_wo[should_refine] = 1
 
         refined_mesh = mesh.refine_marked_elements(markers)
         self.add_mesh(refined_mesh)
@@ -213,7 +221,7 @@ def split_function(self, u, child=True):
 
         ind = 1 if child else 0
         hierarchy_dict = self.submesh_hierarchies[int(level) - ind]
-        parent_mesh = hierarchy_dict[[*hierarchy_dict][0]].meshes[ind].submesh_parent
+        parent_mesh = tuple(hierarchy_dict.values())[0].meshes[ind].submesh_parent
         parent_space = V.reconstruct(parent_mesh)
         u_corr_space = Function(parent_space, val=u.dat)
         key = (u, child)
@@ -261,10 +269,7 @@ def recombine(self, split_funcs, f, child=True):
             split_funcs[[*split_funcs][0]].function_space().mesh().submesh_parent
         )
         V_label = V.reconstruct(mesh=parent_mesh)
-        if isinstance(f, Function):
-            f_label = Function(V_label, val=f.dat)
-        elif isinstance(f, Cofunction):
-            f_label = Cofunction(V_label, val=f.dat)
+        f_label = Function(V_label, val=f.dat)
 
         for split_label, val in split_funcs.items():
             assert val.function_space().mesh().submesh_parent == parent_mesh
@@ -280,55 +285,45 @@ def recombine(self, split_funcs, f, child=True):
         return f
 
 
-def get_c2f_f2c_fd(mesh, coarse_mesh):
+def get_c2f_f2c_fd(fine_mesh, coarse_mesh):
     """
     Construct coarse->fine and fine->coarse relations by mapping netgen elements to firedrake ones
     """
-    ngmesh = mesh.netgen_mesh
-    num_parents = coarse_mesh.num_cells()
-
-    if mesh.topology_dm.getDimension() == 2:
+    fine_dm = fine_mesh.topology_dm
+    tdim = fine_dm.getDimension()
+    ngmesh = fine_mesh.netgen_mesh
+    if tdim == 2:
         parents = ngmesh.parentsurfaceelements.NumPy()
-        elements = ngmesh.Elements2D()
-    elif mesh.topology_dm.getDimension() == 3:
+    elif tdim == 3:
         parents = ngmesh.parentelements.NumPy()
-        elements = ngmesh.Elements3D()
     else:
         raise RuntimeError("Adaptivity not implemented in dimension of mesh")
 
-    c2f = [[] for _ in range(num_parents)]
-    f2c = [[] for _ in range(mesh.num_cells())]
+    fstart, fend = fine_mesh.topology_dm.getDepthStratum(tdim)
+    cstart, cend = coarse_mesh.topology_dm.getDepthStratum(tdim)
+    total_coarse_cells = FunctionSpace(coarse_mesh, "DG", 0).dim()
+
+    c2f = [[] for _ in range(cstart, cend)]
+    f2c = [[] for _ in range(fstart, fend)]
 
     if parents.shape[0] == 0:
         raise RuntimeError("Added mesh has not refined any cells from previous mesh")
-    for l, _ in enumerate(elements):
-        if parents[l][0] == -1 or l < num_parents:
-            f2c[mesh._cell_numbering.getOffset(l)].append(
-                coarse_mesh._cell_numbering.getOffset(l)
-            )
-            c2f[coarse_mesh._cell_numbering.getOffset(l)].append(
-                mesh._cell_numbering.getOffset(l)
-            )
 
-        elif parents[l][0] < num_parents:
-            fine_ind = mesh._cell_numbering.getOffset(l)
-            coarse_ind = coarse_mesh._cell_numbering.getOffset(parents[l][0])
-            f2c[fine_ind].append(coarse_ind)
-            c2f[coarse_ind].append(fine_ind)
+    for l in range(fstart, fend):
+        a = l
+        while a >= total_coarse_cells:
+            a = parents[a][0]
 
-        else:
-            a = parents[parents[l][0]][0]
-            while a >= num_parents:
-                a = parents[a][0]
+        fine_ind = fine_mesh._cell_numbering.getOffset(l)
+        coarse_ind = coarse_mesh._cell_numbering.getOffset(a)
 
-            f2c[mesh._cell_numbering.getOffset(l)].append(
-                coarse_mesh._cell_numbering.getOffset(a)
-            )
-            c2f[coarse_mesh._cell_numbering.getOffset(a)].append(
-                mesh._cell_numbering.getOffset(l)
-            )
+        fine_ind -= fstart
+        coarse_ind -= cstart
 
-    return c2f, np.array(f2c).astype(int)
+        f2c[fine_ind].append(coarse_ind)
+        c2f[coarse_ind].append(fine_ind)
+
+    return c2f, np.array(f2c, dtype=PETSc.IntType)
 
 
 def split_to_submesh(mesh, coarse_mesh, c2f, f2c):
@@ -337,10 +332,7 @@ def split_to_submesh(mesh, coarse_mesh, c2f, f2c):
     Returns splits which are Functions denoting whether elements
     belong to the corresponing submesh (bool)
     """
-    if mesh.topological_dimension <= 2:
-        max_children = 4
-    else:
-        max_children = 16
+    max_children = 2 ** mesh.topological_dimension
     V = FunctionSpace(mesh, "DG", 0)
     V2 = FunctionSpace(coarse_mesh, "DG", 0)
     coarse_splits = {
@@ -349,29 +341,28 @@ def split_to_submesh(mesh, coarse_mesh, c2f, f2c):
     fine_splits = {
         i: Function(V, name=f"{i}_elements") for i in range(1, max_children + 1)
     }
-    num_children = np.zeros((len(c2f)))
-
+    num_children = np.zeros((len(c2f),))
     for i, children in enumerate(c2f):
         n = len(children)
         if 1 <= n <= max_children:
-            coarse_splits[n].dat.data[i] = 1
             num_children[i] = n
 
     for i in range(1, max_children + 1):
-        fine_splits[i].dat.data[num_children[f2c.squeeze()] == i] = 1
+        coarse_splits[i].dat.data_wo_with_halos[num_children == i] = 1
+        fine_splits[i].dat.data_wo_with_halos[num_children[f2c.squeeze()] == i] = 1
 
     return coarse_splits, fine_splits, num_children
 
 
-def full_to_sub(mesh, submesh):
+def get_full_to_sub_numbering(mesh, submesh):
     """
-    Returns the submesh element id associated with the full mesh element id
+    Returns the submesh cell id associated with the full mesh cell id
     """
     V1 = FunctionSpace(mesh, "DG", 0)
     V2 = FunctionSpace(submesh, "DG", 0)
     u1 = Function(V1)
     u2 = Function(V2)
-    u2.dat.data[:] = np.arange(len(u2.dat.data))
+    u2.dat.data_wo[:] = np.arange(submesh.cell_set.size)
     u1.assign(u2, allow_missing_dofs=True)
 
-    return u1.dat.data.astype(int)
+    return u1.dat.data_ro_with_halos.astype(PETSc.IntType)
diff --git a/firedrake/mg/adaptive_transfer_manager.py b/firedrake/mg/adaptive_transfer_manager.py
index 895794711a..5fa0e92f01 100644
--- a/firedrake/mg/adaptive_transfer_manager.py
+++ b/firedrake/mg/adaptive_transfer_manager.py
@@ -6,6 +6,7 @@
 from firedrake.function import Function
 from firedrake.mg.embedded import TransferManager
 from firedrake.mg.utils import get_level
+from firedrake.petsc import PETSc
 
 
 __all__ = ("AdaptiveTransferManager",)
@@ -67,7 +68,7 @@ def generic_transfer(self, source, target, transfer_op):
                 source_function_splits = amh.split_function(curr_source, child=True)
                 target_function_splits = amh.split_function(curr_target, child=False)
 
-            for split_label, _ in source_function_splits.items():
+            for split_label in source_function_splits:
                 if split_label == 1:
                     # we don't want to transfer across unsplit parts,
                     # instead we copy dofs
@@ -122,10 +123,11 @@ def get_perm(self, unsplit_source, unsplit_target, transfer_op):
         except KeyError:
             source_nodes = Function(key[0])
             permutation = Function(key[1])
-            source_nodes.dat.data[:] = np.arange(len(source_nodes.dat.data))
+            source_nodes.dat.data_wo[:] = np.arange(len(source_nodes.dat.data_ro))
             transfer_op(source_nodes, permutation)
+
             return self.perm_cache.setdefault(
-                key, np.rint(permutation.dat.data_ro).astype(int)
+                key, np.rint(permutation.dat.data_ro).astype(PETSc.IntType)
             )
 
     def prolong(self, uc, uf):
diff --git a/tests/firedrake/multigrid/test_adaptive_multigrid.py b/tests/firedrake/multigrid/test_adaptive_multigrid.py
index cc9d7fee32..b2d88c1216 100644
--- a/tests/firedrake/multigrid/test_adaptive_multigrid.py
+++ b/tests/firedrake/multigrid/test_adaptive_multigrid.py
@@ -3,7 +3,6 @@
 and AdaptiveTransferManager
 """
 
-import random
 import pytest
 import numpy as np
 from firedrake import *
@@ -17,34 +16,38 @@ def amh(request):
     """
     from netgen.occ import WorkPlane, OCCGeometry, Box, Pnt
     dim = request.param
-    random.seed(1234)
     if dim == 2:
         wp = WorkPlane()
-        wp.Rectangle(2, 2)
+        wp.Rectangle(1, 1)
         face = wp.Face()
         geo = OCCGeometry(face, dim=2)
         maxh = 0.5
     else:
-        cube = Box(Pnt(0, 0, 0), Pnt(2, 2, 2))
+        cube = Box(Pnt(0, 0, 0), Pnt(1, 1, 1))
         geo = OCCGeometry(cube, dim=3)
         maxh = 0.5
 
     ngmesh = geo.GenerateMesh(maxh=maxh)
-    base = Mesh(ngmesh)
+    base = Mesh(ngmesh, distribution_parameters={"overlap_type": (DistributedMeshOverlapType.VERTEX, 2)})
     amh_test = AdaptiveMeshHierarchy(base)
 
-    if dim == 2:
-        els = ngmesh.Elements2D()
-    else:
-        els = ngmesh.Elements3D()
+    rg = RandomGenerator(PCG64(seed=0))
+    for l in range(2):
+        mesh = amh_test[-1]
+        DG = FunctionSpace(mesh, "DG", 0)
+        should_refine = rg.uniform(DG, 0, 1).dat.global_data
+
+        ngmesh = mesh.netgen_mesh
+        if dim == 2:
+            els = ngmesh.Elements2D()
+        else:
+            els = ngmesh.Elements3D()
+        for i, el in enumerate(els):
+            el.refine = 1 if should_refine[i] < 0.5 else 0
 
-    for _ in range(2):
-        for _, el in enumerate(els):
-            el.refine = 0
-            if random.random() < 0.5:
-                el.refine = 1
         ngmesh.Refine(adaptive=True)
         mesh = Mesh(ngmesh)
+
         amh_test.add_mesh(mesh)
     return amh_test
 
@@ -63,7 +66,7 @@ def mh_res():
     ngmesh = geo.GenerateMesh(maxh=maxh)
     base = Mesh(ngmesh)
     mesh2 = Mesh(ngmesh)
-    amh_unif = AdaptiveMeshHierarchy([base])
+    amh_unif = AdaptiveMeshHierarchy(base)
     for _ in range(2):
         refs = np.ones(len(ngmesh.Elements2D()))
         amh_unif.refine(refs)
@@ -84,6 +87,7 @@ def tm():
     return TransferManager()
 
 
+@pytest.mark.parallel([1])
 @pytest.mark.skipnetgen
 @pytest.mark.parametrize("operator", ["prolong", "inject"])
 def test_DG0(amh, atm, operator):  # pylint: disable=W0621
@@ -117,6 +121,7 @@ def test_DG0(amh, atm, operator):  # pylint: disable=W0621
         assert errornorm(stepc, u_coarse) <= 1e-12
 
 
+@pytest.mark.parallel([1])
 @pytest.mark.skipnetgen
 @pytest.mark.parametrize("operator", ["prolong", "inject"])
 def test_CG1(amh, atm, operator):  # pylint: disable=W0621

From 5c23e5ebf2ec13c6f1ddd75a8051a92c3acae575 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 19 Jan 2026 19:16:33 +0000
Subject: [PATCH 06/37] Remove Submesh

---
 firedrake/mg/adaptive_hierarchy.py            | 272 +-----------------
 firedrake/mg/adaptive_transfer_manager.py     | 205 +++++--------
 .../multigrid/test_adaptive_multigrid.py      |  15 +-
 3 files changed, 87 insertions(+), 405 deletions(-)

diff --git a/firedrake/mg/adaptive_hierarchy.py b/firedrake/mg/adaptive_hierarchy.py
index 29b0b33dad..4fc7bee3e2 100644
--- a/firedrake/mg/adaptive_hierarchy.py
+++ b/firedrake/mg/adaptive_hierarchy.py
@@ -5,15 +5,12 @@
 
 from fractions import Fraction
 from collections import defaultdict
-import numpy as np
 
 from firedrake.cofunction import Cofunction
 from firedrake.function import Function
-from firedrake.functionspace import FunctionSpace
-from firedrake.mesh import Mesh, Submesh, RelabeledMesh
+from firedrake.mesh import Mesh
 from firedrake.mg import HierarchyBase
-from firedrake.mg.utils import set_level, get_level
-from firedrake.petsc import PETSc
+from firedrake.mg.utils import set_level
 
 __all__ = ["AdaptiveMeshHierarchy"]
 
@@ -40,121 +37,12 @@ def add_mesh(self, mesh):
         Then computes the coarse_to_fine and fine_to_coarse mappings.
         Constructs intermediate submesh hierarchies with this.
         """
-        max_children = 2 ** mesh.topological_dimension
-        coarse_mesh = self.meshes[-1]
         self._meshes += (mesh,)
         self.meshes += (mesh,)
         level = len(self.meshes)
         set_level(self.meshes[-1], self, level - 1)
         self._shared_data_cache = defaultdict(dict)
 
-        # extract parent child relationships from netgen meshes
-        (c2f, f2c) = get_c2f_f2c_fd(mesh, coarse_mesh)
-        c2f_global_key = Fraction(len(self.meshes) - 2, 1)
-        f2c_global_key = Fraction(len(self.meshes) - 1, 1)
-        self.coarse_to_fine_cells[c2f_global_key] = c2f
-        self.fine_to_coarse_cells[f2c_global_key] = np.array(f2c)
-
-        # split both the fine and coarse meshes into the submeshes
-        (coarse_splits, fine_splits, num_children) = split_to_submesh(
-            mesh, coarse_mesh, c2f, f2c
-        )
-        for i in range(1, max_children + 1):
-            coarse_mesh.mark_entities(coarse_splits[i], i)
-            mesh.mark_entities(fine_splits[i], int(f"10{i}"))
-
-        coarse_labels = list(range(1, max_children + 1))
-        coarse_indicators = [coarse_splits[i] for i in coarse_labels]
-
-        has_children = {}
-        for label in coarse_labels:
-            with coarse_splits[label].dat.vec as v:
-                has_children[label] = v.norm() > 1E-12
-
-        coarse_mesh = RelabeledMesh(
-            coarse_mesh,
-            coarse_indicators,
-            coarse_labels,
-            name="Relabeled_coarse",
-        )
-        c_subm = {
-            j: Submesh(coarse_mesh, coarse_mesh.topological_dimension, j)
-            for j in range(1, max_children + 1)
-            if has_children[j]
-        }
-        set_level(coarse_mesh, self, level - 2)
-
-        fine_labels = list(range(1, max_children + 1))
-        fine_indicators = [fine_splits[i] for i in fine_labels]
-        mesh = RelabeledMesh(
-            mesh,
-            fine_indicators,
-            fine_labels,
-            name="Relabeled_fine",
-        )
-        f_subm = {
-            int(str(j)[-2:]): Submesh(mesh, mesh.topological_dimension, j)
-            for j in [int("10" + str(i)) for i in range(1, max_children + 1)]
-            if has_children[int(str(j)[-2:])]
-        }
-        set_level(mesh, self, level - 1)
-
-        # stores number of parents for each amount of children
-        parents_per_child_count = [
-            len([el for el in c2f if len(el) == j])
-            for j in range(1, max_children + 1)
-        ]
-
-        # update c2f and f2c for submeshes by mapping numberings
-        # on full mesh to numberings on coarse mesh
-        c2f_adjusted = {
-            j: np.zeros((num_parents, j))
-            for j, num_parents in enumerate(parents_per_child_count, 1)
-            if num_parents != 0
-        }
-        f2c_adjusted = {
-            j: np.zeros((num_parents * j, 1))
-            for j, num_parents in enumerate(parents_per_child_count, 1)
-            if num_parents != 0
-        }
-
-        coarse_full_to_sub_map = {
-            i: get_full_to_sub_numbering(coarse_mesh, c_subm[i])
-            for i in c_subm
-        }
-        fine_full_to_sub_map = {
-            j: get_full_to_sub_numbering(mesh, f_subm[j])
-            for j in f_subm
-        }
-
-        for i, children in enumerate(c2f):
-            n = len(children)
-            if n in coarse_full_to_sub_map:
-                coarse_id_sub = coarse_full_to_sub_map[n][i]
-                fine_id_sub = fine_full_to_sub_map[n][np.array(children)]
-                c2f_adjusted[n][coarse_id_sub] = fine_id_sub
-
-        for j, parent in enumerate(f2c):
-            n = num_children[parent].item()
-            if n in coarse_full_to_sub_map:
-                coarse_id_sub = coarse_full_to_sub_map[n][parent.item()]
-                fine_id_sub = fine_full_to_sub_map[n][j]
-                f2c_adjusted[n][fine_id_sub, 0] = coarse_id_sub
-
-        c2f_subm = {
-            i: {Fraction(0, 1): c2f_adjusted[i].astype(PETSc.IntType)}
-            for i in c2f_adjusted
-        }
-        f2c_subm = {i: {Fraction(1, 1): f2c_adjusted[i]} for i in f2c_adjusted}
-
-        hierarchy_dict = {
-            i: HierarchyBase(
-                [c_subm[i], f_subm[i]], c2f_subm[i], f2c_subm[i], nested=True
-            )
-            for i in c_subm
-        }
-        self.submesh_hierarchies.append(hierarchy_dict)
-
     def refine(self, refinements):
         """
         Refines and adds mesh if input a boolean vector corresponding to cells
@@ -210,159 +98,3 @@ def adapt(self, eta: Function | Cofunction, theta: float):
         refined_mesh = mesh.refine_marked_elements(markers)
         self.add_mesh(refined_mesh)
         return refined_mesh
-
-    def split_function(self, u, child=True):
-        """
-        Split input function across submeshes
-        """
-        V = u.function_space()
-        full_mesh = V.mesh()
-        _, level = get_level(full_mesh)
-
-        ind = 1 if child else 0
-        hierarchy_dict = self.submesh_hierarchies[int(level) - ind]
-        parent_mesh = tuple(hierarchy_dict.values())[0].meshes[ind].submesh_parent
-        parent_space = V.reconstruct(parent_mesh)
-        u_corr_space = Function(parent_space, val=u.dat)
-        key = (u, child)
-        try:
-            split_functions = self.split_cache[key]
-        except KeyError:
-            split_functions = self.split_cache.setdefault(key, {})
-
-        for i in hierarchy_dict:
-            try:
-                f = split_functions[i].zero()
-            except KeyError:
-                V_split = V.reconstruct(mesh=hierarchy_dict[i].meshes[ind])
-                assert (
-                    V_split.mesh().submesh_parent
-                    == u_corr_space.function_space().mesh()
-                )
-                f = split_functions.setdefault(
-                    i,
-                    Function(V_split, name=str(i))
-                )
-
-            f.assign(u_corr_space)
-        return split_functions
-
-    def use_weight(self, V, child):
-        """
-        Counts DoFs across submeshes, computes partition of unity
-        """
-        w = Function(V).assign(1)
-        splits = self.split_function(w, child)
-
-        self.recombine(splits, w, child)
-        with w.dat.vec as wvec:
-            wvec.reciprocal()
-        return w
-
-    def recombine(self, split_funcs, f, child=True):
-        """
-        Recombines functions on submeshes back to the parent mesh
-        """
-        V = f.function_space()
-        f.zero()
-        parent_mesh = (
-            split_funcs[[*split_funcs][0]].function_space().mesh().submesh_parent
-        )
-        V_label = V.reconstruct(mesh=parent_mesh)
-        f_label = Function(V_label, val=f.dat)
-
-        for split_label, val in split_funcs.items():
-            assert val.function_space().mesh().submesh_parent == parent_mesh
-            if child:
-                split_label = int("10" + str(split_label))
-            if isinstance(f_label, Function):
-                f_label.assign(val, allow_missing_dofs=True)
-            else:
-                curr = Function(f_label.function_space()).assign(
-                    val, allow_missing_dofs=True
-                )
-                f_label.assign(f_label + curr)  # partition of unity for restriction
-        return f
-
-
-def get_c2f_f2c_fd(fine_mesh, coarse_mesh):
-    """
-    Construct coarse->fine and fine->coarse relations by mapping netgen elements to firedrake ones
-    """
-    fine_dm = fine_mesh.topology_dm
-    tdim = fine_dm.getDimension()
-    ngmesh = fine_mesh.netgen_mesh
-    if tdim == 2:
-        parents = ngmesh.parentsurfaceelements.NumPy()
-    elif tdim == 3:
-        parents = ngmesh.parentelements.NumPy()
-    else:
-        raise RuntimeError("Adaptivity not implemented in dimension of mesh")
-
-    fstart, fend = fine_mesh.topology_dm.getDepthStratum(tdim)
-    cstart, cend = coarse_mesh.topology_dm.getDepthStratum(tdim)
-    total_coarse_cells = FunctionSpace(coarse_mesh, "DG", 0).dim()
-
-    c2f = [[] for _ in range(cstart, cend)]
-    f2c = [[] for _ in range(fstart, fend)]
-
-    if parents.shape[0] == 0:
-        raise RuntimeError("Added mesh has not refined any cells from previous mesh")
-
-    for l in range(fstart, fend):
-        a = l
-        while a >= total_coarse_cells:
-            a = parents[a][0]
-
-        fine_ind = fine_mesh._cell_numbering.getOffset(l)
-        coarse_ind = coarse_mesh._cell_numbering.getOffset(a)
-
-        fine_ind -= fstart
-        coarse_ind -= cstart
-
-        f2c[fine_ind].append(coarse_ind)
-        c2f[coarse_ind].append(fine_ind)
-
-    return c2f, np.array(f2c, dtype=PETSc.IntType)
-
-
-def split_to_submesh(mesh, coarse_mesh, c2f, f2c):
-    """
-    Computes submesh split from full mesh.
-    Returns splits which are Functions denoting whether elements
-    belong to the corresponing submesh (bool)
-    """
-    max_children = 2 ** mesh.topological_dimension
-    V = FunctionSpace(mesh, "DG", 0)
-    V2 = FunctionSpace(coarse_mesh, "DG", 0)
-    coarse_splits = {
-        i: Function(V2, name=f"{i}_elements") for i in range(1, max_children + 1)
-    }
-    fine_splits = {
-        i: Function(V, name=f"{i}_elements") for i in range(1, max_children + 1)
-    }
-    num_children = np.zeros((len(c2f),))
-    for i, children in enumerate(c2f):
-        n = len(children)
-        if 1 <= n <= max_children:
-            num_children[i] = n
-
-    for i in range(1, max_children + 1):
-        coarse_splits[i].dat.data_wo_with_halos[num_children == i] = 1
-        fine_splits[i].dat.data_wo_with_halos[num_children[f2c.squeeze()] == i] = 1
-
-    return coarse_splits, fine_splits, num_children
-
-
-def get_full_to_sub_numbering(mesh, submesh):
-    """
-    Returns the submesh cell id associated with the full mesh cell id
-    """
-    V1 = FunctionSpace(mesh, "DG", 0)
-    V2 = FunctionSpace(submesh, "DG", 0)
-    u1 = Function(V1)
-    u2 = Function(V2)
-    u2.dat.data_wo[:] = np.arange(submesh.cell_set.size)
-    u1.assign(u2, allow_missing_dofs=True)
-
-    return u1.dat.data_ro_with_halos.astype(PETSc.IntType)
diff --git a/firedrake/mg/adaptive_transfer_manager.py b/firedrake/mg/adaptive_transfer_manager.py
index 5fa0e92f01..d6ce290f5a 100644
--- a/firedrake/mg/adaptive_transfer_manager.py
+++ b/firedrake/mg/adaptive_transfer_manager.py
@@ -2,11 +2,15 @@
 This module contains the AdaptiveTransferManager used to perform
 transfer operations on AdaptiveMeshHierarchies
 """
-import numpy as np
-from firedrake.function import Function
 from firedrake.mg.embedded import TransferManager
-from firedrake.mg.utils import get_level
-from firedrake.petsc import PETSc
+from firedrake.ufl_expr import action, TrialFunction
+from firedrake.functionspace import FunctionSpace, TensorFunctionSpace
+from firedrake.interpolation import interpolate
+from firedrake.preconditioners.bddc import is_lagrange
+from finat.quadrature import QuadratureRule
+from functools import partial
+
+import numpy
 
 
 __all__ = ("AdaptiveTransferManager",)
@@ -18,132 +22,73 @@ class AdaptiveTransferManager(TransferManager):
     """
     def __init__(self, *, native_transfers=None, use_averaging=True):
         super().__init__(native_transfers=native_transfers, use_averaging=use_averaging)
-        self.tm = TransferManager()
-        self.weight_cache = {}
-        self.work_function_cache = {}
-        self.perm_cache = {}
-
-    def generic_transfer(self, source, target, transfer_op):
-        """
-        Generalized implementation of transfer operations by wrapping
-        transfer operations from TransferManager()
-        """
-        amh, source_level = get_level(source.function_space().mesh())
-        _, target_level = get_level(target.function_space().mesh())
-
-        # decide order of iteration depending on coarse -> fine or fine -> coarse
-        order = 1
-        if target_level < source_level:
-            order = -1
-
-        curr_source = source
-        if source_level == target_level:
-            target.assign(source)
-            return
-
-        for level in range(source_level, target_level, order):
-            if level + order == target_level:
-                curr_target = target
-            else:
-                target_mesh = amh.meshes[level + order]
-                curr_space = curr_source.function_space()
-                target_space = curr_space.reconstruct(mesh=target_mesh)
-                curr_target = self.get_work_function(target_space)
-
-            if transfer_op == self.tm.restrict:
-                w = self.get_weight(curr_source.function_space())
-                wsource = self.get_work_function(curr_source.function_space())
-                with (
-                    curr_source.dat.vec as svec,
-                    w.dat.vec as wvec,
-                    wsource.dat.vec as wsvec,
-                ):
-                    wsvec.pointwiseMult(svec, wvec)
-                curr_source = wsource
-
-            if order == 1:
-                source_function_splits = amh.split_function(curr_source, child=False)
-                target_function_splits = amh.split_function(curr_target, child=True)
-            else:
-                source_function_splits = amh.split_function(curr_source, child=True)
-                target_function_splits = amh.split_function(curr_target, child=False)
-
-            for split_label in source_function_splits:
-                if split_label == 1:
-                    # we don't want to transfer across unsplit parts,
-                    # instead we copy dofs
-                    us_func = source_function_splits[1]
-                    ut_func = target_function_splits[1]
-                    permutations = self.get_perm(
-                        us_func,
-                        ut_func,
-                        transfer_op
-                    )
-                    ut_func.dat.data_wo[permutations] = us_func.dat.data_ro
-                else:
-                    transfer_op(
-                        source_function_splits[split_label],
-                        target_function_splits[split_label],
-                    )
-
-            amh.recombine(target_function_splits, curr_target, child=order + 1)
-            curr_source = curr_target
-
-    def get_work_function(self, func_space):
-        """
-        Cache for function on function space
-        """
-        try:
-            return self.work_function_cache[func_space]
-        except KeyError:
-            return self.work_function_cache.setdefault(func_space, Function(func_space))
+        self.cache = {}
 
-    def get_weight(self, V_source):
-        """
-        Cache for weights from partition of unity used during restriction
-        """
-        try:
-            return self.weight_cache[V_source]
-        except KeyError:
-            amh, _ = get_level(V_source.mesh())
-            return self.weight_cache.setdefault(
-                V_source, amh.use_weight(V_source, child=True)
-            )
-
-    def get_perm(self, unsplit_source, unsplit_target, transfer_op):
-        """
-        Cache permutations of DoFs from unsplit source
-        to unsplit target. This is used to skip transfer
-        across unsplit mesh hierarchies
-        """
-        key = (unsplit_source.function_space(),
-               unsplit_target.function_space())
+    def get_operators(self, Vc, Vf):
+        key = (Vc, Vf)
         try:
-            return self.perm_cache[key]
+            return self.cache[key]
         except KeyError:
-            source_nodes = Function(key[0])
-            permutation = Function(key[1])
-            source_nodes.dat.data_wo[:] = np.arange(len(source_nodes.dat.data_ro))
-            transfer_op(source_nodes, permutation)
-
-            return self.perm_cache.setdefault(
-                key, np.rint(permutation.dat.data_ro).astype(PETSc.IntType)
-            )
-
-    def prolong(self, uc, uf):
-        """
-        Prolongation of AdaptiveMeshHierarchy
-        """
-        self.generic_transfer(uc, uf, transfer_op=self.tm.prolong)
-
-    def inject(self, uf, uc):
-        """
-        Injection of AdaptiveMeshHierarchy
-        """
-        self.generic_transfer(uf, uc, transfer_op=self.tm.inject)
-
-    def restrict(self, source, target):
-        """
-        Restriction of AdaptiveMeshHierarchy
-        """
-        self.generic_transfer(source, target, transfer_op=self.tm.restrict)
+            ops = get_mg_interpolator(Vc, Vf)
+            return self.cache.setdefault(key, ops)
+
+    def forward(self, uc, uf):
+        from firedrake.assemble import assemble
+        Vc = uc.function_space()
+        Vf = uf.function_space()
+        ops = self.get_operators(Vc, Vf)
+
+        expr = uc
+        for op in ops:
+            expr = action(op, expr)
+        return assemble(expr, tensor=uf)
+
+    def adjoint(self, uf, uc):
+        from firedrake.assemble import assemble
+        Vc = uc.function_space().dual()
+        Vf = uf.function_space().dual()
+        ops = self.get_operators(Vc, Vf)
+
+        expr = uf
+        for op in reversed(ops):
+            expr = action(expr, op)
+        return assemble(expr, tensor=uc)
+
+    def prolong(self, uf, uc):
+        return self.forward(uf, uc)
+
+    def inject(self, uc, uf):
+        return self.forward(uc, uf)
+
+    def restrict(self, uc, uf):
+        return self.adjoint(uc, uf)
+
+
+def make_quadrature_space(V):
+    fe = V.finat_element
+    _, ps = fe.dual_basis
+    wts = numpy.full(len(ps.points), numpy.nan)
+    scheme = QuadratureRule(ps, wts, ref_el=fe.cell)
+    if V.value_shape == ():
+        make_space = FunctionSpace
+    else:
+        make_space = partial(TensorFunctionSpace, shape=V.value_shape)
+    return make_space(V.mesh(), "Quadrature", degree=fe.degree, quad_scheme=scheme)
+
+
+def get_mg_interpolator(V1, V2):
+    from firedrake.assemble import assemble
+    if is_lagrange(V2.finat_element):
+        spaces = (V1, V2)
+    else:
+        Q2 = make_quadrature_space(V2)
+        spaces = (V1, Q2, V2)
+
+    ops = []
+    for i in range(len(spaces)-1):
+        Vsrc = spaces[i]
+        Vdest = spaces[i+1]
+        Iexpr = interpolate(TrialFunction(Vsrc), Vdest)
+        op = assemble(Iexpr, mat_type="aij")
+        ops.append(op)
+    return ops
diff --git a/tests/firedrake/multigrid/test_adaptive_multigrid.py b/tests/firedrake/multigrid/test_adaptive_multigrid.py
index b2d88c1216..1eb966e555 100644
--- a/tests/firedrake/multigrid/test_adaptive_multigrid.py
+++ b/tests/firedrake/multigrid/test_adaptive_multigrid.py
@@ -27,8 +27,9 @@ def amh(request):
         geo = OCCGeometry(cube, dim=3)
         maxh = 0.5
 
+    dparams = {"overlap_type": (DistributedMeshOverlapType.VERTEX, 1)}
     ngmesh = geo.GenerateMesh(maxh=maxh)
-    base = Mesh(ngmesh, distribution_parameters={"overlap_type": (DistributedMeshOverlapType.VERTEX, 2)})
+    base = Mesh(ngmesh, distribution_parameters=dparams)
     amh_test = AdaptiveMeshHierarchy(base)
 
     rg = RandomGenerator(PCG64(seed=0))
@@ -46,8 +47,7 @@ def amh(request):
             el.refine = 1 if should_refine[i] < 0.5 else 0
 
         ngmesh.Refine(adaptive=True)
-        mesh = Mesh(ngmesh)
-
+        mesh = Mesh(ngmesh, distribution_parameters=dparams)
         amh_test.add_mesh(mesh)
     return amh_test
 
@@ -87,7 +87,7 @@ def tm():
     return TransferManager()
 
 
-@pytest.mark.parallel([1])
+@pytest.mark.parallel([1, 2])
 @pytest.mark.skipnetgen
 @pytest.mark.parametrize("operator", ["prolong", "inject"])
 def test_DG0(amh, atm, operator):  # pylint: disable=W0621
@@ -121,7 +121,7 @@ def test_DG0(amh, atm, operator):  # pylint: disable=W0621
         assert errornorm(stepc, u_coarse) <= 1e-12
 
 
-@pytest.mark.parallel([1])
+@pytest.mark.parallel([1, 2])
 @pytest.mark.skipnetgen
 @pytest.mark.parametrize("operator", ["prolong", "inject"])
 def test_CG1(amh, atm, operator):  # pylint: disable=W0621
@@ -149,6 +149,7 @@ def test_CG1(amh, atm, operator):  # pylint: disable=W0621
         assert errornorm(xc, u_coarse) <= 1e-12
 
 
+@pytest.mark.parallel([1, 2])
 @pytest.mark.skipnetgen
 def test_restrict_consistency(mh_res, atm, tm):  # pylint: disable=W0621
     """
@@ -195,6 +196,7 @@ def test_restrict_consistency(mh_res, atm, tm):  # pylint: disable=W0621
     ) <= 1e-12
 
 
+@pytest.mark.parallel([1, 2])
 @pytest.mark.skipnetgen
 def test_restrict_CG1(amh, atm):  # pylint: disable=W0621
     """
@@ -220,6 +222,7 @@ def test_restrict_CG1(amh, atm):  # pylint: disable=W0621
     )
 
 
+@pytest.mark.parallel([1, 2])
 @pytest.mark.skipnetgen
 def test_restrict_DG0(amh, atm):  # pylint: disable=W0621
     """
@@ -245,6 +248,7 @@ def test_restrict_DG0(amh, atm):  # pylint: disable=W0621
     )
 
 
+@pytest.mark.parallel([1, 2])
 @pytest.mark.skipnetgen
 def test_mg_jacobi(amh, atm):  # pylint: disable=W0621
     """
@@ -283,6 +287,7 @@ def test_mg_jacobi(amh, atm):  # pylint: disable=W0621
     assert errornorm(u_ex, u) <= 1e-8
 
 
+@pytest.mark.parallel([1, 2])
 @pytest.mark.skipnetgen
 @pytest.mark.parametrize("params", ["jacobi", "asm", "patch"])
 def test_mg_patch(amh, atm, params):  # pylint: disable=W0621

From 433a4b0935d7d93140ce3370c948847c8ceb9f68 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 19 Jan 2026 22:58:12 +0000
Subject: [PATCH 07/37] Fix tests

---
 tests/firedrake/multigrid/test_adaptive_multigrid.py | 9 ++-------
 1 file changed, 2 insertions(+), 7 deletions(-)

diff --git a/tests/firedrake/multigrid/test_adaptive_multigrid.py b/tests/firedrake/multigrid/test_adaptive_multigrid.py
index 1eb966e555..4e3c151f4f 100644
--- a/tests/firedrake/multigrid/test_adaptive_multigrid.py
+++ b/tests/firedrake/multigrid/test_adaptive_multigrid.py
@@ -6,7 +6,6 @@
 import pytest
 import numpy as np
 from firedrake import *
-from firedrake.utils import complex_mode
 
 
 @pytest.fixture(params=[2, 3])
@@ -112,12 +111,8 @@ def test_DG0(amh, atm, operator):  # pylint: disable=W0621
     if operator == "inject":
         u_fine.interpolate(stepf)
         assert errornorm(stepf, u_fine) <= 1e-12
-        if complex_mode:
-            with pytest.raises(NotImplementedError):
-                atm.inject(u_fine, u_coarse)
-            return
-        else:
-            atm.inject(u_fine, u_coarse)
+
+        atm.inject(u_fine, u_coarse)
         assert errornorm(stepc, u_coarse) <= 1e-12
 
 

From 706e303d05eb6febd89090f52c41f0c8bb6d7c26 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Tue, 20 Jan 2026 15:18:59 +0000
Subject: [PATCH 08/37] Apply suggestions from code review

---
 firedrake/mg/adaptive_hierarchy.py | 14 --------------
 1 file changed, 14 deletions(-)

diff --git a/firedrake/mg/adaptive_hierarchy.py b/firedrake/mg/adaptive_hierarchy.py
index 4fc7bee3e2..0489291a2b 100644
--- a/firedrake/mg/adaptive_hierarchy.py
+++ b/firedrake/mg/adaptive_hierarchy.py
@@ -34,8 +34,6 @@ def __init__(self, base_mesh, refinements_per_level=1, nested=True):
     def add_mesh(self, mesh):
         """
         Adds newly refined mesh into hierarchy.
-        Then computes the coarse_to_fine and fine_to_coarse mappings.
-        Constructs intermediate submesh hierarchies with this.
         """
         self._meshes += (mesh,)
         self.meshes += (mesh,)
@@ -43,18 +41,6 @@ def add_mesh(self, mesh):
         set_level(self.meshes[-1], self, level - 1)
         self._shared_data_cache = defaultdict(dict)
 
-    def refine(self, refinements):
-        """
-        Refines and adds mesh if input a boolean vector corresponding to cells
-        """
-        ngmesh = self.meshes[-1].netgen_mesh
-        for i, el in enumerate(ngmesh.Elements2D()):
-            el.refine = refinements[i]
-
-        ngmesh.Refine(adaptive=True)
-        mesh = Mesh(ngmesh)
-        self.add_mesh(mesh)
-
     def adapt(self, eta: Function | Cofunction, theta: float):
         """
         Add a refinement level to the hierarchy by local refinement

From 8dd0e66c6c9390e0c86975c6a7f188dc555520cc Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Tue, 20 Jan 2026 16:48:52 +0000
Subject: [PATCH 09/37] fixup

---
 firedrake/mg/adaptive_hierarchy.py            |  1 -
 .../multigrid/test_adaptive_multigrid.py      | 31 +++++++++++--------
 2 files changed, 18 insertions(+), 14 deletions(-)

diff --git a/firedrake/mg/adaptive_hierarchy.py b/firedrake/mg/adaptive_hierarchy.py
index 0489291a2b..d1dcceefe1 100644
--- a/firedrake/mg/adaptive_hierarchy.py
+++ b/firedrake/mg/adaptive_hierarchy.py
@@ -8,7 +8,6 @@
 
 from firedrake.cofunction import Cofunction
 from firedrake.function import Function
-from firedrake.mesh import Mesh
 from firedrake.mg import HierarchyBase
 from firedrake.mg.utils import set_level
 
diff --git a/tests/firedrake/multigrid/test_adaptive_multigrid.py b/tests/firedrake/multigrid/test_adaptive_multigrid.py
index 4e3c151f4f..de9197f974 100644
--- a/tests/firedrake/multigrid/test_adaptive_multigrid.py
+++ b/tests/firedrake/multigrid/test_adaptive_multigrid.py
@@ -26,8 +26,9 @@ def amh(request):
         geo = OCCGeometry(cube, dim=3)
         maxh = 0.5
 
-    dparams = {"overlap_type": (DistributedMeshOverlapType.VERTEX, 1)}
     ngmesh = geo.GenerateMesh(maxh=maxh)
+
+    dparams = {"overlap_type": (DistributedMeshOverlapType.VERTEX, 1)}
     base = Mesh(ngmesh, distribution_parameters=dparams)
     amh_test = AdaptiveMeshHierarchy(base)
 
@@ -52,7 +53,7 @@ def amh(request):
 
 
 @pytest.fixture
-def mh_res():
+def mh_uniform():
     """
     Generate MeshHierarchy for reference
     """
@@ -63,15 +64,20 @@ def mh_res():
     geo = OCCGeometry(face, dim=2)
     maxh = 0.5
     ngmesh = geo.GenerateMesh(maxh=maxh)
-    base = Mesh(ngmesh)
-    mesh2 = Mesh(ngmesh)
-    amh_unif = AdaptiveMeshHierarchy(base)
-    for _ in range(2):
-        refs = np.ones(len(ngmesh.Elements2D()))
-        amh_unif.refine(refs)
-    mh = MeshHierarchy(mesh2, 2)
 
-    return amh_unif, mh
+    dparams = {"overlap_type": (DistributedMeshOverlapType.VERTEX, 1)}
+    base1 = Mesh(ngmesh, distribution_parameters=dparams)
+    mh = MeshHierarchy(base1, 2)
+
+    base2 = Mesh(ngmesh, distribution_parameters=dparams)
+    amh = AdaptiveMeshHierarchy(base2)
+    for _ in range(2):
+        mesh = amh[-1]
+        ngmesh = mesh.netgen_mesh
+        ngmesh.Refine()
+        mesh = Mesh(ngmesh, distribution_parameters=dparams)
+        amh.add_mesh(mesh)
+    return amh, mh
 
 
 @pytest.fixture
@@ -146,12 +152,11 @@ def test_CG1(amh, atm, operator):  # pylint: disable=W0621
 
 @pytest.mark.parallel([1, 2])
 @pytest.mark.skipnetgen
-def test_restrict_consistency(mh_res, atm, tm):  # pylint: disable=W0621
+def test_restrict_consistency(mh_uniform, atm, tm):  # pylint: disable=W0621
     """
     Test restriction consistency of amh with uniform refinement vs mh
     """
-    amh_unif = mh_res[0]
-    mh = mh_res[1]
+    amh_unif, mh = mh_uniform
 
     V_coarse = FunctionSpace(amh_unif[0], "DG", 0)
     V_fine = FunctionSpace(amh_unif[-1], "DG", 0)

From c40a36b313ce7e5c9909bf7c1ae0859aa008e847 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 26 Jan 2026 09:55:11 +0000
Subject: [PATCH 10/37] Apply suggestions from code review

---
 firedrake/mg/adaptive_hierarchy.py | 16 +++++-----------
 1 file changed, 5 insertions(+), 11 deletions(-)

diff --git a/firedrake/mg/adaptive_hierarchy.py b/firedrake/mg/adaptive_hierarchy.py
index d1dcceefe1..e307d24ad9 100644
--- a/firedrake/mg/adaptive_hierarchy.py
+++ b/firedrake/mg/adaptive_hierarchy.py
@@ -3,7 +3,6 @@
 related helper functions
 """
 
-from fractions import Fraction
 from collections import defaultdict
 
 from firedrake.cofunction import Cofunction
@@ -19,23 +18,18 @@ class AdaptiveMeshHierarchy(HierarchyBase):
     HierarchyBase for hierarchies of adaptively refined meshes
     """
     def __init__(self, base_mesh, refinements_per_level=1, nested=True):
-        self.meshes = (base_mesh,)
-        self._meshes = (base_mesh,)
-        self.submesh_hierarchies = []
-        self.coarse_to_fine_cells = {}
-        self.fine_to_coarse_cells = {}
-        self.fine_to_coarse_cells[Fraction(0, 1)] = None
+        self.meshes = []
+        self._meshes = []
         self.refinements_per_level = refinements_per_level
         self.nested = nested
-        set_level(base_mesh, self, 0)
-        self.split_cache = {}
+        self.add_mesh(base_mesh)
 
     def add_mesh(self, mesh):
         """
         Adds newly refined mesh into hierarchy.
         """
-        self._meshes += (mesh,)
-        self.meshes += (mesh,)
+        self._meshes.append(mesh)
+        self.meshes.append(mesh)
         level = len(self.meshes)
         set_level(self.meshes[-1], self, level - 1)
         self._shared_data_cache = defaultdict(dict)

From 76c2d59f005fb9960a49e86b0522ed292eb5d124 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Fri, 19 Dec 2025 16:35:20 +0000
Subject: [PATCH 11/37] enable tests

---
 .../regression/test_interpolate_cross_mesh.py    | 16 ++++------------
 1 file changed, 4 insertions(+), 12 deletions(-)

diff --git a/tests/firedrake/regression/test_interpolate_cross_mesh.py b/tests/firedrake/regression/test_interpolate_cross_mesh.py
index 32179172f7..303e95182c 100644
--- a/tests/firedrake/regression/test_interpolate_cross_mesh.py
+++ b/tests/firedrake/regression/test_interpolate_cross_mesh.py
@@ -47,14 +47,7 @@ def make_high_order(m_low_order, degree):
         "unitsquare_vfs",
         "unitsquare_tfs",
         "unitsquare_N1curl_source",
-        pytest.param(
-            "unitsquare_SminusDiv_destination",
-            marks=pytest.mark.xfail(
-                # CalledProcessError is so the parallel tests correctly xfail
-                raises=(subprocess.CalledProcessError, NotImplementedError),
-                reason="Can only interpolate into spaces with point evaluation nodes",
-            ),
-        ),
+        "unitsquare_SminusDiv_destination",
         "unitsquare_Regge_source",
         # This test fails in complex mode
         pytest.param("spheresphere", marks=pytest.mark.skipcomplex),
@@ -436,10 +429,9 @@ def test_interpolate_cross_mesh_not_point_eval():
     atol = 1e-8  # default
     # This might not make much mathematical sense, but it should test if we get
     # the not implemented error for non-point evaluation nodes!
-    with pytest.raises(NotImplementedError):
-        interpolate_function(
-            m_src, m_dest, V_src, V_dest, dest_eval, expected, expr_src, expr_dest, atol
-        )
+    interpolate_function(
+        m_src, m_dest, V_src, V_dest, dest_eval, expected, expr_src, expr_dest, atol
+    )
 
 
 def interpolate_function(

From 67a5a0cb44688bf63de6265605429f677413dd0b Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Sun, 21 Dec 2025 18:30:05 +0000
Subject: [PATCH 12/37] remove check

---
 firedrake/interpolation.py | 11 -----------
 1 file changed, 11 deletions(-)

diff --git a/firedrake/interpolation.py b/firedrake/interpolation.py
index 3159e7221c..aa5043670c 100644
--- a/firedrake/interpolation.py
+++ b/firedrake/interpolation.py
@@ -424,17 +424,6 @@ def __init__(self, expr: Interpolate):
         else:
             self.access = op2.WRITE
 
-        # TODO check V.finat_element.is_lagrange() once https://github.com/firedrakeproject/fiat/pull/200 is released
-        target_element = self.target_space.ufl_element()
-        if not ((isinstance(target_element, MixedElement)
-                 and all(sub.mapping() == "identity" for sub in target_element.sub_elements))
-                or target_element.mapping() == "identity"):
-            # Identity mapping between reference cell and physical coordinates
-            # implies point evaluation nodes.
-            raise NotImplementedError(
-                "Can only cross-mesh interpolate into spaces with point evaluation nodes."
-            )
-
         if self.allow_missing_dofs:
             self.missing_points_behaviour = MissingPointsBehaviour.IGNORE
         else:

From e637d6f28f5ffb3f0211c0ae602b7ec48ecb21a1 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Sun, 21 Dec 2025 19:27:35 +0000
Subject: [PATCH 13/37] add test

---
 .../test_cross_mesh_non_lagrange.py           | 78 +++++++++++++++++++
 1 file changed, 78 insertions(+)
 create mode 100644 tests/firedrake/regression/test_cross_mesh_non_lagrange.py

diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
new file mode 100644
index 0000000000..2540930274
--- /dev/null
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -0,0 +1,78 @@
+from firedrake import *
+import pytest
+import numpy as np
+from finat.quadrature import QuadratureRule
+
+def make_quadrature_space(V):
+    """Builds a quadrature space on the target mesh.
+    
+    This space has point evaluation dofs at the quadrature PointSet
+    of the target space's element
+
+    """
+    fe = V.finat_element
+    _, ps = fe.dual_basis
+    wts = np.full(len(ps.points), np.nan)  # These can be any number since we never integrate
+    scheme = QuadratureRule(ps, wts, ref_el=fe.cell)
+    element = FiniteElement("Quadrature", degree=fe.degree, quad_scheme=scheme)
+    return VectorFunctionSpace(V.mesh(), element)
+
+
+@pytest.fixture
+def mesh1():
+    return UnitSquareMesh(5, 5)
+
+@pytest.fixture
+def mesh2():
+    return UnitSquareMesh(3, 3)
+
+@pytest.mark.parametrize("element,degree", [("RT", 1), ("RT", 2), ("RT", 3),
+                                            ("BDM", 1), ("BDM", 2), ("BDM", 3),
+                                            ("BDFM", 2)])
+def test_hdiv_cross_mesh_oneform(mesh1, mesh2, element, degree):
+    V_source = VectorFunctionSpace(mesh1, "CG", 2)
+    V_target = FunctionSpace(mesh2, element, degree)
+
+    x, y = SpatialCoordinate(mesh1)
+    f_source = Function(V_source).interpolate(as_vector([x, y]))
+
+    # Make intermediate Quadrature space on target mesh
+    Q_target = make_quadrature_space(V_target)
+
+    # Interp V_source -> Q
+    I1 = interpolate(f_source, Q_target)
+    f_quadrature = assemble(I1)
+
+    # Interp Q -> V_target
+    I2 = interpolate(f_quadrature, V_target)
+    f_target = assemble(I2)
+
+    x1, y1 = SpatialCoordinate(mesh2)
+    f_direct = Function(V_target).interpolate(as_vector([x1, y1]))
+
+    assert np.allclose(f_target.dat.data_ro, f_direct.dat.data_ro)
+
+@pytest.mark.parametrize("element,degree", [("N1curl", 1), ("N1curl", 2), ("N1curl", 3),
+                                            ("N2curl", 1), ("N2curl", 2), ("N2curl", 3),])
+def test_hcurl_cross_mesh_oneform(mesh1, mesh2, element, degree):
+    V_source = VectorFunctionSpace(mesh1, "CG", 2)
+    V_target = FunctionSpace(mesh2, element, degree)
+
+    x, y = SpatialCoordinate(mesh1)
+    f_source = Function(V_source).interpolate(as_vector([x, y]))
+
+    # Make intermediate Quadrature space on target mesh
+    Q_target = make_quadrature_space(V_target)
+
+    # Interp V_source -> Q
+    I1 = interpolate(f_source, Q_target)
+    f_quadrature = assemble(I1)
+
+    # Interp Q -> V_target
+    I2 = interpolate(f_quadrature, V_target)
+    f_target = assemble(I2)
+
+    x1, y1 = SpatialCoordinate(mesh2)
+    f_direct = Function(V_target).interpolate(as_vector([x1, y1]))
+
+    assert np.allclose(f_target.dat.data_ro, f_direct.dat.data_ro)

From 4d23dc28c4f4aa06c35958bf8f96389d659811a8 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Mon, 22 Dec 2025 17:13:55 +0000
Subject: [PATCH 14/37] update test

---
 .../test_cross_mesh_non_lagrange.py           | 80 +++++++++----------
 1 file changed, 40 insertions(+), 40 deletions(-)

diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 2540930274..6c88cf57f9 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -2,6 +2,18 @@
 import pytest
 import numpy as np
 from finat.quadrature import QuadratureRule
+from functools import partial
+
+
+def fs_shape(V):
+    shape = V.ufl_function_space().value_shape
+    if len(shape) == 1:
+        fs_type = partial(VectorFunctionSpace, dim=shape[0])
+    elif len(shape) == 2:
+        fs_type = partial(TensorFunctionSpace, shape=shape)
+    else:
+        raise ValueError("Invalid function space shape")
+    return fs_type
 
 def make_quadrature_space(V):
     """Builds a quadrature space on the target mesh.
@@ -15,51 +27,40 @@ def make_quadrature_space(V):
     wts = np.full(len(ps.points), np.nan)  # These can be any number since we never integrate
     scheme = QuadratureRule(ps, wts, ref_el=fe.cell)
     element = FiniteElement("Quadrature", degree=fe.degree, quad_scheme=scheme)
-    return VectorFunctionSpace(V.mesh(), element)
-
+    return fs_shape(V)(V.mesh(), element)
 
-@pytest.fixture
-def mesh1():
-    return UnitSquareMesh(5, 5)
 
-@pytest.fixture
-def mesh2():
-    return UnitSquareMesh(3, 3)
+@pytest.fixture(params=[("RT", 1), ("RT", 2), ("RT", 3), ("BDM", 1), ("BDM", 2), ("BDM", 3),
+                        ("BDFM", 2), ("HHJ", 2),("N1curl", 1), ("N1curl", 2), ("N1curl", 3),
+                        ("N2curl", 1), ("N2curl", 2), ("N2curl", 3), ("GLS", 1), ("GLS", 2),
+                        ("GLS", 3), ("GLS2", 1), ("GLS2", 2), ("GLS2", 3)],
+                        ids=lambda x: f"{x[0]}_{x[1]}")
+def V_target(request):
+    element, degree = request.param
+    mesh = UnitSquareMesh(3, 3)
+    return FunctionSpace(mesh, element, degree)
 
-@pytest.mark.parametrize("element,degree", [("RT", 1), ("RT", 2), ("RT", 3),
-                                            ("BDM", 1), ("BDM", 2), ("BDM", 3),
-                                            ("BDFM", 2)])
-def test_hdiv_cross_mesh_oneform(mesh1, mesh2, element, degree):
-    V_source = VectorFunctionSpace(mesh1, "CG", 2)
-    V_target = FunctionSpace(mesh2, element, degree)
 
+def test_cross_mesh_oneform(V_target):
+    mesh1 = UnitSquareMesh(5, 5)
+    mesh2 = V_target.mesh()
     x, y = SpatialCoordinate(mesh1)
-    f_source = Function(V_source).interpolate(as_vector([x, y]))
-
-    # Make intermediate Quadrature space on target mesh
-    Q_target = make_quadrature_space(V_target)
-
-    # Interp V_source -> Q
-    I1 = interpolate(f_source, Q_target)
-    f_quadrature = assemble(I1)
-
-    # Interp Q -> V_target
-    I2 = interpolate(f_quadrature, V_target)
-    f_target = assemble(I2)
-
     x1, y1 = SpatialCoordinate(mesh2)
-    f_direct = Function(V_target).interpolate(as_vector([x1, y1]))
-
-    assert np.allclose(f_target.dat.data_ro, f_direct.dat.data_ro)
-
-@pytest.mark.parametrize("element,degree", [("N1curl", 1), ("N1curl", 2), ("N1curl", 3),
-                                            ("N2curl", 1), ("N2curl", 2), ("N2curl", 3),])
-def test_hcurl_cross_mesh_oneform(mesh1, mesh2, element, degree):
-    V_source = VectorFunctionSpace(mesh1, "CG", 2)
-    V_target = FunctionSpace(mesh2, element, degree)
 
-    x, y = SpatialCoordinate(mesh1)
-    f_source = Function(V_source).interpolate(as_vector([x, y]))
+    shape = V_target.ufl_function_space().value_shape
+    if len(shape) == 1:
+        fs_type = partial(VectorFunctionSpace, dim=shape[0])
+        expr1 = as_vector([x, y])
+        expr2 = as_vector([x1, y1])
+    elif len(shape) == 2:
+        fs_type = partial(TensorFunctionSpace, shape=shape)
+        expr1 = as_tensor([[x, x*y], [x*y, y]])
+        expr2 = as_tensor([[x1, x1*y1], [x1*y1, y1]])
+    else:
+        raise ValueError("Unsupported target space shape")
+    
+    V_source = fs_type(mesh1, "CG", 2)
+    f_source = Function(V_source).interpolate(expr1)
 
     # Make intermediate Quadrature space on target mesh
     Q_target = make_quadrature_space(V_target)
@@ -72,7 +73,6 @@ def test_hcurl_cross_mesh_oneform(mesh1, mesh2, element, degree):
     I2 = interpolate(f_quadrature, V_target)
     f_target = assemble(I2)
 
-    x1, y1 = SpatialCoordinate(mesh2)
-    f_direct = Function(V_target).interpolate(as_vector([x1, y1]))
+    f_direct = Function(V_target).interpolate(expr2)
 
     assert np.allclose(f_target.dat.data_ro, f_direct.dat.data_ro)

From 5c84b1232ef4836ef24882702da339006b05a6e8 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Mon, 5 Jan 2026 17:56:10 +0000
Subject: [PATCH 15/37] update tests

---
 .../test_cross_mesh_non_lagrange.py           | 48 ++++++++++++++++---
 .../regression/test_interpolate_cross_mesh.py |  4 +-
 2 files changed, 43 insertions(+), 9 deletions(-)

diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 6c88cf57f9..65032bdddb 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -35,19 +35,21 @@ def make_quadrature_space(V):
                         ("N2curl", 1), ("N2curl", 2), ("N2curl", 3), ("GLS", 1), ("GLS", 2),
                         ("GLS", 3), ("GLS2", 1), ("GLS2", 2), ("GLS2", 3)],
                         ids=lambda x: f"{x[0]}_{x[1]}")
-def V_target(request):
+def V(request):
     element, degree = request.param
     mesh = UnitSquareMesh(3, 3)
     return FunctionSpace(mesh, element, degree)
 
+# V_source -> Q -> V_target
+# V_target^* -> Q^* -> V_source^*
 
-def test_cross_mesh_oneform(V_target):
+def test_cross_mesh_oneform(V):
     mesh1 = UnitSquareMesh(5, 5)
-    mesh2 = V_target.mesh()
+    mesh2 = V.mesh()
     x, y = SpatialCoordinate(mesh1)
     x1, y1 = SpatialCoordinate(mesh2)
 
-    shape = V_target.ufl_function_space().value_shape
+    shape = V.ufl_function_space().value_shape
     if len(shape) == 1:
         fs_type = partial(VectorFunctionSpace, dim=shape[0])
         expr1 = as_vector([x, y])
@@ -63,16 +65,48 @@ def test_cross_mesh_oneform(V_target):
     f_source = Function(V_source).interpolate(expr1)
 
     # Make intermediate Quadrature space on target mesh
-    Q_target = make_quadrature_space(V_target)
+    Q_target = make_quadrature_space(V)
 
     # Interp V_source -> Q
     I1 = interpolate(f_source, Q_target)
     f_quadrature = assemble(I1)
 
     # Interp Q -> V_target
-    I2 = interpolate(f_quadrature, V_target)
+    I2 = interpolate(f_quadrature, V)
     f_target = assemble(I2)
 
-    f_direct = Function(V_target).interpolate(expr2)
+    f_direct = Function(V).interpolate(expr2)
 
     assert np.allclose(f_target.dat.data_ro, f_direct.dat.data_ro)
+
+
+def test_cross_mesh_oneform_adjoint(V):
+    # Can already do Lagrange -> RT adjoint
+    # V^* -> Q^* -> V_target^*
+    mesh1 = UnitSquareMesh(7, 7)
+    x1 = SpatialCoordinate(mesh1)
+    V_target = fs_shape(V)(mesh1, "CG", 2)
+
+    mesh2 = V.mesh()
+    x2 = SpatialCoordinate(mesh2)
+
+    oneform_V = inner(x2, TestFunction(V)) * dx
+
+    Q_target = make_quadrature_space(V)
+
+    # Interp V^* -> Q^*
+    I1_adj = interpolate(TestFunction(Q_target), oneform_V)
+    cofunc_Q = assemble(I1_adj)
+
+    # Interp Q^* -> V_target^*
+    I2_adj = interpolate(TestFunction(V_target), cofunc_Q)
+    cofunc_V = assemble(I2_adj)
+
+    # cofunc_V = assemble(interpolate(TestFunction(V_target), oneform_target))  # V^* -> V_target^*
+
+    cofunc_V_direct = assemble(inner(x1, TestFunction(V_target)) * dx)
+
+    assert np.allclose(cofunc_V.dat.data_ro, cofunc_V_direct.dat.data_ro)
+
+if __name__ == "__main__":
+    pytest.main([__file__ + "::test_cross_mesh_oneform_adjoint[RT_1]"])
\ No newline at end of file
diff --git a/tests/firedrake/regression/test_interpolate_cross_mesh.py b/tests/firedrake/regression/test_interpolate_cross_mesh.py
index 303e95182c..23dfb742b8 100644
--- a/tests/firedrake/regression/test_interpolate_cross_mesh.py
+++ b/tests/firedrake/regression/test_interpolate_cross_mesh.py
@@ -185,8 +185,8 @@ def parameters(request):
         expr_dest = 2 * SpatialCoordinate(m_dest)
         expected = 2 * coords
         V_src = VectorFunctionSpace(m_src, "CG", 2)
-        V_dest = FunctionSpace(m_dest, "SminusDiv", 2)  # Not point evaluation nodes
-        V_dest_2 = FunctionSpace(m_dest, "SminusCurl", 2)  # Not point evaluation nodes
+        V_dest = FunctionSpace(m_dest, "RT", 2)  # Not point evaluation nodes
+        V_dest_2 = FunctionSpace(m_dest, "N1Curl", 2)  # Not point evaluation nodes
     elif request.param == "unitsquare_Regge_source":
         m_src, m_dest, coords = unitsquaresetup()
         expr_src = outer(SpatialCoordinate(m_src), SpatialCoordinate(m_src))

From d8a1ac30f32e1131c6ed5941701d1e7c31809502 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Mon, 5 Jan 2026 17:56:28 +0000
Subject: [PATCH 16/37] WIP

need to map replaced coefficient correctly
---
 tsfc/driver.py | 22 ++++++++++++++--------
 1 file changed, 14 insertions(+), 8 deletions(-)

diff --git a/tsfc/driver.py b/tsfc/driver.py
index fb3abe9775..6cc8646178 100644
--- a/tsfc/driver.py
+++ b/tsfc/driver.py
@@ -247,17 +247,26 @@ def compile_expression_dual_evaluation(expression, to_element, ufl_element, *,
 
     if isinstance(to_element, (PhysicallyMappedElement, DirectlyDefinedElement)):
         raise NotImplementedError("Don't know how to interpolate onto zany spaces, sorry")
+    
+    if domain is None:
+        domain = extract_unique_domain(expression)
+    assert domain is not None
 
     orig_coefficients = extract_coefficients(expression)
-    if isinstance(expression, ufl.Interpolate):
-        v, operand = expression.argument_slots()
-    else:
-        operand = expression
-        v = ufl.FunctionSpace(extract_unique_domain(operand), ufl_element)
+    v, operand = expression.argument_slots()
 
     # Map into reference space
     operand = apply_mapping(operand, ufl_element, domain)
 
+    if ufl_element.mapping() != "identity":
+        # Need to map dual argument for adjoint interpolation
+        ref_element = finat.ufl.WithMapping(ufl_element, "identity")
+        V = ufl.FunctionSpace(domain, ref_element)
+        if isinstance(v, ufl.Coargument):
+            v = ufl.Coargument(V.dual(), v.number())
+        else:
+            v = ufl.Cofunction(V.dual())
+
     # Apply UFL preprocessing
     operand = ufl_utils.preprocess_expression(operand, complex_mode=complex_mode)
 
@@ -276,9 +285,6 @@ def compile_expression_dual_evaluation(expression, to_element, ufl_element, *,
     assert len(argument_multiindices) == len(arguments)
 
     # Replace coordinates (if any) unless otherwise specified by kwarg
-    if domain is None:
-        domain = extract_unique_domain(expression)
-    assert domain is not None
     builder._domain_integral_type_map = {domain: "cell"}
     builder._entity_ids = {domain: (0,)}
 

From 8c255e240717adadd4706cce6b327093abbc1780 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Tue, 20 Jan 2026 10:10:50 +0000
Subject: [PATCH 17/37] fix test

fixes
---
 .../test_cross_mesh_non_lagrange.py           | 33 ++++++++++++-------
 tsfc/driver.py                                | 22 +++++--------
 2 files changed, 30 insertions(+), 25 deletions(-)

diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 65032bdddb..6651eada43 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -4,6 +4,8 @@
 from finat.quadrature import QuadratureRule
 from functools import partial
 
+from ufl.compound_expressions import deviatoric_expr_2x2
+
 
 def fs_shape(V):
     shape = V.ufl_function_space().value_shape
@@ -30,14 +32,14 @@ def make_quadrature_space(V):
     return fs_shape(V)(V.mesh(), element)
 
 
-@pytest.fixture(params=[("RT", 1), ("RT", 2), ("RT", 3), ("BDM", 1), ("BDM", 2), ("BDM", 3),
-                        ("BDFM", 2), ("HHJ", 2),("N1curl", 1), ("N1curl", 2), ("N1curl", 3),
-                        ("N2curl", 1), ("N2curl", 2), ("N2curl", 3), ("GLS", 1), ("GLS", 2),
-                        ("GLS", 3), ("GLS2", 1), ("GLS2", 2), ("GLS2", 3)],
+@pytest.fixture(params=[("RT", 2), ("RT", 3), ("RT", 4), ("BDM", 1), ("BDM", 2), ("BDM", 3),
+                        ("BDFM", 2), ("HHJ", 2),("N1curl", 2), ("N1curl", 3), ("N1curl", 4),
+                        ("N2curl", 1), ("N2curl", 2), ("N2curl", 3), ("GLS", 2), ("GLS", 3),
+                        ("GLS", 4), ("GLS2", 1), ("GLS2", 2), ("GLS2", 3)],
                         ids=lambda x: f"{x[0]}_{x[1]}")
 def V(request):
     element, degree = request.param
-    mesh = UnitSquareMesh(3, 3)
+    mesh = UnitSquareMesh(8, 8)
     return FunctionSpace(mesh, element, degree)
 
 # V_source -> Q -> V_target
@@ -83,14 +85,24 @@ def test_cross_mesh_oneform(V):
 def test_cross_mesh_oneform_adjoint(V):
     # Can already do Lagrange -> RT adjoint
     # V^* -> Q^* -> V_target^*
-    mesh1 = UnitSquareMesh(7, 7)
+    mesh1 = UnitSquareMesh(2, 2)
     x1 = SpatialCoordinate(mesh1)
-    V_target = fs_shape(V)(mesh1, "CG", 2)
+    V_target = fs_shape(V)(mesh1, "CG", 1)
 
     mesh2 = V.mesh()
     x2 = SpatialCoordinate(mesh2)
 
-    oneform_V = inner(x2, TestFunction(V)) * dx
+    if len(V.value_shape) > 1:
+        expr = outer(x2, x2)
+        target_expr = outer(x1, x1)
+        if V.ufl_element().mapping() == "covariant contravariant Piola":
+            expr = deviatoric_expr_2x2(expr)
+            target_expr = deviatoric_expr_2x2(target_expr)
+    else:
+        expr = x2
+        target_expr = x1
+
+    oneform_V = inner(expr, TestFunction(V)) * dx
 
     Q_target = make_quadrature_space(V)
 
@@ -104,9 +116,8 @@ def test_cross_mesh_oneform_adjoint(V):
 
     # cofunc_V = assemble(interpolate(TestFunction(V_target), oneform_target))  # V^* -> V_target^*
 
-    cofunc_V_direct = assemble(inner(x1, TestFunction(V_target)) * dx)
-
+    cofunc_V_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
     assert np.allclose(cofunc_V.dat.data_ro, cofunc_V_direct.dat.data_ro)
 
 if __name__ == "__main__":
-    pytest.main([__file__ + "::test_cross_mesh_oneform_adjoint[RT_1]"])
\ No newline at end of file
+    pytest.main([__file__ + "::test_cross_mesh_oneform_adjoint[GLS_3]"])
\ No newline at end of file
diff --git a/tsfc/driver.py b/tsfc/driver.py
index 6cc8646178..fb3abe9775 100644
--- a/tsfc/driver.py
+++ b/tsfc/driver.py
@@ -247,26 +247,17 @@ def compile_expression_dual_evaluation(expression, to_element, ufl_element, *,
 
     if isinstance(to_element, (PhysicallyMappedElement, DirectlyDefinedElement)):
         raise NotImplementedError("Don't know how to interpolate onto zany spaces, sorry")
-    
-    if domain is None:
-        domain = extract_unique_domain(expression)
-    assert domain is not None
 
     orig_coefficients = extract_coefficients(expression)
-    v, operand = expression.argument_slots()
+    if isinstance(expression, ufl.Interpolate):
+        v, operand = expression.argument_slots()
+    else:
+        operand = expression
+        v = ufl.FunctionSpace(extract_unique_domain(operand), ufl_element)
 
     # Map into reference space
     operand = apply_mapping(operand, ufl_element, domain)
 
-    if ufl_element.mapping() != "identity":
-        # Need to map dual argument for adjoint interpolation
-        ref_element = finat.ufl.WithMapping(ufl_element, "identity")
-        V = ufl.FunctionSpace(domain, ref_element)
-        if isinstance(v, ufl.Coargument):
-            v = ufl.Coargument(V.dual(), v.number())
-        else:
-            v = ufl.Cofunction(V.dual())
-
     # Apply UFL preprocessing
     operand = ufl_utils.preprocess_expression(operand, complex_mode=complex_mode)
 
@@ -285,6 +276,9 @@ def compile_expression_dual_evaluation(expression, to_element, ufl_element, *,
     assert len(argument_multiindices) == len(arguments)
 
     # Replace coordinates (if any) unless otherwise specified by kwarg
+    if domain is None:
+        domain = extract_unique_domain(expression)
+    assert domain is not None
     builder._domain_integral_type_map = {domain: "cell"}
     builder._entity_ids = {domain: (0,)}
 

From 40c125265277b6319f21b01324a00dd5bd78a585 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Wed, 21 Jan 2026 18:29:57 +0000
Subject: [PATCH 18/37] WIP

---
 firedrake/interpolation.py                    | 124 +++++++++++++++---
 .../test_cross_mesh_non_lagrange.py           |  79 ++++++++++-
 .../regression/test_interpolate_cross_mesh.py |   4 +
 3 files changed, 187 insertions(+), 20 deletions(-)

diff --git a/firedrake/interpolation.py b/firedrake/interpolation.py
index aa5043670c..0fb7a15f24 100644
--- a/firedrake/interpolation.py
+++ b/firedrake/interpolation.py
@@ -22,7 +22,7 @@
 from FIAT.reference_element import Point
 
 from finat.element_factory import create_element, as_fiat_cell
-from finat.ufl import TensorElement, VectorElement, MixedElement
+from finat.ufl import TensorElement, VectorElement, MixedElement, FiniteElement, FiniteElementBase
 from finat.fiat_elements import ScalarFiatElement
 from finat.quadrature import QuadratureRule
 from finat.quadrature_element import QuadratureElement
@@ -36,7 +36,7 @@
 
 from firedrake.utils import IntType, ScalarType, cached_property, known_pyop2_safe, tuplify
 from firedrake.tsfc_interface import extract_numbered_coefficients, _cachedir
-from firedrake.ufl_expr import Argument, Coargument, action
+from firedrake.ufl_expr import Argument, Coargument, TrialFunction, TestFunction, action
 from firedrake.mesh import MissingPointsBehaviour, VertexOnlyMeshTopology, MeshGeometry, MeshTopology, VertexOnlyMesh
 from firedrake.petsc import PETSc
 from firedrake.halo import _get_mtype
@@ -432,7 +432,21 @@ def __init__(self, expr: Interpolate):
         if self.source_mesh.geometric_dimension != self.target_mesh.geometric_dimension:
             raise ValueError("Geometric dimensions of source and destination meshes must match.")
 
-        dest_element = self.target_space.ufl_element()
+    def _get_element(self, V: WithGeometry) -> FiniteElementBase:
+        """Return the element of the function space V. If V is tensor/vector valued,
+        return the base scalar element.
+
+        Parameters
+        ----------
+        V
+            A :class:`.WithGeometry` function space.
+
+        Returns
+        -------
+        FiniteElementBase
+            The base element of V.
+        """
+        dest_element = V.ufl_element()
         if isinstance(dest_element, MixedElement):
             if isinstance(dest_element, VectorElement | TensorElement):
                 # In this case all sub elements are equal
@@ -442,16 +456,68 @@ def __init__(self, expr: Interpolate):
                         "Can't yet cross-mesh interpolate onto function spaces made from VectorElements "
                         "or TensorElements made from sub elements with value shape other than ()."
                     )
-                self.dest_element = base_element
+                return base_element
             else:
                 raise NotImplementedError("Interpolation with MixedFunctionSpace requires MixedInterpolator.")
         else:
             # scalar fiat/finat element
-            self.dest_element = dest_element
+            return dest_element
+
+    def _fs_type(self, V: WithGeometry) -> Callable[..., WithGeometry]:
+        """Returns a callable which returns a function space matching the type of V.
+
+        Parameters
+        ----------
+        V
+            A :class:`.WithGeometry` function space.
+
+        Returns
+        -------
+        Callable
+            A callable which returns a :class:`.WithGeometry` matching the type of V.
+        """
+        # Get the correct type of function space
+        shape = V.ufl_function_space().value_shape
+        if len(shape) == 0:
+            return FunctionSpace
+        elif len(shape) == 1:
+            return partial(VectorFunctionSpace, dim=shape[0])
+        else:
+            symmetry = V.ufl_element().symmetry()
+            return partial(TensorFunctionSpace, shape=shape, symmetry=symmetry)
+
+    def _get_quadrature_space(self, V: WithGeometry) -> WithGeometry:
+        """Return a FunctionSpace whose element is a quadrature element with 
+        points at the quadrature points of V's element.
+
+        Used as an intermediate space for interpolating into a space which doesn't
+        have point evaluation dofs.
+
+        Parameters
+        ----------
+        V : WithGeometry
+            A :class:`.WithGeometry` function space to build the quadrature space for.
+
+        Returns
+        -------
+        WithGeometry
+            A :class:`.WithGeometry` function space with quadrature element.
+        """  
+        V_element = V.finat_element
+        _, points = V_element.dual_basis
+        weights = numpy.full(len(points.points), numpy.nan)  # These can be any number since we never integrate
+        quad_scheme = QuadratureRule(points, weights, ref_el=V_element.cell)
+        element = FiniteElement("Quadrature", degree=V_element.degree, quad_scheme=quad_scheme)
+        return self._fs_type(V)(V.mesh(), element)
+
+    def _get_symbolic_expressions(self, target_space: WithGeometry) -> tuple[Interpolate, Interpolate]:
+        """Return symbolic ``Interpolate`` expressions for point evaluation of the `target_space`s
+        dofs in the source mesh, and the corresponding input-ordering interpolation.
 
-    def _get_symbolic_expressions(self) -> tuple[Interpolate, Interpolate]:
-        """Return the symbolic ``Interpolate`` expressions for point evaluation and
-        re-ordering into the input-ordering VertexOnlyMesh.
+        Parameters
+        ----------
+        target_space
+            The :class:`.WithGeometry` function space which we are interpolating into.
 
         Returns
         -------
@@ -462,13 +528,19 @@ def _get_symbolic_expressions(self) -> tuple[Interpolate, Interpolate]:
         Raises
         ------
         DofNotDefinedError
-            If any DoFs in the target mesh cannot be defined in the source mesh.
+            If any of the target spaces dofs cannot be defined in the source mesh.
+        ValueError
+            If the target space does not have point-evaluation dofs.
         """
         from firedrake.assemble import assemble
+        if target_space.ufl_element().mapping() != "identity":
+            raise ValueError(f"FunctionSpace {target_space} must have point-evaluation dofs.")
+
         # Immerse coordinates of target space point evaluation dofs in src_mesh
-        target_space_vec = VectorFunctionSpace(self.target_mesh, self.dest_element)
-        f_dest_node_coords = assemble(interpolate(self.target_mesh.coordinates, target_space_vec))
-        dest_node_coords = f_dest_node_coords.dat.data_ro.reshape(-1, self.target_mesh.geometric_dimension)
+        target_mesh = target_space.mesh().unique()
+        target_space_vec = VectorFunctionSpace(target_mesh, self._get_element(target_space))
+        f_dest_node_coords = assemble(interpolate(target_mesh.coordinates, target_space_vec))
+        dest_node_coords = f_dest_node_coords.dat.data_ro.reshape(-1, target_mesh.geometric_dimension)
         try:
             vom = VertexOnlyMesh(
                 self.source_mesh,
@@ -494,6 +566,7 @@ def _get_symbolic_expressions(self) -> tuple[Interpolate, Interpolate]:
             fs_type = partial(TensorFunctionSpace, shape=shape, symmetry=symmetry)
 
         # Get expression for point evaluation at the dest_node_coords
+        fs_type = self._fs_type(target_space)
         P0DG_vom = fs_type(vom, "DG", 0)
         point_eval = interpolate(self.operand, P0DG_vom)
 
@@ -512,9 +585,16 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
 
         # self.ufl_interpolate.function_space() is None in the 0-form case
         V_dest = self.ufl_interpolate.function_space() or self.target_space
-        f = tensor or Function(V_dest)
 
-        point_eval, point_eval_input_ordering = self._get_symbolic_expressions()
+        # Interpolate into intermediate quadrature space for non-identity mapped elements
+        if into_quadrature_space := V_dest.ufl_element().mapping() != "identity":
+            Q_dest = self._get_quadrature_space(V_dest)
+        else:
+            Q_dest = None
+        target_space = Q_dest or V_dest
+        f = tensor or Function(target_space)
+
+        point_eval, point_eval_input_ordering = self._get_symbolic_expressions(target_space)
         P0DG_vom_input_ordering = point_eval_input_ordering.argument_slots()[0].function_space().dual()
 
         if self.rank == 2:
@@ -523,12 +603,22 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
             # `self.point_eval_interpolate` and the permutation
             # given by `self.to_input_ordering_interpolate`.
             if self.ufl_interpolate.is_adjoint:
-                symbolic = action(point_eval, point_eval_input_ordering)
+                interp_expr = action(point_eval, point_eval_input_ordering)
             else:
-                symbolic = action(point_eval_input_ordering, point_eval)
+                interp_expr = action(point_eval_input_ordering, point_eval)
 
             def callable() -> PETSc.Mat:
-                return assemble(symbolic, mat_type=mat_type).petscmat
+                res = assemble(interp_expr, mat_type=mat_type).petscmat
+                if into_quadrature_space:
+                    if self.ufl_interpolate.is_adjoint:
+                        I = AssembledMatrix((Argument(Q_dest, 0), Argument(V_dest.dual(), 1)), None, res)
+                        return assemble(action(interpolate(TestFunction(V_dest), Q_dest))).petscmat
+                    else:
+                        I = AssembledMatrix((Argument(Q_dest.dual(), 0), Argument(V_dest, 1)), None, res)  # V_dest x Q_dest^* -> R
+                        return assemble(action(interpolate(TrialFunction(Q_dest), V_dest), I)).petscmat  # Q_dest x V_dest^* -> R
+                else:
+                    return res
+
         elif self.ufl_interpolate.is_adjoint:
             assert self.rank == 1
             # f_src is a cofunction on V_dest.dual
diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 6651eada43..3d0e1b0e35 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -70,11 +70,11 @@ def test_cross_mesh_oneform(V):
     Q_target = make_quadrature_space(V)
 
     # Interp V_source -> Q
-    I1 = interpolate(f_source, Q_target)
+    I1 = interpolate(f_source, Q_target)  # CrossMeshInterpolator
     f_quadrature = assemble(I1)
 
     # Interp Q -> V_target
-    I2 = interpolate(f_quadrature, V)
+    I2 = interpolate(f_quadrature, V)  # SameMeshInterpolator
     f_target = assemble(I2)
 
     f_direct = Function(V).interpolate(expr2)
@@ -82,6 +82,42 @@ def test_cross_mesh_oneform(V):
     assert np.allclose(f_target.dat.data_ro, f_direct.dat.data_ro)
 
 
+def test_cross_mesh_twoform(V):
+    mesh1 = UnitSquareMesh(5, 5)
+    mesh2 = V.mesh()
+    x, y = SpatialCoordinate(mesh1)
+    x1, y1 = SpatialCoordinate(mesh2)
+
+    shape = V.ufl_function_space().value_shape
+    if len(shape) == 1:
+        fs_type = partial(VectorFunctionSpace, dim=shape[0])
+        expr1 = as_vector([x, y])
+        expr2 = as_vector([x1, y1])
+    elif len(shape) == 2:
+        fs_type = partial(TensorFunctionSpace, shape=shape)
+        expr1 = as_tensor([[x, x*y], [x*y, y]])
+        expr2 = as_tensor([[x1, x1*y1], [x1*y1, y1]])
+    else:
+        raise ValueError("Unsupported target space shape")
+    
+    V_source = fs_type(mesh1, "CG", 2)
+    Q_target = make_quadrature_space(V)
+
+    I1 = interpolate(TrialFunction(V_source), Q_target)  # V_source x Q_target^* -> R
+    I2 = interpolate(TrialFunction(Q_target), V)  # Q_target x V^* -> R
+
+    I = assemble(action(I2, I1))  # V_source x V^* -> R
+
+    I_direct = assemble(interpolate(TrialFunction(V_source), V))  # V_source x V^* -> R
+
+    f_source = Function(V_source).interpolate(expr1)
+    f_direct = Function(V).interpolate(expr2)
+
+    f_interpolated = assemble(action(I_direct, f_source))
+    assert np.allclose(f_interpolated.dat.data_ro, f_direct.dat.data_ro)
+
+
+
 def test_cross_mesh_oneform_adjoint(V):
     # Can already do Lagrange -> RT adjoint
     # V^* -> Q^* -> V_target^*
@@ -119,5 +155,42 @@ def test_cross_mesh_oneform_adjoint(V):
     cofunc_V_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
     assert np.allclose(cofunc_V.dat.data_ro, cofunc_V_direct.dat.data_ro)
 
+
+def test_cross_mesh_twoform_adjoint(V):
+    # V^* -> Q^* -> V_target^*
+    mesh1 = UnitSquareMesh(2, 2)
+    x1 = SpatialCoordinate(mesh1)
+    V_target = fs_shape(V)(mesh1, "CG", 1)
+    mesh2 = V.mesh()
+    x2 = SpatialCoordinate(mesh2)
+
+    if len(V.value_shape) > 1:
+        expr = outer(x2, x2)
+        target_expr = outer(x1, x1)
+        if V.ufl_element().mapping() == "covariant contravariant Piola":
+            expr = deviatoric_expr_2x2(expr)
+            target_expr = deviatoric_expr_2x2(target_expr)
+    else:
+        expr = x2
+        target_expr = x1
+
+    oneform_V = inner(expr, TestFunction(V)) * dx
+
+    Q = make_quadrature_space(V)
+
+    I1 = interpolate(TestFunction(Q), V)  # V^* x Q -> R
+    I2 = interpolate(TestFunction(V_target), Q)  # Q^* x V_target -> R
+    I = assemble(action(I2, I1))  # V^* x V_target -> R
+
+    # I_direct = assemble(interpolate(TestFunction(V_target), V))  # V^* x V_target -> R
+    # assert I_direct.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
+
+    cofunc_V = assemble(action(I, oneform_V))
+    cofunc_V_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
+
+    assert np.allclose(cofunc_V.dat.data_ro, cofunc_V_direct.dat.data_ro)
+
+
+
 if __name__ == "__main__":
-    pytest.main([__file__ + "::test_cross_mesh_oneform_adjoint[GLS_3]"])
\ No newline at end of file
+    pytest.main([__file__ + "::test_cross_mesh_twoform[RT_2]"])
\ No newline at end of file
diff --git a/tests/firedrake/regression/test_interpolate_cross_mesh.py b/tests/firedrake/regression/test_interpolate_cross_mesh.py
index 23dfb742b8..310cb1abf4 100644
--- a/tests/firedrake/regression/test_interpolate_cross_mesh.py
+++ b/tests/firedrake/regression/test_interpolate_cross_mesh.py
@@ -743,3 +743,7 @@ def test_interpolate_cross_mesh_interval(periodic):
     f_dest = Function(V_dest).interpolate(f_src)
     x_dest, = SpatialCoordinate(m_dest)
     assert abs(assemble((f_dest - (-(x_dest - .5) ** 2)) ** 2 * dx)) < 1.e-16
+
+
+if __name__ == "__main__":
+    pytest.main([__file__ + "::test_interpolate_matrix_cross_mesh_adjoint"])
\ No newline at end of file

From b5ebf328feba89c8b8fc542bf1a7ad2787c6053d Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Wed, 21 Jan 2026 18:53:50 +0000
Subject: [PATCH 19/37] fix target_space

---
 firedrake/interpolation.py                        | 15 ++++++++-------
 .../regression/test_cross_mesh_non_lagrange.py    |  6 +++---
 2 files changed, 11 insertions(+), 10 deletions(-)

diff --git a/firedrake/interpolation.py b/firedrake/interpolation.py
index 0fb7a15f24..35221df397 100644
--- a/firedrake/interpolation.py
+++ b/firedrake/interpolation.py
@@ -585,14 +585,15 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
 
         # self.ufl_interpolate.function_space() is None in the 0-form case
         V_dest = self.ufl_interpolate.function_space() or self.target_space
+        f = tensor or Function(V_dest)
 
         # Interpolate into intermediate quadrature space for non-identity mapped elements
-        if into_quadrature_space := V_dest.ufl_element().mapping() != "identity":
+        if into_quadrature_space := self.target_space.ufl_element().mapping() != "identity":
             Q_dest = self._get_quadrature_space(V_dest)
         else:
             Q_dest = None
-        target_space = Q_dest or V_dest
-        f = tensor or Function(target_space)
+        target_space = Q_dest or self.target_space
+        
 
         point_eval, point_eval_input_ordering = self._get_symbolic_expressions(target_space)
         P0DG_vom_input_ordering = point_eval_input_ordering.argument_slots()[0].function_space().dual()
@@ -611,11 +612,11 @@ def callable() -> PETSc.Mat:
                 res = assemble(interp_expr, mat_type=mat_type).petscmat
                 if into_quadrature_space:
                     if self.ufl_interpolate.is_adjoint:
-                        I = AssembledMatrix((Argument(Q_dest, 0), Argument(V_dest.dual(), 1)), None, res)
-                        return assemble(action(interpolate(TestFunction(V_dest), Q_dest))).petscmat
+                        I = AssembledMatrix((Argument(Q_dest, 0), Argument(target_space.dual(), 1)), None, res)
+                        return assemble(action(interpolate(TestFunction(target_space), Q_dest), I)).petscmat
                     else:
-                        I = AssembledMatrix((Argument(Q_dest.dual(), 0), Argument(V_dest, 1)), None, res)  # V_dest x Q_dest^* -> R
-                        return assemble(action(interpolate(TrialFunction(Q_dest), V_dest), I)).petscmat  # Q_dest x V_dest^* -> R
+                        I = AssembledMatrix((Argument(Q_dest.dual(), 0), Argument(target_space, 1)), None, res)  # target_space x Q_dest^* -> R
+                        return assemble(action(interpolate(TrialFunction(Q_dest), target_space), I)).petscmat  # Q_dest x target_space^* -> R
                 else:
                     return res
 
diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 3d0e1b0e35..114470d83a 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -182,10 +182,10 @@ def test_cross_mesh_twoform_adjoint(V):
     I2 = interpolate(TestFunction(V_target), Q)  # Q^* x V_target -> R
     I = assemble(action(I2, I1))  # V^* x V_target -> R
 
-    # I_direct = assemble(interpolate(TestFunction(V_target), V))  # V^* x V_target -> R
-    # assert I_direct.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
+    I_direct = assemble(interpolate(TestFunction(V_target), V))  # V^* x V_target -> R
+    assert I_direct.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
 
-    cofunc_V = assemble(action(I, oneform_V))
+    cofunc_V = assemble(action(I_direct, oneform_V))
     cofunc_V_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
 
     assert np.allclose(cofunc_V.dat.data_ro, cofunc_V_direct.dat.data_ro)

From a2789dadd945ea6fd9e1fd88fb11845b13a1146d Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Wed, 21 Jan 2026 19:14:13 +0000
Subject: [PATCH 20/37] two-form assembly working

---
 firedrake/interpolation.py                                | 8 ++++----
 .../firedrake/regression/test_cross_mesh_non_lagrange.py  | 3 ++-
 2 files changed, 6 insertions(+), 5 deletions(-)

diff --git a/firedrake/interpolation.py b/firedrake/interpolation.py
index 35221df397..f7d02e7cb0 100644
--- a/firedrake/interpolation.py
+++ b/firedrake/interpolation.py
@@ -589,7 +589,7 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
 
         # Interpolate into intermediate quadrature space for non-identity mapped elements
         if into_quadrature_space := self.target_space.ufl_element().mapping() != "identity":
-            Q_dest = self._get_quadrature_space(V_dest)
+            Q_dest = self._get_quadrature_space(self.target_space)
         else:
             Q_dest = None
         target_space = Q_dest or self.target_space
@@ -613,10 +613,10 @@ def callable() -> PETSc.Mat:
                 if into_quadrature_space:
                     if self.ufl_interpolate.is_adjoint:
                         I = AssembledMatrix((Argument(Q_dest, 0), Argument(target_space.dual(), 1)), None, res)
-                        return assemble(action(interpolate(TestFunction(target_space), Q_dest), I)).petscmat
+                        return assemble(action(I, interpolate(TestFunction(Q_dest), self.target_space))).petscmat
                     else:
-                        I = AssembledMatrix((Argument(Q_dest.dual(), 0), Argument(target_space, 1)), None, res)  # target_space x Q_dest^* -> R
-                        return assemble(action(interpolate(TrialFunction(Q_dest), target_space), I)).petscmat  # Q_dest x target_space^* -> R
+                        I = AssembledMatrix((Argument(Q_dest.dual(), 0), Argument(target_space, 1)), None, res)
+                        return assemble(action(interpolate(TrialFunction(Q_dest), self.target_space), I)).petscmat
                 else:
                     return res
 
diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 114470d83a..4e9cd6f122 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -181,6 +181,7 @@ def test_cross_mesh_twoform_adjoint(V):
     I1 = interpolate(TestFunction(Q), V)  # V^* x Q -> R
     I2 = interpolate(TestFunction(V_target), Q)  # Q^* x V_target -> R
     I = assemble(action(I2, I1))  # V^* x V_target -> R
+    assert I.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
 
     I_direct = assemble(interpolate(TestFunction(V_target), V))  # V^* x V_target -> R
     assert I_direct.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
@@ -193,4 +194,4 @@ def test_cross_mesh_twoform_adjoint(V):
 
 
 if __name__ == "__main__":
-    pytest.main([__file__ + "::test_cross_mesh_twoform[RT_2]"])
\ No newline at end of file
+    pytest.main([__file__ + "::test_cross_mesh_twoform_adjoint[RT_2]"])
\ No newline at end of file

From 986ce11a5a42e39c75f37d639415234d19450209 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 26 Jan 2026 14:15:51 +0000
Subject: [PATCH 21/37] Apply suggestions from code review

---
 .../multigrid/test_adaptive_multigrid.py      | 34 ++++++++-----------
 1 file changed, 14 insertions(+), 20 deletions(-)

diff --git a/tests/firedrake/multigrid/test_adaptive_multigrid.py b/tests/firedrake/multigrid/test_adaptive_multigrid.py
index de9197f974..1235c95bd5 100644
--- a/tests/firedrake/multigrid/test_adaptive_multigrid.py
+++ b/tests/firedrake/multigrid/test_adaptive_multigrid.py
@@ -254,14 +254,12 @@ def test_mg_jacobi(amh, atm):  # pylint: disable=W0621
     """
     Test multigrid with jacobi smoothers
     """
-    V_J = FunctionSpace(amh[-1], "CG", 1)
+    V = FunctionSpace(amh[-1], "CG", 1)
     x = SpatialCoordinate(amh[-1])
-    u_ex = Function(V_J, name="u_fine_real").interpolate(
-        sin(2 * pi * x[0]) * sin(2 * pi * x[1])
-    )
-    u = Function(V_J)
-    v = TestFunction(V_J)
-    bc = DirichletBC(V_J, u_ex, "on_boundary")
+    u_ex = Function(V).interpolate(sin(2 * pi * x[0]) * sin(2 * pi * x[1]))
+    u = Function(V)
+    v = TestFunction(V)
+    bc = DirichletBC(V, u_ex, "on_boundary")
     F = inner(grad(u - u_ex), grad(v)) * dx
 
     params = {
@@ -305,10 +303,9 @@ def test_mg_patch(amh, atm, params):  # pylint: disable=W0621
                 "pc_type": "jacobi",
             },
             "mg_coarse": {
+                "mat_type": "aij",
                 "ksp_type": "preonly",
-                "pc_type": "python",
-                "pc_python_type": "firedrake.AssembledPC",
-                "assembled": {"ksp_type": "preonly", "pc_type": "lu"},
+                "pc_type": "lu",
             },
         }
     elif params == "patch":
@@ -335,10 +332,9 @@ def test_mg_patch(amh, atm, params):  # pylint: disable=W0621
                 },
             },
             "mg_coarse": {
+                "mat_type": "aij",
                 "ksp_type": "preonly",
-                "pc_type": "python",
-                "pc_python_type": "firedrake.AssembledPC",
-                "assembled": {"ksp_type": "preonly", "pc_type": "lu"},
+                "pc_type": "lu",
             },
         }
     else:
@@ -356,14 +352,12 @@ def test_mg_patch(amh, atm, params):  # pylint: disable=W0621
             "mg_coarse": {"ksp_type": "preonly", "pc_type": "lu"},
         }
 
-    V_J = FunctionSpace(amh[-1], "CG", 1)
+    V = FunctionSpace(amh[-1], "CG", 1)
     x = SpatialCoordinate(amh[-1])
-    u_ex = Function(V_J, name="u_fine_real").interpolate(
-        sin(2 * pi * x[0]) * sin(2 * pi * x[1])
-    )
-    u = Function(V_J)
-    v = TestFunction(V_J)
-    bc = DirichletBC(V_J, u_ex, "on_boundary")
+    u_ex = Function(V).interpolate(sin(2 * pi * x[0]) * sin(2 * pi * x[1]))
+    u = Function(V)
+    v = TestFunction(V)
+    bc = DirichletBC(V, u_ex, "on_boundary")
     F = inner(grad(u - u_ex), grad(v)) * dx
 
     problem = NonlinearVariationalProblem(F, u, bc)

From e1dba6bf9e843542e20ecd510c505e9e0f8d7750 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Mon, 26 Jan 2026 14:44:18 +0000
Subject: [PATCH 22/37] docs

---
 firedrake/mg/adaptive_hierarchy.py        | 44 +++++++++++++----------
 firedrake/mg/adaptive_transfer_manager.py |  2 ++
 2 files changed, 27 insertions(+), 19 deletions(-)

diff --git a/firedrake/mg/adaptive_hierarchy.py b/firedrake/mg/adaptive_hierarchy.py
index e307d24ad9..fc6c2f728a 100644
--- a/firedrake/mg/adaptive_hierarchy.py
+++ b/firedrake/mg/adaptive_hierarchy.py
@@ -1,10 +1,4 @@
-"""
-This module contains the class for the AdaptiveMeshHierarchy and
-related helper functions
-"""
-
-from collections import defaultdict
-
+from firedrake.mesh import MeshGeometry
 from firedrake.cofunction import Cofunction
 from firedrake.function import Function
 from firedrake.mg import HierarchyBase
@@ -15,29 +9,41 @@
 
 class AdaptiveMeshHierarchy(HierarchyBase):
     """
-    HierarchyBase for hierarchies of adaptively refined meshes
+    HierarchyBase for hierarchies of adaptively refined meshes.
+
+    Parameters
+    ----------
+    base_mesh
+        The coarsest mesh in the hierarchy.
+    nested: bool
+        A flag to indicate whether the meshes are nested.
+
     """
-    def __init__(self, base_mesh, refinements_per_level=1, nested=True):
+    def __init__(self, base_mesh: MeshGeometry, nested: bool = True):
         self.meshes = []
         self._meshes = []
-        self.refinements_per_level = refinements_per_level
         self.nested = nested
         self.add_mesh(base_mesh)
 
-    def add_mesh(self, mesh):
+    def add_mesh(self, mesh: MeshGeometry):
         """
-        Adds newly refined mesh into hierarchy.
+        Adds a mesh into the hierarchy.
+
+        Parameters
+        ----------
+        mesh
+            The mesh to be added to the finest level.
         """
+        level = len(self.meshes)
         self._meshes.append(mesh)
         self.meshes.append(mesh)
-        level = len(self.meshes)
-        set_level(self.meshes[-1], self, level - 1)
-        self._shared_data_cache = defaultdict(dict)
+        set_level(mesh, self, level)
 
     def adapt(self, eta: Function | Cofunction, theta: float):
         """
-        Add a refinement level to the hierarchy by local refinement
-        with a simplified variant of Dorfler marking.
+        Adds a new mesh to the hierarchy by locally refining the finest mesh
+        with a simplified variant of Dorfler marking. The finest mesh must
+        come from a netgen mesh.
 
         Parameters
         ----------
@@ -65,8 +71,8 @@ def adapt(self, eta: Function | Cofunction, theta: float):
             raise ValueError("eta must be defined on the finest mesh of the hierarchy")
 
         # Take the maximum over all processes
-        with eta.dat.vec_ro as eta_:
-            eta_max = eta_.max()[1]
+        with eta.dat.vec_ro as evec:
+            _, eta_max = evec.max()
 
         threshold = theta * eta_max
         should_refine = eta.dat.data_ro > threshold
diff --git a/firedrake/mg/adaptive_transfer_manager.py b/firedrake/mg/adaptive_transfer_manager.py
index d6ce290f5a..05b1feca4b 100644
--- a/firedrake/mg/adaptive_transfer_manager.py
+++ b/firedrake/mg/adaptive_transfer_manager.py
@@ -21,6 +21,8 @@ class AdaptiveTransferManager(TransferManager):
     TransferManager for adaptively refined mesh hierarchies
     """
     def __init__(self, *, native_transfers=None, use_averaging=True):
+        if native_transfers is not None:
+            raise NotImplementedError("Custom transfers not implemented.")
         super().__init__(native_transfers=native_transfers, use_averaging=use_averaging)
         self.cache = {}
 

From 10f8516fab868ec8acdf43868a49e42df2a5c55d Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Mon, 26 Jan 2026 16:02:09 +0000
Subject: [PATCH 23/37] forward and adjoint one-form

---
 firedrake/interpolation.py                    | 52 +++++++--------
 .../test_cross_mesh_non_lagrange.py           | 66 ++++++-------------
 .../regression/test_interpolate_cross_mesh.py | 12 ++--
 3 files changed, 51 insertions(+), 79 deletions(-)

diff --git a/firedrake/interpolation.py b/firedrake/interpolation.py
index f7d02e7cb0..244876b8d4 100644
--- a/firedrake/interpolation.py
+++ b/firedrake/interpolation.py
@@ -22,7 +22,7 @@
 from FIAT.reference_element import Point
 
 from finat.element_factory import create_element, as_fiat_cell
-from finat.ufl import TensorElement, VectorElement, MixedElement, FiniteElement, FiniteElementBase
+from finat.ufl import TensorElement, VectorElement, MixedElement, FiniteElementBase
 from finat.fiat_elements import ScalarFiatElement
 from finat.quadrature import QuadratureRule
 from finat.quadrature_element import QuadratureElement
@@ -477,7 +477,7 @@ def _fs_type(self, V: WithGeometry) -> Callable[..., WithGeometry]:
             A callable which returns a :class:`.WithGeometry` matching the type of V.
         """
         # Get the correct type of function space
-        shape = V.ufl_function_space().value_shape
+        shape = V.value_shape
         if len(shape) == 0:
             return FunctionSpace
         elif len(shape) == 1:
@@ -487,7 +487,7 @@ def _fs_type(self, V: WithGeometry) -> Callable[..., WithGeometry]:
             return partial(TensorFunctionSpace, shape=shape, symmetry=symmetry)
 
     def _get_quadrature_space(self, V: WithGeometry) -> WithGeometry:
-        """Return a FunctionSpace whose element is a quadrature element with 
+        """Return a FunctionSpace whose element is a quadrature element with
         points at the quadrature points of V's element.
 
         Used as an intermediate space for interpolating into a space which doesn't
@@ -502,13 +502,12 @@ def _get_quadrature_space(self, V: WithGeometry) -> WithGeometry:
         -------
         WithGeometry
             A :class:`.WithGeometry` function space with quadrature element.
-        """  
+        """
         V_element = V.finat_element
         _, points = V_element.dual_basis
         weights = numpy.full(len(points.points), numpy.nan)  # These can be any number since we never integrate
         quad_scheme = QuadratureRule(points, weights, ref_el=V_element.cell)
-        element = FiniteElement("Quadrature", degree=V_element.degree, quad_scheme=quad_scheme)
-        return self._fs_type(V)(V.mesh(), element)
+        return self._fs_type(V)(V.mesh(), "Quadrature", degree=V_element.degree, quad_scheme=quad_scheme)
 
     def _get_symbolic_expressions(self, target_space: WithGeometry) -> tuple[Interpolate, Interpolate]:
         """Return symbolic ``Interpolate`` expressions for point evaluation of the `target_space`s
@@ -555,16 +554,6 @@ def _get_symbolic_expressions(self, target_space: WithGeometry) -> tuple[Interpo
                                      "This may be because the target mesh covers a larger domain than the "
                                      "source mesh. To disable this error, set allow_missing_dofs=True.")
 
-        # Get the correct type of function space
-        shape = self.target_space.ufl_function_space().value_shape
-        if len(shape) == 0:
-            fs_type = FunctionSpace
-        elif len(shape) == 1:
-            fs_type = partial(VectorFunctionSpace, dim=shape[0])
-        else:
-            symmetry = self.target_space.ufl_element().symmetry()
-            fs_type = partial(TensorFunctionSpace, shape=shape, symmetry=symmetry)
-
         # Get expression for point evaluation at the dest_node_coords
         fs_type = self._fs_type(target_space)
         P0DG_vom = fs_type(vom, "DG", 0)
@@ -583,17 +572,15 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
             raise NotImplementedError("bcs not implemented for cross-mesh interpolation.")
         mat_type = mat_type or "aij"
 
-        # self.ufl_interpolate.function_space() is None in the 0-form case
-        V_dest = self.ufl_interpolate.function_space() or self.target_space
-        f = tensor or Function(V_dest)
-
         # Interpolate into intermediate quadrature space for non-identity mapped elements
         if into_quadrature_space := self.target_space.ufl_element().mapping() != "identity":
             Q_dest = self._get_quadrature_space(self.target_space)
         else:
             Q_dest = None
         target_space = Q_dest or self.target_space
-        
+
+        # self.ufl_interpolate.function_space() is None in the 0-form case
+        f = tensor or Function(target_space.dual() if self.ufl_interpolate.is_adjoint else target_space)
 
         point_eval, point_eval_input_ordering = self._get_symbolic_expressions(target_space)
         P0DG_vom_input_ordering = point_eval_input_ordering.argument_slots()[0].function_space().dual()
@@ -622,8 +609,12 @@ def callable() -> PETSc.Mat:
 
         elif self.ufl_interpolate.is_adjoint:
             assert self.rank == 1
-            # f_src is a cofunction on V_dest.dual
-            cofunc = self.dual_arg
+
+            if into_quadrature_space:
+                cofunc = assemble(interpolate(TestFunction(target_space), self.dual_arg))
+            else:
+                cofunc = self.dual_arg
+
             assert isinstance(cofunc, Cofunction)
 
             # Our first adjoint operation is to assign the dat values to a
@@ -638,8 +629,9 @@ def callable() -> PETSc.Mat:
             # and all points from the input ordering VOM are in the original.
             def callable() -> Cofunction:
                 f_src_at_src_node_coords = assemble(action(point_eval_input_ordering, f_input_ordering))
-                assemble(action(point_eval, f_src_at_src_node_coords), tensor=f)
-                return f
+                f_target = Cofunction(point_eval.function_space()) if into_quadrature_space else f
+                assemble(action(point_eval, f_src_at_src_node_coords), tensor=f_target)
+                return f_target
         else:
             assert self.rank in {0, 1}
             # We create the input-ordering Function before interpolating so we can
@@ -663,12 +655,18 @@ def callable() -> Function | Number:
                 else:
                     f.dat.data_wo[:] = f_point_eval_input_ordering.dat.data_ro[:]
 
+                if into_quadrature_space:
+                    f_target = Function(self.target_space)
+                    assemble(interpolate(f, self.target_space), tensor=f_target)
+                else:
+                    f_target = f
+
                 if self.rank == 0:
                     # We take the action of the dual_arg on the interpolated function
                     assert isinstance(self.dual_arg, Cofunction)
-                    return assemble(action(self.dual_arg, f))
+                    return assemble(action(self.dual_arg, f_target))
                 else:
-                    return f
+                    return f_target
         return callable
 
     @property
diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 4e9cd6f122..837d3ce1eb 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -17,9 +17,10 @@ def fs_shape(V):
         raise ValueError("Invalid function space shape")
     return fs_type
 
+
 def make_quadrature_space(V):
     """Builds a quadrature space on the target mesh.
-    
+
     This space has point evaluation dofs at the quadrature PointSet
     of the target space's element
 
@@ -33,17 +34,15 @@ def make_quadrature_space(V):
 
 
 @pytest.fixture(params=[("RT", 2), ("RT", 3), ("RT", 4), ("BDM", 1), ("BDM", 2), ("BDM", 3),
-                        ("BDFM", 2), ("HHJ", 2),("N1curl", 2), ("N1curl", 3), ("N1curl", 4),
+                        ("BDFM", 2), ("HHJ", 2), ("N1curl", 2), ("N1curl", 3), ("N1curl", 4),
                         ("N2curl", 1), ("N2curl", 2), ("N2curl", 3), ("GLS", 2), ("GLS", 3),
                         ("GLS", 4), ("GLS2", 1), ("GLS2", 2), ("GLS2", 3)],
-                        ids=lambda x: f"{x[0]}_{x[1]}")
+                ids=lambda x: f"{x[0]}_{x[1]}")
 def V(request):
     element, degree = request.param
     mesh = UnitSquareMesh(8, 8)
     return FunctionSpace(mesh, element, degree)
 
-# V_source -> Q -> V_target
-# V_target^* -> Q^* -> V_source^*
 
 def test_cross_mesh_oneform(V):
     mesh1 = UnitSquareMesh(5, 5)
@@ -62,21 +61,11 @@ def test_cross_mesh_oneform(V):
         expr2 = as_tensor([[x1, x1*y1], [x1*y1, y1]])
     else:
         raise ValueError("Unsupported target space shape")
-    
+
     V_source = fs_type(mesh1, "CG", 2)
     f_source = Function(V_source).interpolate(expr1)
 
-    # Make intermediate Quadrature space on target mesh
-    Q_target = make_quadrature_space(V)
-
-    # Interp V_source -> Q
-    I1 = interpolate(f_source, Q_target)  # CrossMeshInterpolator
-    f_quadrature = assemble(I1)
-
-    # Interp Q -> V_target
-    I2 = interpolate(f_quadrature, V)  # SameMeshInterpolator
-    f_target = assemble(I2)
-
+    f_target = assemble(interpolate(f_source, V))
     f_direct = Function(V).interpolate(expr2)
 
     assert np.allclose(f_target.dat.data_ro, f_direct.dat.data_ro)
@@ -99,25 +88,18 @@ def test_cross_mesh_twoform(V):
         expr2 = as_tensor([[x1, x1*y1], [x1*y1, y1]])
     else:
         raise ValueError("Unsupported target space shape")
-    
-    V_source = fs_type(mesh1, "CG", 2)
-    Q_target = make_quadrature_space(V)
 
-    I1 = interpolate(TrialFunction(V_source), Q_target)  # V_source x Q_target^* -> R
-    I2 = interpolate(TrialFunction(Q_target), V)  # Q_target x V^* -> R
-
-    I = assemble(action(I2, I1))  # V_source x V^* -> R
+    V_source = fs_type(mesh1, "CG", 2)
 
-    I_direct = assemble(interpolate(TrialFunction(V_source), V))  # V_source x V^* -> R
+    I = assemble(interpolate(TrialFunction(V_source), V))  # V_source x V^* -> R
 
     f_source = Function(V_source).interpolate(expr1)
     f_direct = Function(V).interpolate(expr2)
 
-    f_interpolated = assemble(action(I_direct, f_source))
+    f_interpolated = assemble(action(I, f_source))
     assert np.allclose(f_interpolated.dat.data_ro, f_direct.dat.data_ro)
 
 
-
 def test_cross_mesh_oneform_adjoint(V):
     # Can already do Lagrange -> RT adjoint
     # V^* -> Q^* -> V_target^*
@@ -140,17 +122,17 @@ def test_cross_mesh_oneform_adjoint(V):
 
     oneform_V = inner(expr, TestFunction(V)) * dx
 
-    Q_target = make_quadrature_space(V)
+    # Q_target = make_quadrature_space(V)
 
-    # Interp V^* -> Q^*
-    I1_adj = interpolate(TestFunction(Q_target), oneform_V)
-    cofunc_Q = assemble(I1_adj)
+    # # Interp V^* -> Q^*
+    # I1_adj = interpolate(TestFunction(Q_target), oneform_V)  # SameMesh
+    # cofunc_Q = assemble(I1_adj)
 
-    # Interp Q^* -> V_target^*
-    I2_adj = interpolate(TestFunction(V_target), cofunc_Q)
-    cofunc_V = assemble(I2_adj)
+    # # Interp Q^* -> V_target^*
+    # I2_adj = interpolate(TestFunction(V_target), cofunc_Q)  # CrossMesh
+    # cofunc_V = assemble(I2_adj)
 
-    # cofunc_V = assemble(interpolate(TestFunction(V_target), oneform_target))  # V^* -> V_target^*
+    cofunc_V = assemble(interpolate(TestFunction(V_target), oneform_V))  # V^* -> V_target^*
 
     cofunc_V_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
     assert np.allclose(cofunc_V.dat.data_ro, cofunc_V_direct.dat.data_ro)
@@ -176,22 +158,14 @@ def test_cross_mesh_twoform_adjoint(V):
 
     oneform_V = inner(expr, TestFunction(V)) * dx
 
-    Q = make_quadrature_space(V)
-
-    I1 = interpolate(TestFunction(Q), V)  # V^* x Q -> R
-    I2 = interpolate(TestFunction(V_target), Q)  # Q^* x V_target -> R
-    I = assemble(action(I2, I1))  # V^* x V_target -> R
+    I = assemble(interpolate(TestFunction(V_target), V))  # V^* x V_target -> R
     assert I.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
 
-    I_direct = assemble(interpolate(TestFunction(V_target), V))  # V^* x V_target -> R
-    assert I_direct.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
-
-    cofunc_V = assemble(action(I_direct, oneform_V))
+    cofunc_V = assemble(action(I, oneform_V))
     cofunc_V_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
 
     assert np.allclose(cofunc_V.dat.data_ro, cofunc_V_direct.dat.data_ro)
 
 
-
 if __name__ == "__main__":
-    pytest.main([__file__ + "::test_cross_mesh_twoform_adjoint[RT_2]"])
\ No newline at end of file
+    pytest.main([__file__ + "::test_cross_mesh_oneform_adjoint[RT_2]"])
diff --git a/tests/firedrake/regression/test_interpolate_cross_mesh.py b/tests/firedrake/regression/test_interpolate_cross_mesh.py
index 310cb1abf4..f02f3b7c1c 100644
--- a/tests/firedrake/regression/test_interpolate_cross_mesh.py
+++ b/tests/firedrake/regression/test_interpolate_cross_mesh.py
@@ -1,10 +1,10 @@
 from firedrake import *
 from firedrake.petsc import DEFAULT_PARTITIONER
 from firedrake.ufl_expr import extract_unique_domain
+from firedrake.mesh import plex_from_cell_list, Mesh
 import numpy as np
 import pytest
 from ufl import product
-import subprocess
 
 
 def allgather(comm, coords):
@@ -606,8 +606,8 @@ def test_line_integral():
     # Create a 1D line mesh in 2D from (0, 0) to (1, 1) with 1 cell
     cells = np.asarray([[0, 1]])
     vertex_coords = np.asarray([[0.0, 0.0], [1.0, 1.0]])
-    plex = mesh.plex_from_cell_list(1, cells, vertex_coords, comm=m.comm)
-    line = mesh.Mesh(plex, dim=2)
+    plex = plex_from_cell_list(1, cells, vertex_coords, comm=m.comm)
+    line = Mesh(plex, dim=2)
     x, y = SpatialCoordinate(line)
     V_line = FunctionSpace(line, "CG", 2)
     f_line = Function(V_line).interpolate(x * y)
@@ -616,8 +616,8 @@ def test_line_integral():
     # Create a 1D line around the unit square (2D) with 4 cells
     cells = np.asarray([[0, 1], [1, 2], [2, 3], [3, 0]])
     vertex_coords = np.asarray([[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [0.0, 1.0]])
-    plex = mesh.plex_from_cell_list(1, cells, vertex_coords, comm=m.comm)
-    line_square = mesh.Mesh(plex, dim=2)
+    plex = plex_from_cell_list(1, cells, vertex_coords, comm=m.comm)
+    line_square = Mesh(plex, dim=2)
     x, y = SpatialCoordinate(line_square)
     V_line_square = FunctionSpace(line_square, "CG", 2)
     f_line_square = Function(V_line_square).interpolate(x * y)
@@ -746,4 +746,4 @@ def test_interpolate_cross_mesh_interval(periodic):
 
 
 if __name__ == "__main__":
-    pytest.main([__file__ + "::test_interpolate_matrix_cross_mesh_adjoint"])
\ No newline at end of file
+    pytest.main([__file__ + "::test_interpolate_matrix_cross_mesh_adjoint"])

From fc17045041a425313eab470065ce4b4c64656361 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Mon, 26 Jan 2026 16:19:35 +0000
Subject: [PATCH 24/37] fixes

---
 firedrake/interpolation.py                    | 35 ++++++++++---------
 .../test_cross_mesh_non_lagrange.py           | 21 +++++------
 2 files changed, 29 insertions(+), 27 deletions(-)

diff --git a/firedrake/interpolation.py b/firedrake/interpolation.py
index 244876b8d4..8bef75f1d9 100644
--- a/firedrake/interpolation.py
+++ b/firedrake/interpolation.py
@@ -580,7 +580,7 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
         target_space = Q_dest or self.target_space
 
         # self.ufl_interpolate.function_space() is None in the 0-form case
-        f = tensor or Function(target_space.dual() if self.ufl_interpolate.is_adjoint else target_space)
+        f = tensor or Function(self.ufl_interpolate.function_space() or self.target_space)
 
         point_eval, point_eval_input_ordering = self._get_symbolic_expressions(target_space)
         P0DG_vom_input_ordering = point_eval_input_ordering.argument_slots()[0].function_space().dual()
@@ -610,26 +610,27 @@ def callable() -> PETSc.Mat:
         elif self.ufl_interpolate.is_adjoint:
             assert self.rank == 1
 
-            if into_quadrature_space:
-                cofunc = assemble(interpolate(TestFunction(target_space), self.dual_arg))
-            else:
-                cofunc = self.dual_arg
+            def callable() -> Cofunction:
+                if into_quadrature_space:
+                    cofunc = assemble(interpolate(TestFunction(target_space), self.dual_arg))
+                    f_target = Cofunction(point_eval.function_space())
+                else:
+                    cofunc = self.dual_arg
+                    f_target = f
 
-            assert isinstance(cofunc, Cofunction)
+                assert isinstance(cofunc, Cofunction)
 
-            # Our first adjoint operation is to assign the dat values to a
-            # P0DG cofunction on our input ordering VOM.
-            f_input_ordering = Cofunction(P0DG_vom_input_ordering.dual())
-            f_input_ordering.dat.data_wo[:] = cofunc.dat.data_ro[:]
+                # Our first adjoint operation is to assign the dat values to a
+                # P0DG cofunction on our input ordering VOM.
+                f_input_ordering = Cofunction(P0DG_vom_input_ordering.dual())
+                f_input_ordering.dat.data_wo[:] = cofunc.dat.data_ro[:]
 
-            # The rest of the adjoint interpolation is the composition
-            # of the adjoint interpolators in the reverse direction.
-            # We don't worry about skipping over missing points here
-            # because we're going from the input ordering VOM to the original VOM
-            # and all points from the input ordering VOM are in the original.
-            def callable() -> Cofunction:
+                # The rest of the adjoint interpolation is the composition
+                # of the adjoint interpolators in the reverse direction.
+                # We don't worry about skipping over missing points here
+                # because we're going from the input ordering VOM to the original VOM
+                # and all points from the input ordering VOM are in the original.
                 f_src_at_src_node_coords = assemble(action(point_eval_input_ordering, f_input_ordering))
-                f_target = Cofunction(point_eval.function_space()) if into_quadrature_space else f
                 assemble(action(point_eval, f_src_at_src_node_coords), tensor=f_target)
                 return f_target
         else:
diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 837d3ce1eb..d5155a7b40 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -122,20 +122,21 @@ def test_cross_mesh_oneform_adjoint(V):
 
     oneform_V = inner(expr, TestFunction(V)) * dx
 
-    # Q_target = make_quadrature_space(V)
+    Q_target = make_quadrature_space(V)
 
-    # # Interp V^* -> Q^*
-    # I1_adj = interpolate(TestFunction(Q_target), oneform_V)  # SameMesh
-    # cofunc_Q = assemble(I1_adj)
+    # Interp V^* -> Q^*
+    I1_adj = interpolate(TestFunction(Q_target), oneform_V)  # SameMesh
+    cofunc_Q = assemble(I1_adj)
 
-    # # Interp Q^* -> V_target^*
-    # I2_adj = interpolate(TestFunction(V_target), cofunc_Q)  # CrossMesh
-    # cofunc_V = assemble(I2_adj)
+    # Interp Q^* -> V_target^*
+    I2_adj = interpolate(TestFunction(V_target), cofunc_Q)  # CrossMesh
+    cofunc_Vtarget_manual = assemble(I2_adj)
 
-    cofunc_V = assemble(interpolate(TestFunction(V_target), oneform_V))  # V^* -> V_target^*
+    cofunc_Vtarget = assemble(interpolate(TestFunction(V_target), oneform_V))  # V^* -> V_target^*
+    assert np.allclose(cofunc_Vtarget_manual.dat.data_ro, cofunc_Vtarget.dat.data_ro)
 
-    cofunc_V_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
-    assert np.allclose(cofunc_V.dat.data_ro, cofunc_V_direct.dat.data_ro)
+    cofunc_Vtarget_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
+    assert np.allclose(cofunc_Vtarget.dat.data_ro, cofunc_Vtarget_direct.dat.data_ro)
 
 
 def test_cross_mesh_twoform_adjoint(V):

From cb92d817dfcf10d2be70f704084275adb9c263ad Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Mon, 26 Jan 2026 23:07:28 +0000
Subject: [PATCH 25/37] working

remove
---
 firedrake/interpolation.py                    |   8 +-
 .../test_cross_mesh_non_lagrange.py           | 152 ++++++++----------
 .../regression/test_interpolate_cross_mesh.py |  18 +--
 3 files changed, 78 insertions(+), 100 deletions(-)

diff --git a/firedrake/interpolation.py b/firedrake/interpolation.py
index 8bef75f1d9..72776fffb7 100644
--- a/firedrake/interpolation.py
+++ b/firedrake/interpolation.py
@@ -579,8 +579,12 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
             Q_dest = None
         target_space = Q_dest or self.target_space
 
-        # self.ufl_interpolate.function_space() is None in the 0-form case
-        f = tensor or Function(self.ufl_interpolate.function_space() or self.target_space)
+        if into_quadrature_space and not self.ufl_interpolate.is_adjoint:
+            f = Function(target_space)
+        else:
+            # self.ufl_interpolate.function_space() is None in the 0-form case
+            f = Function(self.ufl_interpolate.function_space() or target_space)
+        # f = tensor or Function(target_space.dual() if self.ufl_interpolate.is_adjoint else target_space)
 
         point_eval, point_eval_input_ordering = self._get_symbolic_expressions(target_space)
         P0DG_vom_input_ordering = point_eval_input_ordering.argument_slots()[0].function_space().dual()
diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index d5155a7b40..4796cb2452 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -4,8 +4,6 @@
 from finat.quadrature import QuadratureRule
 from functools import partial
 
-from ufl.compound_expressions import deviatoric_expr_2x2
-
 
 def fs_shape(V):
     shape = V.ufl_function_space().value_shape
@@ -44,34 +42,8 @@ def V(request):
     return FunctionSpace(mesh, element, degree)
 
 
-def test_cross_mesh_oneform(V):
-    mesh1 = UnitSquareMesh(5, 5)
-    mesh2 = V.mesh()
-    x, y = SpatialCoordinate(mesh1)
-    x1, y1 = SpatialCoordinate(mesh2)
-
-    shape = V.ufl_function_space().value_shape
-    if len(shape) == 1:
-        fs_type = partial(VectorFunctionSpace, dim=shape[0])
-        expr1 = as_vector([x, y])
-        expr2 = as_vector([x1, y1])
-    elif len(shape) == 2:
-        fs_type = partial(TensorFunctionSpace, shape=shape)
-        expr1 = as_tensor([[x, x*y], [x*y, y]])
-        expr2 = as_tensor([[x1, x1*y1], [x1*y1, y1]])
-    else:
-        raise ValueError("Unsupported target space shape")
-
-    V_source = fs_type(mesh1, "CG", 2)
-    f_source = Function(V_source).interpolate(expr1)
-
-    f_target = assemble(interpolate(f_source, V))
-    f_direct = Function(V).interpolate(expr2)
-
-    assert np.allclose(f_target.dat.data_ro, f_direct.dat.data_ro)
-
-
-def test_cross_mesh_twoform(V):
+@pytest.mark.parametrize("rank", [1, 2])
+def test_cross_mesh(V, rank):
     mesh1 = UnitSquareMesh(5, 5)
     mesh2 = V.mesh()
     x, y = SpatialCoordinate(mesh1)
@@ -90,17 +62,42 @@ def test_cross_mesh_twoform(V):
         raise ValueError("Unsupported target space shape")
 
     V_source = fs_type(mesh1, "CG", 2)
-
-    I = assemble(interpolate(TrialFunction(V_source), V))  # V_source x V^* -> R
-
     f_source = Function(V_source).interpolate(expr1)
     f_direct = Function(V).interpolate(expr2)
 
-    f_interpolated = assemble(action(I, f_source))
-    assert np.allclose(f_interpolated.dat.data_ro, f_direct.dat.data_ro)
-
-
-def test_cross_mesh_oneform_adjoint(V):
+    Q = make_quadrature_space(V)
+
+    if rank == 2:
+        # Assemble the operator
+        I1 = interpolate(TrialFunction(V_source), Q)  # V_source x Q_target^* -> R
+        I2 = interpolate(TrialFunction(Q), V)  # Q_target x V^* -> R
+        I_manual = assemble(action(I2, I1))  # V_source x V^* -> R
+        assert I_manual.arguments() == (TestFunction(V.dual()), TrialFunction(V_source))
+        # Direct assembly
+        I_direct = assemble(interpolate(TrialFunction(V_source), V))  # V_source
+        assert I_direct.arguments() == (TestFunction(V.dual()), TrialFunction(V_source))
+
+        f_interpolated_manual = assemble(action(I_manual, f_source))
+        assert np.allclose(f_interpolated_manual.dat.data_ro, f_direct.dat.data_ro)
+        f_interpolated_direct = assemble(action(I_direct, f_source))
+        assert np.allclose(f_interpolated_direct.dat.data_ro, f_direct.dat.data_ro)
+    elif rank == 1:
+        # Interp V_source -> Q
+        I1 = interpolate(f_source, Q)  # SameMesh
+        f_quadrature = assemble(I1)
+        # Interp Q -> V
+        I2 = interpolate(f_quadrature, V)  # CrossMesh
+        f_interpolated_manual = assemble(I2)
+        assert f_interpolated_manual.function_space() == V
+        assert np.allclose(f_interpolated_manual.dat.data_ro, f_direct.dat.data_ro)
+
+        f_interpolated_direct = assemble(interpolate(f_source, V))
+        assert f_interpolated_direct.function_space() == V
+        assert np.allclose(f_interpolated_direct.dat.data_ro, f_direct.dat.data_ro)
+
+
+@pytest.mark.parametrize("rank", [1, 2])
+def test_cross_mesh_adjoint(V, rank):
     # Can already do Lagrange -> RT adjoint
     # V^* -> Q^* -> V_target^*
     mesh1 = UnitSquareMesh(2, 2)
@@ -114,59 +111,40 @@ def test_cross_mesh_oneform_adjoint(V):
         expr = outer(x2, x2)
         target_expr = outer(x1, x1)
         if V.ufl_element().mapping() == "covariant contravariant Piola":
-            expr = deviatoric_expr_2x2(expr)
-            target_expr = deviatoric_expr_2x2(target_expr)
+            expr = dev(expr)
+            target_expr = dev(target_expr)
     else:
         expr = x2
         target_expr = x1
 
     oneform_V = inner(expr, TestFunction(V)) * dx
-
-    Q_target = make_quadrature_space(V)
-
-    # Interp V^* -> Q^*
-    I1_adj = interpolate(TestFunction(Q_target), oneform_V)  # SameMesh
-    cofunc_Q = assemble(I1_adj)
-
-    # Interp Q^* -> V_target^*
-    I2_adj = interpolate(TestFunction(V_target), cofunc_Q)  # CrossMesh
-    cofunc_Vtarget_manual = assemble(I2_adj)
-
-    cofunc_Vtarget = assemble(interpolate(TestFunction(V_target), oneform_V))  # V^* -> V_target^*
-    assert np.allclose(cofunc_Vtarget_manual.dat.data_ro, cofunc_Vtarget.dat.data_ro)
-
     cofunc_Vtarget_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
-    assert np.allclose(cofunc_Vtarget.dat.data_ro, cofunc_Vtarget_direct.dat.data_ro)
-
-
-def test_cross_mesh_twoform_adjoint(V):
-    # V^* -> Q^* -> V_target^*
-    mesh1 = UnitSquareMesh(2, 2)
-    x1 = SpatialCoordinate(mesh1)
-    V_target = fs_shape(V)(mesh1, "CG", 1)
-    mesh2 = V.mesh()
-    x2 = SpatialCoordinate(mesh2)
-
-    if len(V.value_shape) > 1:
-        expr = outer(x2, x2)
-        target_expr = outer(x1, x1)
-        if V.ufl_element().mapping() == "covariant contravariant Piola":
-            expr = deviatoric_expr_2x2(expr)
-            target_expr = deviatoric_expr_2x2(target_expr)
-    else:
-        expr = x2
-        target_expr = x1
-
-    oneform_V = inner(expr, TestFunction(V)) * dx
-
-    I = assemble(interpolate(TestFunction(V_target), V))  # V^* x V_target -> R
-    assert I.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
-
-    cofunc_V = assemble(action(I, oneform_V))
-    cofunc_V_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
-
-    assert np.allclose(cofunc_V.dat.data_ro, cofunc_V_direct.dat.data_ro)
-
 
-if __name__ == "__main__":
-    pytest.main([__file__ + "::test_cross_mesh_oneform_adjoint[RT_2]"])
+    Q = make_quadrature_space(V)
+
+    if rank == 2:
+        # Assemble the operator
+        I1 = interpolate(TestFunction(Q), V)  # V^* x Q -> R
+        I2 = interpolate(TestFunction(V_target), Q)  # Q^* x V_target -> R
+        I_manual = assemble(action(I2, I1))  # V^* x V_target -> R
+        assert I_manual.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
+        # Direct assembly
+        I_direct = assemble(interpolate(TestFunction(V_target), V))  # V^* x V_target -> R
+        assert I_direct.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
+
+        cofunc_Vtarget_manual = assemble(action(I_manual, oneform_V))
+        assert np.allclose(cofunc_Vtarget_manual.dat.data_ro, cofunc_Vtarget_direct.dat.data_ro)
+        cofunc_Vtarget = assemble(action(I_direct, oneform_V))
+        assert np.allclose(cofunc_Vtarget.dat.data_ro, cofunc_Vtarget_direct.dat.data_ro)
+    elif rank == 1:
+        # Interp V^* -> Q^*
+        I1_adj = interpolate(TestFunction(Q), oneform_V)  # SameMesh
+        cofunc_Q = assemble(I1_adj)
+
+        # Interp Q^* -> V_target^*
+        I2_adj = interpolate(TestFunction(V_target), cofunc_Q)  # CrossMesh
+        cofunc_Vtarget_manual = assemble(I2_adj)
+        assert np.allclose(cofunc_Vtarget_manual.dat.data_ro, cofunc_Vtarget_direct.dat.data_ro)
+
+        cofunc_Vtarget = assemble(interpolate(TestFunction(V_target), oneform_V))  # V^* -> V_target^*
+        assert np.allclose(cofunc_Vtarget.dat.data_ro, cofunc_Vtarget_direct.dat.data_ro)
diff --git a/tests/firedrake/regression/test_interpolate_cross_mesh.py b/tests/firedrake/regression/test_interpolate_cross_mesh.py
index f02f3b7c1c..23fe046485 100644
--- a/tests/firedrake/regression/test_interpolate_cross_mesh.py
+++ b/tests/firedrake/regression/test_interpolate_cross_mesh.py
@@ -15,9 +15,9 @@ def allgather(comm, coords):
     return coords
 
 
-def unitsquaresetup():
+def unitsquaresetup(dest_quad=True):
     m_src = UnitSquareMesh(2, 3)
-    m_dest = UnitSquareMesh(3, 5, quadrilateral=True)
+    m_dest = UnitSquareMesh(3, 5, quadrilateral=dest_quad)
     coords = np.array(
         [[0.56, 0.6], [0.1, 0.9], [0.9, 0.1], [0.9, 0.9], [0.726, 0.6584]]
     )  # fairly arbitrary
@@ -38,7 +38,7 @@ def make_high_order(m_low_order, degree):
 
 @pytest.fixture(
     params=[
-        "unitsquare",
+        "unitsquare_RT_N1curl_destination",
         "circlemanifold",
         "circlemanifold_to_high_order",
         "unitsquare_from_high_order",
@@ -47,7 +47,7 @@ def make_high_order(m_low_order, degree):
         "unitsquare_vfs",
         "unitsquare_tfs",
         "unitsquare_N1curl_source",
-        "unitsquare_SminusDiv_destination",
+        "unitsquare_RT_N1curl_destination",
         "unitsquare_Regge_source",
         # This test fails in complex mode
         pytest.param("spheresphere", marks=pytest.mark.skipcomplex),
@@ -179,14 +179,14 @@ def parameters(request):
         V_src = FunctionSpace(m_src, "N1curl", 2)  # Not point evaluation nodes
         V_dest = VectorFunctionSpace(m_dest, "CG", 4)
         V_dest_2 = VectorFunctionSpace(m_dest, "DQ", 2)
-    elif request.param == "unitsquare_SminusDiv_destination":
-        m_src, m_dest, coords = unitsquaresetup()
+    elif request.param == "unitsquare_RT_N1curl_destination":
+        m_src, m_dest, coords = unitsquaresetup(dest_quad=False)
         expr_src = 2 * SpatialCoordinate(m_src)
         expr_dest = 2 * SpatialCoordinate(m_dest)
         expected = 2 * coords
         V_src = VectorFunctionSpace(m_src, "CG", 2)
         V_dest = FunctionSpace(m_dest, "RT", 2)  # Not point evaluation nodes
-        V_dest_2 = FunctionSpace(m_dest, "N1Curl", 2)  # Not point evaluation nodes
+        V_dest_2 = FunctionSpace(m_dest, "N1curl", 2)  # Not point evaluation nodes
     elif request.param == "unitsquare_Regge_source":
         m_src, m_dest, coords = unitsquaresetup()
         expr_src = outer(SpatialCoordinate(m_src), SpatialCoordinate(m_src))
@@ -743,7 +743,3 @@ def test_interpolate_cross_mesh_interval(periodic):
     f_dest = Function(V_dest).interpolate(f_src)
     x_dest, = SpatialCoordinate(m_dest)
     assert abs(assemble((f_dest - (-(x_dest - .5) ** 2)) ** 2 * dx)) < 1.e-16
-
-
-if __name__ == "__main__":
-    pytest.main([__file__ + "::test_interpolate_matrix_cross_mesh_adjoint"])

From cf1d7475df2e27af6a3055fb5f9bc07c0d408a68 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Tue, 27 Jan 2026 13:01:49 +0000
Subject: [PATCH 26/37] add zero-form test

---
 .../regression/test_cross_mesh_non_lagrange.py        | 11 +++++++++--
 1 file changed, 9 insertions(+), 2 deletions(-)

diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 4796cb2452..85485fd3a0 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -96,7 +96,7 @@ def test_cross_mesh(V, rank):
         assert np.allclose(f_interpolated_direct.dat.data_ro, f_direct.dat.data_ro)
 
 
-@pytest.mark.parametrize("rank", [1, 2])
+@pytest.mark.parametrize("rank", [0, 1, 2])
 def test_cross_mesh_adjoint(V, rank):
     # Can already do Lagrange -> RT adjoint
     # V^* -> Q^* -> V_target^*
@@ -117,7 +117,7 @@ def test_cross_mesh_adjoint(V, rank):
         expr = x2
         target_expr = x1
 
-    oneform_V = inner(expr, TestFunction(V)) * dx
+    oneform_V = inner(expr, TestFunction(V)) * dx  # V^*
     cofunc_Vtarget_direct = assemble(inner(target_expr, TestFunction(V_target)) * dx)
 
     Q = make_quadrature_space(V)
@@ -148,3 +148,10 @@ def test_cross_mesh_adjoint(V, rank):
 
         cofunc_Vtarget = assemble(interpolate(TestFunction(V_target), oneform_V))  # V^* -> V_target^*
         assert np.allclose(cofunc_Vtarget.dat.data_ro, cofunc_Vtarget_direct.dat.data_ro)
+    elif rank == 0:
+        res = assemble(interpolate(target_expr, oneform_V))
+        actual = assemble(inner(expr, expr) * dx)
+        assert np.isclose(res, actual)
+
+if __name__ == "__main__":
+    pytest.main([__file__ + "::test_cross_mesh_adjoint[RT_2-1]"])

From 170bcb4e6fed9b9433cd0285826b9816e335365d Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Tue, 27 Jan 2026 13:57:22 +0000
Subject: [PATCH 27/37] fix tests

---
 firedrake/assemble.py                         |  2 +-
 .../test_cross_mesh_non_lagrange.py           | 30 +++++++++++++++----
 2 files changed, 25 insertions(+), 7 deletions(-)

diff --git a/firedrake/assemble.py b/firedrake/assemble.py
index 95fa61418c..c0590bd9d3 100644
--- a/firedrake/assemble.py
+++ b/firedrake/assemble.py
@@ -873,7 +873,7 @@ def restructure_base_form(expr, visited=None):
         # If F: V3 x V2 -> R, then
         # Interpolate(TestFunction(V1), F) <=> Action(Interpolate(TestFunction(V1), TrialFunction(V2.dual())), F).
         # The result is a two-form V3 x V1 -> R.
-        if isinstance(expr, ufl.Interpolate) and isinstance(expr.argument_slots()[0], ufl.form.Form):
+        if isinstance(expr, ufl.Interpolate) and isinstance(expr.argument_slots()[0], ufl.form.Form) and len(expr.argument_slots()[0].arguments()) == 2:
             form, operand = expr.argument_slots()
             vstar = firedrake.Argument(form.arguments()[0].function_space().dual(), 1)
             expr = expr._ufl_expr_reconstruct_(operand, v=vstar)
diff --git a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
index 85485fd3a0..5153e36dee 100644
--- a/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
+++ b/tests/firedrake/regression/test_cross_mesh_non_lagrange.py
@@ -5,6 +5,13 @@
 from functools import partial
 
 
+def mat_equals(a, b) -> bool:
+    """Check that two Matrices are equal."""
+    a = a.petscmat.copy()
+    a.axpy(-1.0, b.petscmat)
+    return a.norm(norm_type=PETSc.NormType.NORM_FROBENIUS) < 1e-14
+
+
 def fs_shape(V):
     shape = V.ufl_function_space().value_shape
     if len(shape) == 1:
@@ -31,9 +38,9 @@ def make_quadrature_space(V):
     return fs_shape(V)(V.mesh(), element)
 
 
-@pytest.fixture(params=[("RT", 2), ("RT", 3), ("RT", 4), ("BDM", 1), ("BDM", 2), ("BDM", 3),
-                        ("BDFM", 2), ("HHJ", 2), ("N1curl", 2), ("N1curl", 3), ("N1curl", 4),
-                        ("N2curl", 1), ("N2curl", 2), ("N2curl", 3), ("GLS", 2), ("GLS", 3),
+@pytest.fixture(params=[("RT", 1), ("RT", 2), ("RT", 3), ("RT", 4), ("BDM", 1), ("BDM", 2), ("BDM", 3),
+                        ("BDFM", 2), ("HHJ", 2), ("N1curl", 1), ("N1curl", 2), ("N1curl", 3), ("N1curl", 4),
+                        ("N2curl", 1), ("N2curl", 2), ("N2curl", 3), ("GLS", 1), ("GLS", 2), ("GLS", 3),
                         ("GLS", 4), ("GLS2", 1), ("GLS2", 2), ("GLS2", 3)],
                 ids=lambda x: f"{x[0]}_{x[1]}")
 def V(request):
@@ -42,6 +49,7 @@ def V(request):
     return FunctionSpace(mesh, element, degree)
 
 
+@pytest.mark.parallel([1, 3])
 @pytest.mark.parametrize("rank", [1, 2])
 def test_cross_mesh(V, rank):
     mesh1 = UnitSquareMesh(5, 5)
@@ -76,6 +84,7 @@ def test_cross_mesh(V, rank):
         # Direct assembly
         I_direct = assemble(interpolate(TrialFunction(V_source), V))  # V_source
         assert I_direct.arguments() == (TestFunction(V.dual()), TrialFunction(V_source))
+        assert mat_equals(I_manual, I_direct)
 
         f_interpolated_manual = assemble(action(I_manual, f_source))
         assert np.allclose(f_interpolated_manual.dat.data_ro, f_direct.dat.data_ro)
@@ -96,10 +105,19 @@ def test_cross_mesh(V, rank):
         assert np.allclose(f_interpolated_direct.dat.data_ro, f_direct.dat.data_ro)
 
 
+@pytest.mark.parallel([1, 3])
 @pytest.mark.parametrize("rank", [0, 1, 2])
 def test_cross_mesh_adjoint(V, rank):
     # Can already do Lagrange -> RT adjoint
     # V^* -> Q^* -> V_target^*
+    name = V.ufl_element()._short_name
+    deg = V.ufl_element().degree()
+    if name in ["N1curl", "GLS", "RT"] and deg == 1:
+        if name == "RT" and rank == 0:
+            pass
+        else:
+            pytest.skip(f"Not exact for degree {deg} {name} elements")
+
     mesh1 = UnitSquareMesh(2, 2)
     x1 = SpatialCoordinate(mesh1)
     V_target = fs_shape(V)(mesh1, "CG", 1)
@@ -131,6 +149,7 @@ def test_cross_mesh_adjoint(V, rank):
         # Direct assembly
         I_direct = assemble(interpolate(TestFunction(V_target), V))  # V^* x V_target -> R
         assert I_direct.arguments() == (TestFunction(V_target), TrialFunction(V.dual()))
+        assert mat_equals(I_manual, I_direct)
 
         cofunc_Vtarget_manual = assemble(action(I_manual, oneform_V))
         assert np.allclose(cofunc_Vtarget_manual.dat.data_ro, cofunc_Vtarget_direct.dat.data_ro)
@@ -149,9 +168,8 @@ def test_cross_mesh_adjoint(V, rank):
         cofunc_Vtarget = assemble(interpolate(TestFunction(V_target), oneform_V))  # V^* -> V_target^*
         assert np.allclose(cofunc_Vtarget.dat.data_ro, cofunc_Vtarget_direct.dat.data_ro)
     elif rank == 0:
+        if name == "GLS2" and deg == 1:
+            pytest.skip(f"Not exact for degree {deg} {name} elements")
         res = assemble(interpolate(target_expr, oneform_V))
         actual = assemble(inner(expr, expr) * dx)
         assert np.isclose(res, actual)
-
-if __name__ == "__main__":
-    pytest.main([__file__ + "::test_cross_mesh_adjoint[RT_2-1]"])

From 76a93238da25a3fb2152a4c412c87421ca9ed217 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Tue, 27 Jan 2026 13:57:26 +0000
Subject: [PATCH 28/37] docs

---
 docs/source/interpolation.rst | 12 ++++++++++--
 1 file changed, 10 insertions(+), 2 deletions(-)

diff --git a/docs/source/interpolation.rst b/docs/source/interpolation.rst
index c0cffba159..ddc72d399a 100644
--- a/docs/source/interpolation.rst
+++ b/docs/source/interpolation.rst
@@ -267,8 +267,16 @@ function space has finite elements (as given in the list of
 
 * **Lagrange/CG** (also known as Continuous Galerkin or P elements),
 * **Q** (i.e. Lagrange/CG on lines, quadrilaterals and hexahedra),
-* **Discontinuous Lagrange/DG** (also known as Discontinuous Galerkin or DP elements) and
-* **DQ** (i.e. Discontinuous Lagrange/DG on lines, quadrilaterals and hexahedra).
+* **Discontinuous Lagrange/DG** (also known as Discontinuous Galerkin or DP elements),
+* **DQ** (i.e. Discontinuous Lagrange/DG on lines, quadrilaterals and hexahedra),
+* **Raviart-Thomas/RT**,
+* **Brezzi-Douglas-Marini/BDM**,
+* **Brezzi-Douglas-Fortin-Marini/BDFM**,
+* **Hellan-Herrmann-Johnson/HHJ**,
+* **Nedelec 1st kind/N1curl**,
+* **Nedelec 2nd kind/N2curl**,
+* **Gopalakrishnan-Lederer-Schoberl 1st kind/GLS**,
+* **Gopalakrishnan-Lederer-Schoberl 2nd kind/GLS2**.
 
 Vector, tensor, and mixed function spaces can also be interpolated into from
 other meshes as long as they are constructed from these spaces.

From 5979f71b1b61db09123191e2c3db9c14667e43a0 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Tue, 27 Jan 2026 15:15:35 +0000
Subject: [PATCH 29/37] fixes

---
 firedrake/interpolation.py                    | 50 +++++++++----------
 .../regression/test_interpolate_cross_mesh.py | 25 ----------
 2 files changed, 23 insertions(+), 52 deletions(-)

diff --git a/firedrake/interpolation.py b/firedrake/interpolation.py
index 72776fffb7..142cc987c2 100644
--- a/firedrake/interpolation.py
+++ b/firedrake/interpolation.py
@@ -532,7 +532,7 @@ def _get_symbolic_expressions(self, target_space: WithGeometry) -> tuple[Interpo
             If the target space does not have point-evaluation dofs.
         """
         from firedrake.assemble import assemble
-        if target_space.ufl_element().mapping() != "identity":
+        if target_space.finat_element.mapping != "affine":
             raise ValueError(f"FunctionSpace {target_space} must have point-evaluation dofs.")
 
         # Immerse coordinates of target space point evaluation dofs in src_mesh
@@ -573,18 +573,12 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
         mat_type = mat_type or "aij"
 
         # Interpolate into intermediate quadrature space for non-identity mapped elements
-        if into_quadrature_space := self.target_space.ufl_element().mapping() != "identity":
-            Q_dest = self._get_quadrature_space(self.target_space)
+        if into_quadrature_space := self.target_space.finat_element.mapping != "affine":
+            target_space = self._get_quadrature_space(self.target_space)
+            f = Function(target_space.dual() if self.ufl_interpolate.is_adjoint else target_space)
         else:
-            Q_dest = None
-        target_space = Q_dest or self.target_space
-
-        if into_quadrature_space and not self.ufl_interpolate.is_adjoint:
-            f = Function(target_space)
-        else:
-            # self.ufl_interpolate.function_space() is None in the 0-form case
-            f = Function(self.ufl_interpolate.function_space() or target_space)
-        # f = tensor or Function(target_space.dual() if self.ufl_interpolate.is_adjoint else target_space)
+            target_space = self.target_space
+            f = tensor or Function(self.ufl_interpolate.function_space() or target_space)
 
         point_eval, point_eval_input_ordering = self._get_symbolic_expressions(target_space)
         P0DG_vom_input_ordering = point_eval_input_ordering.argument_slots()[0].function_space().dual()
@@ -602,12 +596,13 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
             def callable() -> PETSc.Mat:
                 res = assemble(interp_expr, mat_type=mat_type).petscmat
                 if into_quadrature_space:
+                    source_space = self.ufl_interpolate.function_space()
                     if self.ufl_interpolate.is_adjoint:
-                        I = AssembledMatrix((Argument(Q_dest, 0), Argument(target_space.dual(), 1)), None, res)
-                        return assemble(action(I, interpolate(TestFunction(Q_dest), self.target_space))).petscmat
+                        I = AssembledMatrix((Argument(source_space, 0), Argument(target_space.dual(), 1)), None, res)
+                        return assemble(action(I, interpolate(TestFunction(target_space), self.target_space))).petscmat
                     else:
-                        I = AssembledMatrix((Argument(Q_dest.dual(), 0), Argument(target_space, 1)), None, res)
-                        return assemble(action(interpolate(TrialFunction(Q_dest), self.target_space), I)).petscmat
+                        I = AssembledMatrix((Argument(target_space.dual(), 0), Argument(source_space, 1)), None, res)
+                        return assemble(action(interpolate(TrialFunction(target_space), self.target_space), I)).petscmat
                 else:
                     return res
 
@@ -639,19 +634,20 @@ def callable() -> Cofunction:
                 return f_target
         else:
             assert self.rank in {0, 1}
-            # We create the input-ordering Function before interpolating so we can
-            # set default missing values if required.
-            f_point_eval_input_ordering = Function(P0DG_vom_input_ordering)
-            if self.default_missing_val is not None:
-                f_point_eval_input_ordering.assign(self.default_missing_val)
-            elif self.allow_missing_dofs:
-                # If we allow missing points there may be points in the target
-                # mesh that are not in the source mesh. If we don't specify a
-                # default missing value we set these to NaN so we can identify
-                # them later.
-                f_point_eval_input_ordering.dat.data_wo[:] = numpy.nan
 
             def callable() -> Function | Number:
+                # We create the input-ordering Function before interpolating so we can
+                # set default missing values if required.
+                f_point_eval_input_ordering = Function(P0DG_vom_input_ordering)
+                if self.default_missing_val is not None:
+                    f_point_eval_input_ordering.assign(self.default_missing_val)
+                elif self.allow_missing_dofs:
+                    # If we allow missing points there may be points in the target
+                    # mesh that are not in the source mesh. If we don't specify a
+                    # default missing value we set these to NaN so we can identify
+                    # them later.
+                    f_point_eval_input_ordering.dat.data_wo[:] = numpy.nan
+
                 assemble(action(point_eval_input_ordering, point_eval), tensor=f_point_eval_input_ordering)
                 # We assign these values to the output function
                 if self.allow_missing_dofs and self.default_missing_val is None:
diff --git a/tests/firedrake/regression/test_interpolate_cross_mesh.py b/tests/firedrake/regression/test_interpolate_cross_mesh.py
index 23fe046485..4c1d53df29 100644
--- a/tests/firedrake/regression/test_interpolate_cross_mesh.py
+++ b/tests/firedrake/regression/test_interpolate_cross_mesh.py
@@ -409,31 +409,6 @@ def test_interpolate_unitsquare_tfs_shape(shape, symmetry):
     assemble(interpolate(f_src, V_dest))
 
 
-def test_interpolate_cross_mesh_not_point_eval():
-    m_src = UnitSquareMesh(2, 3)
-    m_dest = UnitSquareMesh(3, 5, quadrilateral=True)
-    coords = np.array(
-        [[0.56, 0.6], [0.1, 0.9], [0.9, 0.1], [0.9, 0.9], [0.726, 0.6584]]
-    )  # fairly arbitrary
-    # add the coordinates of the mesh vertices to test boundaries
-    vertices_src = allgather(m_src.comm, m_src.coordinates.dat.data_ro)
-    coords = np.concatenate((coords, vertices_src))
-    vertices_dest = allgather(m_dest.comm, m_dest.coordinates.dat.data_ro)
-    coords = np.concatenate((coords, vertices_dest))
-    dest_eval = PointEvaluator(m_dest, coords)
-    expr_src = 2 * SpatialCoordinate(m_src)
-    expr_dest = 2 * SpatialCoordinate(m_dest)
-    expected = 2 * coords
-    V_src = FunctionSpace(m_src, "RT", 2)
-    V_dest = FunctionSpace(m_dest, "RTCE", 2)
-    atol = 1e-8  # default
-    # This might not make much mathematical sense, but it should test if we get
-    # the not implemented error for non-point evaluation nodes!
-    interpolate_function(
-        m_src, m_dest, V_src, V_dest, dest_eval, expected, expr_src, expr_dest, atol
-    )
-
-
 def interpolate_function(
     m_src, m_dest, V_src, V_dest, dest_eval, expected, expr_src, expr_dest, atol
 ):

From 1625b3b469ab90c64f8383c1b096b757b8172842 Mon Sep 17 00:00:00 2001
From: AnuragRao1 <102687968+AnuragRao1@users.noreply.github.com>
Date: Tue, 27 Jan 2026 16:17:59 +0000
Subject: [PATCH 30/37] Added demo for performing adaptive multigrid (#4535)

* added demo for adaptive multigrid
---
 .../adaptive_convergence.png                  | Bin 0 -> 34117 bytes
 .../adaptive_multigrid.py.rst                 | 216 ++++++++++++++++++
 demos/adaptive_multigrid/eta_l15.png          | Bin 0 -> 338521 bytes
 demos/adaptive_multigrid/eta_l3.png           | Bin 0 -> 47146 bytes
 demos/adaptive_multigrid/eta_l6.png           | Bin 0 -> 86368 bytes
 demos/adaptive_multigrid/initial_mesh.png     | Bin 0 -> 44940 bytes
 demos/adaptive_multigrid/solution_l1.png      | Bin 0 -> 32367 bytes
 demos/adaptive_multigrid/solution_l15.png     | Bin 0 -> 155993 bytes
 demos/adaptive_multigrid/solution_l4.png      | Bin 0 -> 104185 bytes
 demos/demo_references.bib                     |  11 +
 docs/source/advanced_tut.rst                  |   1 +
 tests/firedrake/demos/test_demos_run.py       |   1 +
 12 files changed, 229 insertions(+)
 create mode 100644 demos/adaptive_multigrid/adaptive_convergence.png
 create mode 100644 demos/adaptive_multigrid/adaptive_multigrid.py.rst
 create mode 100644 demos/adaptive_multigrid/eta_l15.png
 create mode 100644 demos/adaptive_multigrid/eta_l3.png
 create mode 100644 demos/adaptive_multigrid/eta_l6.png
 create mode 100644 demos/adaptive_multigrid/initial_mesh.png
 create mode 100644 demos/adaptive_multigrid/solution_l1.png
 create mode 100644 demos/adaptive_multigrid/solution_l15.png
 create mode 100644 demos/adaptive_multigrid/solution_l4.png

diff --git a/demos/adaptive_multigrid/adaptive_convergence.png b/demos/adaptive_multigrid/adaptive_convergence.png
new file mode 100644
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diff --git a/demos/adaptive_multigrid/adaptive_multigrid.py.rst b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
new file mode 100644
index 0000000000..43bb2e0a9a
--- /dev/null
+++ b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
@@ -0,0 +1,216 @@
+Adaptive Multigrid Methods using AdaptiveMeshHierarchy
+======================================================
+
+
+Contributed by Anurag Rao
+
+The purpose of this demo is to show how to use Firedrake's multigrid solver on a hierarchy of adaptively refined Netgen meshes.
+We will first have a look at how to use the :class:`.AdaptiveMeshHierarchy` to construct the mesh hierarchy with Netgen meshes, then we will consider a solution to the Poisson problem on an L-shaped domain.
+Finally, we will show how to use the :class:`.AdaptiveMeshHierarchy` and :class:`.AdaptiveTransferManager` to construct a scalable solver. The :class:`.AdaptiveMeshHierarchy` contains information of the mesh hierarchy and the parent child relations between the meshes.
+The :class:`.AdaptiveTransferManager` deals with the transfer operator logic across any given levels in the hierarchy.
+We begin by importing the necessary libraries ::
+
+   from firedrake import *
+   from netgen.occ import *
+   import numpy
+
+Constructing the Mesh Hierarchy
+-------------------------------
+We first must construct the domain over which we will solve the problem. For a more comprehensive demo on how to use Open Cascade Technology (OCC) and Constructive Solid Geometry (CSG),
+see `Netgen integration in Firedrake <netgen_mesh.py>`_. 
+We begin with the L-shaped domain, which we build as the union of two rectangles: ::
+  
+   rect1 = WorkPlane(Axes((0,0,0), n=Z, h=X)).Rectangle(1,2).Face()
+   rect2 = WorkPlane(Axes((0,1,0), n=Z, h=X)).Rectangle(2,1).Face()
+   L = rect1 + rect2
+
+   geo = OCCGeometry(L, dim=2)
+   ngmsh = geo.GenerateMesh(maxh=0.5)
+   mesh = Mesh(ngmsh)
+
+It is important to convert the initial Netgen mesh into a Firedrake mesh before constructing the :class:`.AdaptiveMeshHierarchy`. To call the constructor to the hierarchy, we must pass the initial mesh. Our initial mesh looks like this:
+
+.. figure:: initial_mesh.png
+   :align: center
+   :alt: Initial mesh.
+
+We will also initialize the :class:`.AdaptiveTransferManager` here: ::
+  
+   amh = AdaptiveMeshHierarchy(mesh)
+   atm = AdaptiveTransferManager()
+
+Poisson Problem
+---------------
+Now we can define a simple Poisson problem
+
+.. math::
+
+   - \nabla^2 u = f \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega
+
+Our approach strongly follows the similar problem in this `lecture course <https://github.com/pefarrell/icerm2024>`_. We define the function ``solve_poisson``. The first lines correspond to finding a solution in the CG1 space. The variational problem is formulated with F, where f is the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the :class:`.DirichletBC` object and apply it to the entire boundary: ::
+
+   def solve_poisson(mesh, params):
+      V = FunctionSpace(mesh, "CG", 1)
+      uh = Function(V, name="solution")
+      v = TestFunction(V)
+      bc = DirichletBC(V, 0, "on_boundary")
+      f = Constant(1)
+      F = inner(grad(uh), grad(v))*dx - inner(f, v)*dx
+
+      problem = NonlinearVariationalProblem(F, uh, bc)
+      solver = NonlinearVariationalSolver(problem, solver_parameters=params)
+      solver.set_transfer_manager(atm)
+      solver.solve()
+      return uh
+
+Note the code after the construction of the :class:`.NonlinearVariationalProblem`. To use the :class:`.AdaptiveMeshHierarchy` with the existing Firedrake solver, we have to set the :class:`.AdaptiveTransferManager` as the transfer manager of the multigrid solver.
+Since we are using linear CG elements, we will employ Jacobi as the multigrid relaxation, which we define with ::
+
+   solver_params = {
+      "mat_type": "matfree",
+      "ksp_type": "cg",
+      "pc_type": "mg",
+      "mg_levels": {
+          "ksp_type": "chebyshev",
+          "ksp_max_it": 1,
+          "pc_type": "jacobi",
+      },
+      "mg_coarse": {
+          "mat_type": "aij",
+          "pc_type": "lu",
+      },
+   }
+
+Alternatively for high-order CG elements, it is recommended to use patch relaxation
+to achieve degree-independent multigrid convergence.
+For more information
+see :doc:`Using patch relaxation for multigrid <poisson_mg_patches.py>`. 
+The initial solution is shown below.
+
+.. figure:: solution_l1.png
+   :align: center
+   :alt: Initial Solution from multigrid with initial mesh.
+
+
+Adaptive Mesh Refinement
+------------------------
+In this section we will discuss how to adaptively refine select elements and add the newly refined mesh into the :class:`.AdaptiveMeshHierarchy`.
+For this problem, we will be using the Babuška-Rheinbolt a-posteriori estimate for an element:
+
+.. math::
+   \eta_K^2 = h_K^2 \int_K \| f + \nabla^2 u_h \|^2 \mathrm{d}x + \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \mathrm{d}s,
+
+where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the normal, and :math:`⟦ \cdot ⟧` is the jump operator. The a-posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem 
+
+.. math::
+   \int_\Omega \eta_K^2 w \mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \mathrm{d}x w \mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \mathrm{d}s w \mathrm{d}x
+
+Our approach will be to compute the estimator over all elements and selectively choose to refine only those that contribute most to the error. To compute the error estimator, we use the function below to solve the variational formulation of the error estimator. Since our estimator is a constant per element, we use a DG0 function space.  ::
+
+   def estimate_error(mesh, uh):
+       W = FunctionSpace(mesh, "DG", 0)
+       eta_sq = Function(W)
+       w = TestFunction(W)
+       f = Constant(1)
+
+       # symbols for mesh quantities
+       h = CellDiameter(mesh)
+       n = FacetNormal(mesh)
+       v = CellVolume(mesh)
+   
+       # compute cellwise error estimator
+       G = (inner(eta_sq / v, w)*dx
+            - inner(h**2 * (f + div(grad(uh)))**2, w) * dx
+            - inner(h('+')/2 * jump(grad(uh), n)**2, w('+')) * dS
+            - inner(h('-')/2 * jump(grad(uh), n)**2, w('-')) * dS
+       )
+   
+       sp = {"mat_type": "matfree", "ksp_type": "richardson", "pc_type": "jacobi"}
+       solve(G == 0, eta_sq, solver_parameters=sp)
+       eta = Function(W).interpolate(sqrt(eta_sq))  # compute eta from eta^2
+   
+       with eta.dat.vec_ro as eta_:  # compute estimate for error in energy norm
+           error_est = eta_.norm()
+       return eta, error_est
+
+The next step is to choose which elements to refine. For this we use Dörfler marking :cite:`Dorfler1996`:
+
+.. math::
+   \eta_K \geq \theta \text{max}_L \eta_L
+
+The logic is to select an element :math:`K` to refine if the estimator is greater than some factor :math:`\theta` of the maximum error estimate of the mesh, where :math:`\theta` ranges from 0 to 1. In our code we choose :math:`\theta=0.5`.
+With these helper functions complete, we can solve the system iteratively. In the max_iterations is the number of total levels we want to perform multigrid on. We will solve for 15 levels. At every level :math:`l`, we first compute the solution using multigrid with patch relaxation up till level :math:`l`. We then use the current approximation of the solution to estimate the error across the mesh. Finally, we refine the mesh and repeat. ::
+
+   theta = 0.5
+   max_iterations = 15
+   est_errors = []
+   dofs = []
+   for i in range(max_iterations):
+      print(f"level {i}")
+
+      mesh = amh[-1]
+      uh = solve_poisson(mesh, solver_params)
+      VTKFile(f"output/adaptive_loop_{i}.pvd").write(uh)
+
+      (eta, error_est) = estimate_error(mesh, uh)
+      VTKFile(f"output/eta_{i}.pvd").write(eta)
+
+      est_errors.append(error_est)
+      dofs.append(uh.function_space().dim())
+
+      print(f"  ||u - u_h|| <= C * {error_est}")
+      if len(dofs) > 1:
+         rate = -numpy.diff(numpy.log(est_errors)) / numpy.diff(numpy.log(dofs))
+         print(f"  rate = {rate[-1]}")
+
+      if i != max_iterations - 1:
+         amh.adapt(eta, theta)
+
+   from matplotlib import pyplot as plt
+
+   opt_errors = est_errors[0] * (numpy.array(dofs)/dofs[0]) ** -0.5
+   plt.loglog(dofs, est_errors, '-o', markersize = 3, label="Estimated error")
+   plt.loglog(dofs, opt_errors, '--', markersize = 3, label="Optimal convergence")
+   plt.ylabel("Error estimate of the energy norm")
+   plt.xlabel("Number of degrees of freedom")
+   plt.legend()
+   plt.savefig("output/adaptive_convergence.png")
+
+
+To perform Dörfler marking, refine the current mesh, and add the mesh to the :class:`.AdaptiveMeshHierarchy`, we use the ``amh.adapt(eta, theta)`` method. In this method the input is the recently computed error estimator ``eta`` and the Dörfler marking parameter ``theta``. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy: ``amh.add_mesh(mesh)``. In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy.
+The meshes now refine according to the error estimator. The error estimators at levels 3,5, and 15 are shown below. Zooming into the vertex of the L-shape at level 15 shows the error indicator remains strongest there. Further refinements will focus on that area.
+
++-------------------------------+-------------------------------+-------------------------------+
+| .. figure:: eta_l3.png        | .. figure:: eta_l6.png        | .. figure:: eta_l15.png       |
+|    :align: center             |    :align: center             |    :align: center             |
+|    :height: 250px             |    :height: 250px             |    :height: 250px             |
+|    :alt: Eta at level 3       |    :alt: Eta at level 6       |    :alt: Eta at level 15      |
+|                               |                               |                               |
+|    *Level 3*                  |    *Level 6*                  |    *Level 15*                 |
++-------------------------------+-------------------------------+-------------------------------+
+
+The solutions at level 4 and 15 are shown below.
+
++------------------------------------+------------------------------------+
+| .. figure:: solution_l4.png        | .. figure:: solution_l15.png       |
+|    :align: center                  |    :align: center                  |
+|    :height: 300px                  |    :height: 300px                  |
+|    :alt: Solution, level 4         |    :alt: Solution, level 15        |
+|                                    |                                    |
+|    *MG solution at level 4*        |    *MG solution at level 15*       |
++------------------------------------+------------------------------------+
+
+
+The convergence follows the expected optimal behavior:
+
+.. figure:: adaptive_convergence.png
+   :align: center
+   :alt: Convergence of the error estimator.
+
+
+A runnable python version of this demo can be found :demo:`here<adaptive_multigrid.py>`.
+
+.. rubric:: References
+
+.. bibliography:: demo_references.bib
+   :filter: docname in docnames
diff --git a/demos/adaptive_multigrid/eta_l15.png b/demos/adaptive_multigrid/eta_l15.png
new file mode 100644
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diff --git a/demos/demo_references.bib b/demos/demo_references.bib
index 901abaa917..ce29d3c2e8 100644
--- a/demos/demo_references.bib
+++ b/demos/demo_references.bib
@@ -370,6 +370,17 @@ @article{Brubeck2024
   year          = {2024}
 }
 
+@article{Dorfler1996,
+  title={A convergent adaptive algorithm for Poisson’s equation},
+  author={D{\"o}rfler, Willy},
+  journal={SIAM Journal on Numerical Analysis},
+  volume={33},
+  number={3},
+  pages={1106--1124},
+  year={1996},
+  publisher={SIAM}
+}
+
 @Article{Farrell2015,
   author =       {Patrick E. Farrell and \'Asgeir Birkisson and Simon W. Funke},
   title =        {{Deflation techniques for finding distinct solutions of nonlinear partial differential equations}},
diff --git a/docs/source/advanced_tut.rst b/docs/source/advanced_tut.rst
index 29d8d093a4..5007d12f4b 100644
--- a/docs/source/advanced_tut.rst
+++ b/docs/source/advanced_tut.rst
@@ -26,6 +26,7 @@ element systems.
    Full-waveform inversion: spatial and wave sources parallelism.<demos/full_waveform_inversion.py>
    1D Vlasov-Poisson equation using vertical independent function spaces.<demos/vp1d.py>
    Degree-independent multigrid convergence using patch relaxation.<demos/poisson_mg_patches.py>
+   Multigrid on adaptively-refined mesh hierarchies.<demos/adaptive_multigrid.py>
    Monolithic multigrid with Vanka relaxation for Stokes.<demos/stokes_vanka_patches.py>
    Vertex/edge star multigrid relaxation for H(div).<demos/hdiv_riesz_star.py>
    Auxiliary space patch relaxation multigrid for H(curl).<demos/hcurl_riesz_star.py>
diff --git a/tests/firedrake/demos/test_demos_run.py b/tests/firedrake/demos/test_demos_run.py
index dbb0b938be..dd8ce626bc 100644
--- a/tests/firedrake/demos/test_demos_run.py
+++ b/tests/firedrake/demos/test_demos_run.py
@@ -18,6 +18,7 @@
 DEMO_DIR = join(CWD, "..", "..", "..", "demos")
 
 SERIAL_DEMOS = [
+    Demo(("adaptive_multigrid", "adaptive_multigrid"), ["netgen", "vtk"]),
     Demo(("benney_luke", "benney_luke"), ["vtk"]),
     Demo(("boussinesq", "boussinesq"), []),
     Demo(("burgers", "burgers"), ["vtk"]),

From 76f39423c5fa90113be2c038b3febaabbe897a6e Mon Sep 17 00:00:00 2001
From: Patrick Farrell <patrick.farrell@maths.ox.ac.uk>
Date: Tue, 27 Jan 2026 18:50:13 +0000
Subject: [PATCH 31/37] Edits

---
 .../adaptive_multigrid/adaptive_multigrid.py.rst | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/demos/adaptive_multigrid/adaptive_multigrid.py.rst b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
index 43bb2e0a9a..55e3456845 100644
--- a/demos/adaptive_multigrid/adaptive_multigrid.py.rst
+++ b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
@@ -45,9 +45,9 @@ Now we can define a simple Poisson problem
 
 .. math::
 
-   - \nabla^2 u = f \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega
+   - \nabla^2 u = f \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega.
 
-Our approach strongly follows the similar problem in this `lecture course <https://github.com/pefarrell/icerm2024>`_. We define the function ``solve_poisson``. The first lines correspond to finding a solution in the CG1 space. The variational problem is formulated with F, where f is the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the :class:`.DirichletBC` object and apply it to the entire boundary: ::
+Our approach strongly follows the similar problem in this `lecture course <https://github.com/pefarrell/icerm2024>`_. We define the function ``solve_poisson``. The first lines correspond to finding a solution in the CG1 space. The right-hand side is set to be the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the :class:`.DirichletBC` object and apply it to the entire boundary: ::
 
    def solve_poisson(mesh, params):
       V = FunctionSpace(mesh, "CG", 1)
@@ -64,7 +64,7 @@ Our approach strongly follows the similar problem in this `lecture course <https
       return uh
 
 Note the code after the construction of the :class:`.NonlinearVariationalProblem`. To use the :class:`.AdaptiveMeshHierarchy` with the existing Firedrake solver, we have to set the :class:`.AdaptiveTransferManager` as the transfer manager of the multigrid solver.
-Since we are using linear CG elements, we will employ Jacobi as the multigrid relaxation, which we define with ::
+Since we are using linear Lagrange elements, we will employ Jacobi as the multigrid relaxation, which we define with ::
 
    solver_params = {
       "mat_type": "matfree",
@@ -95,15 +95,15 @@ The initial solution is shown below.
 Adaptive Mesh Refinement
 ------------------------
 In this section we will discuss how to adaptively refine select elements and add the newly refined mesh into the :class:`.AdaptiveMeshHierarchy`.
-For this problem, we will be using the Babuška-Rheinbolt a-posteriori estimate for an element:
+For this problem, we will be using the Babuška-Rheinbolt a posteriori estimate for an element:
 
 .. math::
    \eta_K^2 = h_K^2 \int_K \| f + \nabla^2 u_h \|^2 \mathrm{d}x + \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \mathrm{d}s,
 
-where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the normal, and :math:`⟦ \cdot ⟧` is the jump operator. The a-posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem 
+where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the normal, and :math:`⟦ \cdot ⟧` is the jump operator. The a posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem 
 
 .. math::
-   \int_\Omega \eta_K^2 w \mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \mathrm{d}x w \mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \mathrm{d}s w \mathrm{d}x
+   \int_\Omega \eta_K^2 w \,\mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \,\mathrm{d}x w \,\mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \,\mathrm{d}s w \,\mathrm{d}x
 
 Our approach will be to compute the estimator over all elements and selectively choose to refine only those that contribute most to the error. To compute the error estimator, we use the function below to solve the variational formulation of the error estimator. Since our estimator is a constant per element, we use a DG0 function space.  ::
 
@@ -133,13 +133,13 @@ Our approach will be to compute the estimator over all elements and selectively
            error_est = eta_.norm()
        return eta, error_est
 
-The next step is to choose which elements to refine. For this we use Dörfler marking :cite:`Dorfler1996`:
+The next step is to choose which elements to refine. For this we use a simplified variant of Dörfler marking :cite:`Dorfler1996`:
 
 .. math::
    \eta_K \geq \theta \text{max}_L \eta_L
 
 The logic is to select an element :math:`K` to refine if the estimator is greater than some factor :math:`\theta` of the maximum error estimate of the mesh, where :math:`\theta` ranges from 0 to 1. In our code we choose :math:`\theta=0.5`.
-With these helper functions complete, we can solve the system iteratively. In the max_iterations is the number of total levels we want to perform multigrid on. We will solve for 15 levels. At every level :math:`l`, we first compute the solution using multigrid with patch relaxation up till level :math:`l`. We then use the current approximation of the solution to estimate the error across the mesh. Finally, we refine the mesh and repeat. ::
+With these helper functions complete, we can solve the system iteratively. In the max_iterations is the number of total levels we want to perform multigrid on. We will solve for 15 levels. At every level :math:`l`, we first compute the solution using multigrid up to level :math:`l`. We then use the current approximation of the solution to estimate the error across the mesh. Finally, we adaptively refine the mesh and repeat. ::
 
    theta = 0.5
    max_iterations = 15

From eedf0adece140247b373cfd968d8b2df85b6d6dd Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Wed, 28 Jan 2026 13:51:16 +0000
Subject: [PATCH 32/37] fix test

---
 tests/firedrake/regression/test_interpolate_cross_mesh.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/tests/firedrake/regression/test_interpolate_cross_mesh.py b/tests/firedrake/regression/test_interpolate_cross_mesh.py
index 4c1d53df29..f9b8b5c4af 100644
--- a/tests/firedrake/regression/test_interpolate_cross_mesh.py
+++ b/tests/firedrake/regression/test_interpolate_cross_mesh.py
@@ -38,7 +38,7 @@ def make_high_order(m_low_order, degree):
 
 @pytest.fixture(
     params=[
-        "unitsquare_RT_N1curl_destination",
+        "unitsquare",
         "circlemanifold",
         "circlemanifold_to_high_order",
         "unitsquare_from_high_order",

From 139ec3383b44d01c29e50549b0484e57bad28f06 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Thu, 29 Jan 2026 11:58:45 +0000
Subject: [PATCH 33/37] MG iteration counts

---
 .../adaptive_convergence.png                  | Bin 34117 -> 34117 bytes
 .../adaptive_multigrid.py.rst                 |  92 ++++++++++++------
 tests/firedrake/demos/test_demos_run.py       |   2 +-
 3 files changed, 65 insertions(+), 29 deletions(-)

diff --git a/demos/adaptive_multigrid/adaptive_convergence.png b/demos/adaptive_multigrid/adaptive_convergence.png
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diff --git a/demos/adaptive_multigrid/adaptive_multigrid.py.rst b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
index 55e3456845..2b974e1b7a 100644
--- a/demos/adaptive_multigrid/adaptive_multigrid.py.rst
+++ b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
@@ -61,7 +61,8 @@ Our approach strongly follows the similar problem in this `lecture course <https
       solver = NonlinearVariationalSolver(problem, solver_parameters=params)
       solver.set_transfer_manager(atm)
       solver.solve()
-      return uh
+      its = solver.snes.getLinearSolveIterations()
+      return uh, its
 
 Note the code after the construction of the :class:`.NonlinearVariationalProblem`. To use the :class:`.AdaptiveMeshHierarchy` with the existing Firedrake solver, we have to set the :class:`.AdaptiveTransferManager` as the transfer manager of the multigrid solver.
 Since we are using linear Lagrange elements, we will employ Jacobi as the multigrid relaxation, which we define with ::
@@ -108,28 +109,32 @@ where :math:`K` is the element, :math:`h_K` is the diameter of the element, :mat
 Our approach will be to compute the estimator over all elements and selectively choose to refine only those that contribute most to the error. To compute the error estimator, we use the function below to solve the variational formulation of the error estimator. Since our estimator is a constant per element, we use a DG0 function space.  ::
 
    def estimate_error(mesh, uh):
-       W = FunctionSpace(mesh, "DG", 0)
-       eta_sq = Function(W)
-       w = TestFunction(W)
+       Q = FunctionSpace(mesh, "DG", 0)
+       eta_sq = Function(Q)
+       p = TrialFunction(Q)
+       q = TestFunction(Q)
        f = Constant(1)
+       residual = f + div(grad(uh))
 
        # symbols for mesh quantities
        h = CellDiameter(mesh)
        n = FacetNormal(mesh)
-       v = CellVolume(mesh)
+       vol = CellVolume(mesh)
    
        # compute cellwise error estimator
-       G = (inner(eta_sq / v, w)*dx
-            - inner(h**2 * (f + div(grad(uh)))**2, w) * dx
-            - inner(h('+')/2 * jump(grad(uh), n)**2, w('+')) * dS
-            - inner(h('-')/2 * jump(grad(uh), n)**2, w('-')) * dS
+       a = inner(p, q / vol) * dx
+       L = (inner(residual**2, q * h**2) * dx
+            + inner(jump(grad(uh), n)**2, avg(q * h)) * dS
        )
-   
-       sp = {"mat_type": "matfree", "ksp_type": "richardson", "pc_type": "jacobi"}
-       solve(G == 0, eta_sq, solver_parameters=sp)
-       eta = Function(W).interpolate(sqrt(eta_sq))  # compute eta from eta^2
-   
-       with eta.dat.vec_ro as eta_:  # compute estimate for error in energy norm
+
+       sp = {"mat_type": "matfree", "ksp_type": "preonly", "pc_type": "jacobi"}
+       solve(a == L, eta_sq, solver_parameters=sp)
+
+       # compute eta from eta^2
+       eta = Function(Q).interpolate(sqrt(eta_sq))
+
+       # compute estimate for error in energy norm
+       with eta.dat.vec_ro as eta_:  
            error_est = eta_.norm()
        return eta, error_est
 
@@ -142,33 +147,36 @@ The logic is to select an element :math:`K` to refine if the estimator is greate
 With these helper functions complete, we can solve the system iteratively. In the max_iterations is the number of total levels we want to perform multigrid on. We will solve for 15 levels. At every level :math:`l`, we first compute the solution using multigrid up to level :math:`l`. We then use the current approximation of the solution to estimate the error across the mesh. Finally, we adaptively refine the mesh and repeat. ::
 
    theta = 0.5
-   max_iterations = 15
+   refinements = 15
    est_errors = []
-   dofs = []
-   for i in range(max_iterations):
-      print(f"level {i}")
+   sqrt_dofs = []
+   mg_iterations = []
+   for level in range(refinements):
+      print(f"level {level}")
 
       mesh = amh[-1]
-      uh = solve_poisson(mesh, solver_params)
-      VTKFile(f"output/adaptive_loop_{i}.pvd").write(uh)
+      uh, its = solve_poisson(mesh, solver_params)
+      VTKFile(f"output/adaptive_loop_{level}.pvd").write(uh)
 
       (eta, error_est) = estimate_error(mesh, uh)
-      VTKFile(f"output/eta_{i}.pvd").write(eta)
+      VTKFile(f"output/eta_{level}.pvd").write(eta)
 
       est_errors.append(error_est)
-      dofs.append(uh.function_space().dim())
+      sqrt_dofs.append(uh.function_space().dim() ** 0.5)
+      mg_iterations.append(its)
 
       print(f"  ||u - u_h|| <= C * {error_est}")
-      if len(dofs) > 1:
-         rate = -numpy.diff(numpy.log(est_errors)) / numpy.diff(numpy.log(dofs))
-         print(f"  rate = {rate[-1]}")
+      if len(est_errors) > 1:
+         rates = -numpy.diff(numpy.log(est_errors)) / numpy.diff(numpy.log(sqrt_dofs))
+         print(f"  rate = {rates[-1]}")
 
-      if i != max_iterations - 1:
+      if i != refinements - 1:
          amh.adapt(eta, theta)
 
    from matplotlib import pyplot as plt
 
-   opt_errors = est_errors[0] * (numpy.array(dofs)/dofs[0]) ** -0.5
+   dofs = numpy.array(sqrt_dofs) ** 2
+   opt_errors = est_errors[0] * (sqrt_dofs[0] / numpy.array(sqrt_dofs))
    plt.loglog(dofs, est_errors, '-o', markersize = 3, label="Estimated error")
    plt.loglog(dofs, opt_errors, '--', markersize = 3, label="Optimal convergence")
    plt.ylabel("Error estimate of the energy norm")
@@ -207,6 +215,34 @@ The convergence follows the expected optimal behavior:
    :align: center
    :alt: Convergence of the error estimator.
 
+Moreover, the multigrid iteration count is robust to the level of refinement ::
+
+   print(" Level\t | Iterations")
+   print("---------------------")
+   for level, its in enumerate(mg_iterations):
+       print(f"   {level}\t | {its}")
+
+..
+
+======== ================
+ Level     Iterations
+======== ================
+  0	      2
+  1	      12
+  2	      13
+  3	      13
+  4	      13
+  5	      13
+  6	      13
+  7	      14
+  8	      15
+  9	      15
+  10	      15
+  11	      15
+  12	      16
+  13	      16
+  14	      16
+======== ================
 
 A runnable python version of this demo can be found :demo:`here<adaptive_multigrid.py>`.
 
diff --git a/tests/firedrake/demos/test_demos_run.py b/tests/firedrake/demos/test_demos_run.py
index dd8ce626bc..e3a78532aa 100644
--- a/tests/firedrake/demos/test_demos_run.py
+++ b/tests/firedrake/demos/test_demos_run.py
@@ -18,7 +18,7 @@
 DEMO_DIR = join(CWD, "..", "..", "..", "demos")
 
 SERIAL_DEMOS = [
-    Demo(("adaptive_multigrid", "adaptive_multigrid"), ["netgen", "vtk"]),
+    Demo(("adaptive_multigrid", "adaptive_multigrid"), ["matplotlib", "netgen", "vtk"]),
     Demo(("benney_luke", "benney_luke"), ["vtk"]),
     Demo(("boussinesq", "boussinesq"), []),
     Demo(("burgers", "burgers"), ["vtk"]),

From 862b9435825164c1dc4ea556209dee1f72257540 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Thu, 29 Jan 2026 13:52:26 +0000
Subject: [PATCH 34/37] fixes

---
 .../adaptive_multigrid.py.rst                 | 30 +++++++++----------
 1 file changed, 14 insertions(+), 16 deletions(-)

diff --git a/demos/adaptive_multigrid/adaptive_multigrid.py.rst b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
index 2b974e1b7a..76babbbeec 100644
--- a/demos/adaptive_multigrid/adaptive_multigrid.py.rst
+++ b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
@@ -2,7 +2,7 @@ Adaptive Multigrid Methods using AdaptiveMeshHierarchy
 ======================================================
 
 
-Contributed by Anurag Rao
+Contributed by Anurag Rao.
 
 The purpose of this demo is to show how to use Firedrake's multigrid solver on a hierarchy of adaptively refined Netgen meshes.
 We will first have a look at how to use the :class:`.AdaptiveMeshHierarchy` to construct the mesh hierarchy with Netgen meshes, then we will consider a solution to the Poisson problem on an L-shaped domain.
@@ -101,10 +101,10 @@ For this problem, we will be using the Babuška-Rheinbolt a posteriori estimate
 .. math::
    \eta_K^2 = h_K^2 \int_K \| f + \nabla^2 u_h \|^2 \mathrm{d}x + \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \mathrm{d}s,
 
-where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the normal, and :math:`⟦ \cdot ⟧` is the jump operator. The a posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem 
+where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the outward-facing normal, and :math:`⟦ \cdot ⟧` is the jump operator. The a posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem 
 
 .. math::
-   \int_\Omega \eta_K^2 w \,\mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \,\mathrm{d}x w \,\mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \,\mathrm{d}s w \,\mathrm{d}x
+   \int_\Omega \eta_K^2 q \,\mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \,\mathrm{d}x q \,\mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \,\mathrm{d}s q \,\mathrm{d}x \quad \forall\, q \in \mathrm{DG}_0
 
 Our approach will be to compute the estimator over all elements and selectively choose to refine only those that contribute most to the error. To compute the error estimator, we use the function below to solve the variational formulation of the error estimator. Since our estimator is a constant per element, we use a DG0 function space.  ::
 
@@ -173,18 +173,6 @@ With these helper functions complete, we can solve the system iteratively. In th
       if i != refinements - 1:
          amh.adapt(eta, theta)
 
-   from matplotlib import pyplot as plt
-
-   dofs = numpy.array(sqrt_dofs) ** 2
-   opt_errors = est_errors[0] * (sqrt_dofs[0] / numpy.array(sqrt_dofs))
-   plt.loglog(dofs, est_errors, '-o', markersize = 3, label="Estimated error")
-   plt.loglog(dofs, opt_errors, '--', markersize = 3, label="Optimal convergence")
-   plt.ylabel("Error estimate of the energy norm")
-   plt.xlabel("Number of degrees of freedom")
-   plt.legend()
-   plt.savefig("output/adaptive_convergence.png")
-
-
 To perform Dörfler marking, refine the current mesh, and add the mesh to the :class:`.AdaptiveMeshHierarchy`, we use the ``amh.adapt(eta, theta)`` method. In this method the input is the recently computed error estimator ``eta`` and the Dörfler marking parameter ``theta``. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy: ``amh.add_mesh(mesh)``. In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy.
 The meshes now refine according to the error estimator. The error estimators at levels 3,5, and 15 are shown below. Zooming into the vertex of the L-shape at level 15 shows the error indicator remains strongest there. Further refinements will focus on that area.
 
@@ -208,8 +196,18 @@ The solutions at level 4 and 15 are shown below.
 |    *MG solution at level 4*        |    *MG solution at level 15*       |
 +------------------------------------+------------------------------------+
 
+The convergence follows the expected optimal behavior: ::
+
+   from matplotlib import pyplot as plt
 
-The convergence follows the expected optimal behavior:
+   dofs = numpy.array(sqrt_dofs) ** 2
+   opt_errors = est_errors[0] * (sqrt_dofs[0] / numpy.array(sqrt_dofs))
+   plt.loglog(dofs, est_errors, '-o', markersize = 3, label="Estimated error")
+   plt.loglog(dofs, opt_errors, '--', markersize = 3, label="Optimal convergence")
+   plt.ylabel("Error estimate of the energy norm")
+   plt.xlabel("Number of degrees of freedom")
+   plt.legend()
+   plt.savefig("output/adaptive_convergence.png")
 
 .. figure:: adaptive_convergence.png
    :align: center

From f3352c3dba7a67d93e7bde4cb4c34883f86892b2 Mon Sep 17 00:00:00 2001
From: Leo Collins <collins.leo@proton.me>
Date: Thu, 29 Jan 2026 14:59:33 +0000
Subject: [PATCH 35/37] review suggestions

---
 firedrake/interpolation.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/firedrake/interpolation.py b/firedrake/interpolation.py
index 142cc987c2..4ff0372cfc 100644
--- a/firedrake/interpolation.py
+++ b/firedrake/interpolation.py
@@ -532,7 +532,7 @@ def _get_symbolic_expressions(self, target_space: WithGeometry) -> tuple[Interpo
             If the target space does not have point-evaluation dofs.
         """
         from firedrake.assemble import assemble
-        if target_space.finat_element.mapping != "affine":
+        if not target_space.finat_element.has_pointwise_dual_basis:
             raise ValueError(f"FunctionSpace {target_space} must have point-evaluation dofs.")
 
         # Immerse coordinates of target space point evaluation dofs in src_mesh
@@ -573,7 +573,7 @@ def _get_callable(self, tensor=None, bcs=None, mat_type=None, sub_mat_type=None)
         mat_type = mat_type or "aij"
 
         # Interpolate into intermediate quadrature space for non-identity mapped elements
-        if into_quadrature_space := self.target_space.finat_element.mapping != "affine":
+        if into_quadrature_space := not self.target_space.finat_element.has_pointwise_dual_basis:
             target_space = self._get_quadrature_space(self.target_space)
             f = Function(target_space.dual() if self.ufl_interpolate.is_adjoint else target_space)
         else:

From 0ea4dc721db9e915cd05a2fe620a1f5aca7d9eb3 Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Thu, 29 Jan 2026 17:39:09 +0000
Subject: [PATCH 36/37] fix latex

---
 .../adaptive_multigrid.py.rst                 | 41 +++++++++++--------
 1 file changed, 23 insertions(+), 18 deletions(-)

diff --git a/demos/adaptive_multigrid/adaptive_multigrid.py.rst b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
index 76babbbeec..64e6d77819 100644
--- a/demos/adaptive_multigrid/adaptive_multigrid.py.rst
+++ b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
@@ -5,9 +5,9 @@ Adaptive Multigrid Methods using AdaptiveMeshHierarchy
 Contributed by Anurag Rao.
 
 The purpose of this demo is to show how to use Firedrake's multigrid solver on a hierarchy of adaptively refined Netgen meshes.
-We will first have a look at how to use the :class:`.AdaptiveMeshHierarchy` to construct the mesh hierarchy with Netgen meshes, then we will consider a solution to the Poisson problem on an L-shaped domain.
-Finally, we will show how to use the :class:`.AdaptiveMeshHierarchy` and :class:`.AdaptiveTransferManager` to construct a scalable solver. The :class:`.AdaptiveMeshHierarchy` contains information of the mesh hierarchy and the parent child relations between the meshes.
-The :class:`.AdaptiveTransferManager` deals with the transfer operator logic across any given levels in the hierarchy.
+We will first have a look at how to use the :class:`~.AdaptiveMeshHierarchy` to construct the mesh hierarchy with Netgen meshes, then we will consider a solution to the Poisson problem on an L-shaped domain.
+Finally, we will show how to use the :class:`~.AdaptiveMeshHierarchy` and :class:`~.AdaptiveTransferManager` to construct a scalable solver. The :class:`~.AdaptiveMeshHierarchy` contains information of the mesh hierarchy and the parent child relations between the meshes.
+The :class:`~.AdaptiveTransferManager` deals with the transfer operator logic across any given levels in the hierarchy.
 We begin by importing the necessary libraries ::
 
    from firedrake import *
@@ -28,13 +28,13 @@ We begin with the L-shaped domain, which we build as the union of two rectangles
    ngmsh = geo.GenerateMesh(maxh=0.5)
    mesh = Mesh(ngmsh)
 
-It is important to convert the initial Netgen mesh into a Firedrake mesh before constructing the :class:`.AdaptiveMeshHierarchy`. To call the constructor to the hierarchy, we must pass the initial mesh. Our initial mesh looks like this:
+It is important to convert the initial Netgen mesh into a Firedrake mesh before constructing the :class:`~.AdaptiveMeshHierarchy`. To call the constructor to the hierarchy, we must pass the initial mesh. Our initial mesh looks like this:
 
 .. figure:: initial_mesh.png
    :align: center
    :alt: Initial mesh.
 
-We will also initialize the :class:`.AdaptiveTransferManager` here: ::
+We will also initialize the :class:`~.AdaptiveTransferManager` here: ::
   
    amh = AdaptiveMeshHierarchy(mesh)
    atm = AdaptiveTransferManager()
@@ -47,24 +47,29 @@ Now we can define a simple Poisson problem
 
    - \nabla^2 u = f \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega.
 
-Our approach strongly follows the similar problem in this `lecture course <https://github.com/pefarrell/icerm2024>`_. We define the function ``solve_poisson``. The first lines correspond to finding a solution in the CG1 space. The right-hand side is set to be the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the :class:`.DirichletBC` object and apply it to the entire boundary: ::
+Our approach strongly follows the similar problem in this `lecture course <https://github.com/pefarrell/icerm2024>`_. We define the function ``solve_poisson``. The first lines correspond to finding a solution in the CG1 space. The right-hand side is set to be the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the :class:`~.DirichletBC` object and apply it to the entire boundary: ::
 
    def solve_poisson(mesh, params):
       V = FunctionSpace(mesh, "CG", 1)
-      uh = Function(V, name="solution")
       v = TestFunction(V)
-      bc = DirichletBC(V, 0, "on_boundary")
+      u = TrialFunction(V)
+      uh = Function(V, name="solution")
+      bcs = [DirichletBC(V, 0, "on_boundary")]
       f = Constant(1)
-      F = inner(grad(uh), grad(v))*dx - inner(f, v)*dx
 
-      problem = NonlinearVariationalProblem(F, uh, bc)
-      solver = NonlinearVariationalSolver(problem, solver_parameters=params)
+      a = inner(grad(u), grad(v))*dx
+      L = inner(f, v)*dx
+
+      problem = LinearVariationalProblem(a, L, uh, bcs)
+      solver = LinearVariationalSolver(problem, solver_parameters=params)
+
       solver.set_transfer_manager(atm)
       solver.solve()
+
       its = solver.snes.getLinearSolveIterations()
       return uh, its
 
-Note the code after the construction of the :class:`.NonlinearVariationalProblem`. To use the :class:`.AdaptiveMeshHierarchy` with the existing Firedrake solver, we have to set the :class:`.AdaptiveTransferManager` as the transfer manager of the multigrid solver.
+Note the code after the construction of the :class:`~.LinearVariationalProblem`. To use the :class:`~.AdaptiveMeshHierarchy` with the existing Firedrake solver, we have to set the :class:`~.AdaptiveTransferManager` as the transfer manager of the multigrid solver.
 Since we are using linear Lagrange elements, we will employ Jacobi as the multigrid relaxation, which we define with ::
 
    solver_params = {
@@ -95,16 +100,16 @@ The initial solution is shown below.
 
 Adaptive Mesh Refinement
 ------------------------
-In this section we will discuss how to adaptively refine select elements and add the newly refined mesh into the :class:`.AdaptiveMeshHierarchy`.
-For this problem, we will be using the Babuška-Rheinbolt a posteriori estimate for an element:
+In this section we will discuss how to adaptively refine select elements and add the newly refined mesh into the :class:`~.AdaptiveMeshHierarchy`.
+For this problem, we will be using the Babuška-Rheinbolt a-posteriori estimate for an element:
 
 .. math::
-   \eta_K^2 = h_K^2 \int_K \| f + \nabla^2 u_h \|^2 \mathrm{d}x + \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \mathrm{d}s,
+   \eta_K^2 = h_K^2 \int_K \| f + \nabla^2 u_h \|^2 \mathrm{d}x + \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} \left[[ \nabla u_h \cdot \mathbf{n} \right]]^2 \mathrm{d}s,
 
-where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the outward-facing normal, and :math:`⟦ \cdot ⟧` is the jump operator. The a posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem 
+where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`\mathbf{n}` is the outward-facing normal, and :math:`\left[[ \cdot \right]]` is the jump operator. The a-posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem
 
 .. math::
-   \int_\Omega \eta_K^2 q \,\mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \,\mathrm{d}x q \,\mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \,\mathrm{d}s q \,\mathrm{d}x \quad \forall\, q \in \mathrm{DG}_0
+   \int_\Omega \eta_K^2 q \,\mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \,\mathrm{d}x q \,\mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} \left[[ \nabla u_h \cdot \mathbf{n} \right]]^2 \,\mathrm{d}s q \,\mathrm{d}x \quad \forall\, q \in \mathrm{DG}_0
 
 Our approach will be to compute the estimator over all elements and selectively choose to refine only those that contribute most to the error. To compute the error estimator, we use the function below to solve the variational formulation of the error estimator. Since our estimator is a constant per element, we use a DG0 function space.  ::
 
@@ -173,7 +178,7 @@ With these helper functions complete, we can solve the system iteratively. In th
       if i != refinements - 1:
          amh.adapt(eta, theta)
 
-To perform Dörfler marking, refine the current mesh, and add the mesh to the :class:`.AdaptiveMeshHierarchy`, we use the ``amh.adapt(eta, theta)`` method. In this method the input is the recently computed error estimator ``eta`` and the Dörfler marking parameter ``theta``. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy: ``amh.add_mesh(mesh)``. In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy.
+To perform Dörfler marking, refine the current mesh, and add the mesh to the :class:`~.AdaptiveMeshHierarchy`, we use the ``amh.adapt(eta, theta)`` method. In this method the input is the recently computed error estimator ``eta`` and the Dörfler marking parameter ``theta``. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy: ``amh.add_mesh(mesh)``. In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy.
 The meshes now refine according to the error estimator. The error estimators at levels 3,5, and 15 are shown below. Zooming into the vertex of the L-shape at level 15 shows the error indicator remains strongest there. Further refinements will focus on that area.
 
 +-------------------------------+-------------------------------+-------------------------------+

From 9b46f426d95aefd1096ac71a19263b8244e5b5fa Mon Sep 17 00:00:00 2001
From: Pablo Brubeck <brubeck@protonmail.com>
Date: Thu, 29 Jan 2026 18:09:53 +0000
Subject: [PATCH 37/37] Use native interpolation

---
 .../adaptive_multigrid.py.rst                 | 30 ++++-----
 firedrake/mg/adaptive_transfer_manager.py     | 61 +++----------------
 2 files changed, 25 insertions(+), 66 deletions(-)

diff --git a/demos/adaptive_multigrid/adaptive_multigrid.py.rst b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
index 64e6d77819..44ff4eac0f 100644
--- a/demos/adaptive_multigrid/adaptive_multigrid.py.rst
+++ b/demos/adaptive_multigrid/adaptive_multigrid.py.rst
@@ -230,21 +230,21 @@ Moreover, the multigrid iteration count is robust to the level of refinement ::
 ======== ================
  Level     Iterations
 ======== ================
-  0	      2
-  1	      12
-  2	      13
-  3	      13
-  4	      13
-  5	      13
-  6	      13
-  7	      14
-  8	      15
-  9	      15
-  10	      15
-  11	      15
-  12	      16
-  13	      16
-  14	      16
+   0	     2
+   1	     8
+   2	     8
+   3	     8
+   4	     8
+   5	     8
+   6	     8
+   7	     8
+   8	     8
+   9	     9
+   10	     9
+   11	     9
+   12	     9
+   13	     9
+   14	     9
 ======== ================
 
 A runnable python version of this demo can be found :demo:`here<adaptive_multigrid.py>`.
diff --git a/firedrake/mg/adaptive_transfer_manager.py b/firedrake/mg/adaptive_transfer_manager.py
index 05b1feca4b..149906b845 100644
--- a/firedrake/mg/adaptive_transfer_manager.py
+++ b/firedrake/mg/adaptive_transfer_manager.py
@@ -4,13 +4,7 @@
 """
 from firedrake.mg.embedded import TransferManager
 from firedrake.ufl_expr import action, TrialFunction
-from firedrake.functionspace import FunctionSpace, TensorFunctionSpace
 from firedrake.interpolation import interpolate
-from firedrake.preconditioners.bddc import is_lagrange
-from finat.quadrature import QuadratureRule
-from functools import partial
-
-import numpy
 
 
 __all__ = ("AdaptiveTransferManager",)
@@ -26,35 +20,30 @@ def __init__(self, *, native_transfers=None, use_averaging=True):
         super().__init__(native_transfers=native_transfers, use_averaging=use_averaging)
         self.cache = {}
 
-    def get_operators(self, Vc, Vf):
+    def get_interpolator(self, Vc, Vf):
+        from firedrake.assemble import assemble
         key = (Vc, Vf)
         try:
             return self.cache[key]
         except KeyError:
-            ops = get_mg_interpolator(Vc, Vf)
-            return self.cache.setdefault(key, ops)
+            Iexpr = interpolate(TrialFunction(Vc), Vf)
+            # TODO reusable matfree Interpolator
+            I = assemble(Iexpr, mat_type="aij")
+            return self.cache.setdefault(key, I)
 
     def forward(self, uc, uf):
         from firedrake.assemble import assemble
         Vc = uc.function_space()
         Vf = uf.function_space()
-        ops = self.get_operators(Vc, Vf)
-
-        expr = uc
-        for op in ops:
-            expr = action(op, expr)
-        return assemble(expr, tensor=uf)
+        I = self.get_interpolator(Vc, Vf)
+        return assemble(action(I, uc), tensor=uf)
 
     def adjoint(self, uf, uc):
         from firedrake.assemble import assemble
         Vc = uc.function_space().dual()
         Vf = uf.function_space().dual()
-        ops = self.get_operators(Vc, Vf)
-
-        expr = uf
-        for op in reversed(ops):
-            expr = action(expr, op)
-        return assemble(expr, tensor=uc)
+        I = self.get_interpolator(Vc, Vf)
+        return assemble(action(uf, I), tensor=uc)
 
     def prolong(self, uf, uc):
         return self.forward(uf, uc)
@@ -64,33 +53,3 @@ def inject(self, uc, uf):
 
     def restrict(self, uc, uf):
         return self.adjoint(uc, uf)
-
-
-def make_quadrature_space(V):
-    fe = V.finat_element
-    _, ps = fe.dual_basis
-    wts = numpy.full(len(ps.points), numpy.nan)
-    scheme = QuadratureRule(ps, wts, ref_el=fe.cell)
-    if V.value_shape == ():
-        make_space = FunctionSpace
-    else:
-        make_space = partial(TensorFunctionSpace, shape=V.value_shape)
-    return make_space(V.mesh(), "Quadrature", degree=fe.degree, quad_scheme=scheme)
-
-
-def get_mg_interpolator(V1, V2):
-    from firedrake.assemble import assemble
-    if is_lagrange(V2.finat_element):
-        spaces = (V1, V2)
-    else:
-        Q2 = make_quadrature_space(V2)
-        spaces = (V1, Q2, V2)
-
-    ops = []
-    for i in range(len(spaces)-1):
-        Vsrc = spaces[i]
-        Vdest = spaces[i+1]
-        Iexpr = interpolate(TrialFunction(Vsrc), Vdest)
-        op = assemble(Iexpr, mat_type="aij")
-        ops.append(op)
-    return ops