diff --git a/firedrake/mesh.py b/firedrake/mesh.py
index c8cd4fd0f4..27b9585fc3 100644
--- a/firedrake/mesh.py
+++ b/firedrake/mesh.py
@@ -1431,6 +1431,341 @@ def cell_set(self):
         size = list(self._entity_classes[self.cell_dimension(), :])
         return op2.Set(size, "Cells", comm=self.comm)
 
+    @cached_property
+    def cell_facet_neighbours(self):
+        """Map each cell to its neighbouring cells across local facets.
+
+        Returns a :class:`pyop2.types.dat.Dat` with one entry per local facet of
+        each cell.  For cell ``c`` and local facet ``i``,
+        ``cell_facet_neighbours[c][i]`` is the Firedrake cell number of the cell
+        adjacent to ``c`` across that facet, or ``-1`` if the facet lies on the
+        boundary.
+        """
+        num_facets = self.ufl_cell().num_facets
+
+        dset = op2.DataSet(self.cell_set, dim=num_facets)
+
+        # Create a local numpy buffer to store the neighbours of each cell
+        cell_neighbours = np.full((self.num_cells(), num_facets), -1, dtype=IntType)
+
+        # Populate the local buffer by iterating over the cells and querying the DMPlex for local cell information
+        plex = self.topology_dm
+        cStart, cEnd = plex.getHeightStratum(0)  # range of DMPlex point numbers representing cells
+
+        for c_plex_point in range(cStart, cEnd):
+            cid = self._cell_numbering.getOffset(c_plex_point)  # get Firedrake cell ID of plex cell point number
+
+            # Instead of iterating over the plex facet points, iterate over FIAT's local facet IDs
+            for lf in range(num_facets):
+                lf_plex_point = self._cell_facet_point(cid, lf)  # get plex point number corresponding to facet lf
+                support = plex.getSupport(lf_plex_point)
+                if len(support) == 2:
+                    # an interior facet has 2 adjacent cells
+                    # so the neighbouring cell corresponds to the adjacent cell that's not the current cell
+                    other_c_plex_point = support[0] if support[1] == c_plex_point else support[1]
+                    cell_neighbours[cid, lf] = self._cell_numbering.getOffset(other_c_plex_point)
+
+        cell_facet_neighbours = op2.Dat(
+            dset,
+            cell_neighbours,
+            dtype=IntType,
+            name="cell-facet-neighbours-dat"
+        )
+
+        # NOTE: A fresh Dat has its halo entries marked as invalid -> next with_halos access can trigger a collective exchange.
+        # An exchange can overrite a rank's data with the owning rank's local data which is wrong.
+        # We avoid that by marking halos as valid.
+        cell_facet_neighbours.halo_valid = True
+
+        return cell_facet_neighbours
+
+    # helper methods
+
+    @cached_property
+    def _cell_closure_facet_offset(self):
+        """Returns the correct column in cell_closure for a given local facet."""
+        facet_dim = self.cell_dimension() - 1
+        topology = FIAT.ufc_cell(self.ufl_cell()).get_topology()
+        return sum(len(topology[d]) for d in range(facet_dim))
+
+    def _cell_facet_point(self, cell, local_facet):
+        """Returns the DMPlex point for a local facet of a cell."""
+        return self.cell_closure[cell, self._cell_closure_facet_offset + local_facet]
+
+    def _get_facet_embedding_maps(self):
+        """
+        For every local facet `lf` of the reference cell, FIAT provides an entity transform
+        that maps facet-local reference coordinates (X_lf) into the cell's reference coordinates (X):
+
+        X = A_lf @ X_lf + b_lf.
+
+        Returns
+        -------
+        tuple of lists (A, b, A_inv) each indexed by local facet ID.
+            (A[lf], b[lf]) defines the affine embedding of facet reference coordinates into the cell reference frame,
+            and A_inv[lf] defines the pseudoinverse for mapping cell coordinates on the facet back to facet-local coordinates.
+        """
+        import FIAT
+        ref_cell = FIAT.ufc_cell(self.ufl_cell())
+        facet_dim = ref_cell.get_spatial_dimension() - 1
+        num_facets = len(ref_cell.get_topology()[facet_dim])
+
+        A = []
+        b = []
+        A_inv = []
+
+        if facet_dim == 0:
+            vertices = np.asarray(ref_cell.get_vertices())
+            for lf in range(num_facets):
+                A_lf = np.empty((ref_cell.get_spatial_dimension(), 0), dtype=RealType)
+                b_lf = np.asarray(vertices[lf], dtype=RealType)
+                A.append(A_lf)
+                b.append(b_lf)
+                A_inv.append(np.empty((0, ref_cell.get_spatial_dimension()), dtype=RealType))
+            return A, b, A_inv
+
+        for lf in range(num_facets):
+            phi = ref_cell.get_entity_transform(facet_dim, lf)
+            facet_on_ref_cell = ref_cell.construct_subelement(facet_dim)
+            facet_verts = facet_on_ref_cell.get_vertices()
+
+            # Get (A_lf, b_lf) by evaluating the transform at the facet vertices
+            mapped_facet_verts = np.array([phi(v) for v in facet_verts])
+            b_lf = mapped_facet_verts[0]  # first vertex is always the origin
+            A_lf = (mapped_facet_verts[1:facet_dim+1] - b_lf).T
+            A.append(A_lf)
+            b.append(b_lf)
+            A_inv.append(np.linalg.pinv(A_lf))
+
+        return (A, b, A_inv)
+
+    def _get_facet_orientation_coord_maps(self):
+        """
+        When two adjacent cells share a facet, the ordering of the facet
+        vertices (and hence facet-local reference coordinates) may differ between
+        the two cells. This difference is encoded in the local facet orientation.
+
+        This method constructs, for every possible facet orientation `o`, an affine
+        map acting on facet-local reference coordinates:
+
+        X_f_o = Q_o @ X_f_can + t_o,
+
+        where X_f_can defines the facet-local coordinates when the facet is in the canonical orientation and
+        X_f_o defines the facet-local coordinates of the same point when the facet is in orientation `o`.
+
+        The FIAT `make_entity_permutations_simplex` method can be used to derive vertex permutations
+        for each possible facet permutation. We convert each such permutation into the corresponding affine coordinate transform using
+        `FIAT.reference_element.make_affine_mapping`.
+
+        For tensor-product facets (e.g. quadrilateral facets of a hexahedra), FIAT
+        returns orientation keys as tuples (eo, i0, ..., ik), which are converted
+        into Firedrake's integer orientation before storing.
+
+        Returns
+        -------
+        dict
+            Mapping from facet orientation integer to (Q, t), where Q is the matrix
+            and t is the translation vector acting on facet-local coordinates.
+        """
+        # First, identify facet type (simplex or tensor-product)
+        import FIAT
+        ref_cell = FIAT.ufc_cell(self.ufl_cell())
+        facet_dim = ref_cell.get_spatial_dimension() - 1
+
+        if facet_dim == 0:
+            return {
+                0: (
+                    np.empty((0, 0), dtype=RealType),
+                    np.empty((0,), dtype=RealType),
+                )
+            }
+
+        facet_ref = ref_cell.construct_subelement(facet_dim)
+        facet_verts = np.asarray(facet_ref.get_vertices())  # FIAT's canonical facet vertices
+
+        # Second, build orientation -> vertex permutation map
+        if isinstance(facet_ref, FIAT.reference_element.SimplicialComplex):
+            # 2D: quads and triangles (facet is an interval)
+            # 3D: tetrahedra (facet is a triangle)
+            o_p_maps = FIAT.orientation_utils.make_entity_permutations_simplex(dim=facet_dim, npoints=2)
+
+        elif isinstance(facet_ref, FIAT.reference_element.Hypercube):
+            # 3D: hexahedra (facet is a quad)
+            factors = facet_ref.product.cells  # interval in a quad
+            factor_dims = [f.get_spatial_dimension() for f in factors]
+
+            # NOTE: We assume below that all factors are identical (and have the same dim),
+            # otherwise, we need to build factor permutations for each possible dim.
+            same_factor_dims = all(d == factor_dims[0] for d in factor_dims)
+            if same_factor_dims:
+                factor_o_p_maps = FIAT.orientation_utils.make_entity_permutations_simplex(dim=factor_dims[0], npoints=2)
+                o_p_maps = FIAT.orientation_utils.make_entity_permutations_tensorproduct(factors, factor_dims, list([factor_o_p_maps]*len(factors)))
+        else:
+            raise NotImplementedError("Facet permutation maps not yet implemented for facet type %s" % type(facet_ref))
+
+        # Finally, convert each vertex permutation into an affine map
+        o_coord_maps = {}
+        for o, p in o_p_maps.items():
+            permuted_facet_verts = facet_verts[p]
+            if facet_verts.shape[0] == facet_dim + 1:
+                # Simplex facet: use all vertices
+                xs = facet_verts
+                ys = permuted_facet_verts
+                o_int = o
+            else:
+                # Tensor-product facet: use a fixed simplex subset
+                subset = [0, 1, 2]  # any non-collinear triple works on a non-degenerate quad
+                xs = facet_verts[subset]
+                ys = permuted_facet_verts[subset]
+
+                # For a tensor-product facet FIAT returns orientation as a tuple (eo, io1, io2..., iok)
+                # so we need to convert it to an integer (facet orientations stored on the mesh are expressed as integers).
+                # The integer orientation is obtained as o_int = (2**dim) * eo + io (as written in cython.dmcommon.pyx _compute_orientation_interval_tensor_product)
+                # intrinsic orientation is a binary encoding (each factor contributes one bit)
+                # io is obtained by converting this encoding into an integer (with factor 0 being the most significant bit)
+
+                eo = o[0]
+                bits = o[1:]
+                io = 0
+                for bit in bits:
+                    io = 2*io + bit
+                o_int = (2**len(bits))*eo + io
+
+            Q, t = FIAT.reference_element.make_affine_mapping(xs, ys)
+            o_coord_maps[o_int] = (Q, t)
+
+        return o_coord_maps
+
+    @cached_property
+    def cell_facet_coord_transforms(self):
+        """Returns affine reference-coordinate transforms between neighbouring cells across a given facet.
+
+        For each cell `c` and each local facet `i` of this cell, this method constructs the affine
+        map
+
+            X' = A[c,i] @ X + b[c,i],
+
+        where `X` are reference coordinates of a point lying on facet `i` in cell `c`
+        and `X'` are the reference coordinates of the same point expressed in the neighbouring cell
+        across facet `i`.
+
+        Returns
+        -------
+        tuple
+            (A_dat, b_dat) indexed by global cell ID and local facet ID in each cell.
+            The two `pyop2.Dat` objects store the matrix and translation vector of
+            each cell-to-cell reference coordinate transforms.
+        """
+        num_facets = self.ufl_cell().num_facets
+        ref_dim = self.ufl_cell().topological_dimension
+        A_dset = op2.DataSet(self.cell_set, dim=(num_facets, ref_dim, ref_dim))
+        b_dset = op2.DataSet(self.cell_set, dim=(num_facets, ref_dim))
+
+        # Create local numpy buffers
+        A_transform = np.full((self.num_cells(), num_facets, ref_dim, ref_dim), np.nan)
+        b_transform = np.full((self.num_cells(), num_facets, ref_dim), np.nan)
+
+        # Populate the local buffers by iterating over interior facets
+        # and extracting cell adjacency information from the mesh topology directly
+        facet_cells = self.interior_facets.facet_cell  # all local facets (owned + halos)
+        local_facets = self.interior_facets.local_facet_dat.data_ro_with_halos
+
+        local_facet_orientations = self.interior_facets.local_facet_orientation_dat.data_ro_with_halos
+        A, b, A_inv = self._get_facet_embedding_maps()
+        o_coord_maps = self._get_facet_orientation_coord_maps()
+
+        NODATAVAL = np.iinfo(local_facet_orientations.dtype).max
+
+        for f_idx in range(facet_cells.shape[0]):
+            c0, c1 = facet_cells[f_idx]  # adjacent cells
+            lf0, lf1 = local_facets[f_idx]  # local facet ID in each adjacent cell
+            o0, o1 = local_facet_orientations[f_idx]  # local orientation in each adjacent cell
+            if o0 == NODATAVAL or o1 == NODATAVAL:
+                # skip outer halo facets where adjacency data isn't available
+                continue
+
+            # Compute coords. map from o0 -> o1
+            Q0, t0 = o_coord_maps[o0]  # map canonical orientation -> o0
+            Q1, t1 = o_coord_maps[o1]  # map canonical orientation -> o1
+
+            # inverse coord. map
+            # o0 -> canonical orientation
+            Q0_inv = np.linalg.inv(Q0)
+            t0_inv = -Q0_inv @ t0
+
+            # o1 -> canonical orientation
+            Q1_inv = np.linalg.inv(Q1)
+            t1_inv = -Q1_inv @ t1
+
+            # compose maps both ways
+            # o0 -> canonical orientation with canonical orientation -> o1
+            Q01 = Q1 @ Q0_inv
+            t01 = Q1 @ t0_inv + t1
+
+            # o1 -> canonical orientation with canonical orientation -> o0
+            Q10 = Q0 @ Q1_inv
+            t10 = Q0 @ t1_inv + t0
+
+            # Compute forward transform c0 -> c1 and store on c0
+            A_transform[c0, lf0] = A[lf1] @ Q01 @ A_inv[lf0]
+            b_transform[c0, lf0] = b[lf1] + A[lf1] @ t01 - A_transform[c0, lf0]  @ b[lf0]
+
+            # Compute backward transform c1 -> c0 and store on c1
+            A_transform[c1, lf1] = A[lf0] @ Q10 @ A_inv[lf1]
+            b_transform[c1, lf1] = b[lf0] + A[lf0] @ t10 - A_transform[c1, lf1] @ b[lf1]
+
+        cell_facet_A = op2.Dat(
+            A_dset,
+            A_transform,
+            dtype=RealType,
+            name="cell-facet-coord-transforms-A-dat"
+        )
+
+        cell_facet_A.halo_valid = True
+
+        cell_facet_b = op2.Dat(
+            b_dset,
+            b_transform,
+            dtype=RealType,
+            name="cell-facet-coord-transforms-b-dat"
+        )
+        cell_facet_b.halo_valid = True
+
+        return cell_facet_A, cell_facet_b
+
+    @cached_property
+    def cell_facet_exterior_mask(self):
+        """Identify exterior facets of each local cell, including halo cells.
+
+        Returns a boolean array of shape ``(num_cells_with_halos, num_facets)``.
+        The entry ``cell_facet_exterior_mask[c, i]`` is ``True`` if local facet
+        ``i`` of local cell ``c`` lies on the domain boundary, and ``False``
+        otherwise.
+        """
+        num_facets = self.ufl_cell().num_facets
+        mask = np.zeros((self.cell_set.total_size, num_facets), dtype=bool)
+
+        plex = self.topology_dm
+        if plex.getStratumSize("exterior_facets", 1) > 0:
+            ext_plex_points = frozenset(
+                plex.getStratumIS("exterior_facets", 1).getIndices().tolist()
+            )
+        else:
+            ext_plex_points = frozenset()
+
+        cstart, cend = plex.getHeightStratum(0)
+        for c_plex in range(cstart, cend):
+            cid = self._cell_numbering.getOffset(c_plex)
+
+            if cid < 0 or cid >= self.cell_set.total_size:
+                continue
+
+            for lf in range(num_facets):
+                f_point = self._cell_facet_point(cid, lf)
+                mask[cid, lf] = f_point in ext_plex_points
+
+        return mask
+
     @PETSc.Log.EventDecorator()
     def _set_partitioner(self, plex, distribute, partitioner_type=None):
         """Set partitioner for (re)distributing underlying plex over comm.
@@ -2414,6 +2749,8 @@ def __init__(self, coordinates):
         V = functionspaceimpl.WithGeometry(coordinates.function_space(), self)
         self._coordinates_function = function.Function(V, val=coordinates)
 
+        self._topology_version = 0
+
     def _ufl_signature_data_(self, *args, **kwargs):
         return (type(self), self.extruded, self.variable_layers,
                 super()._ufl_signature_data_(*args, **kwargs))
diff --git a/tests/firedrake/regression/test_cell_facet_topology.py b/tests/firedrake/regression/test_cell_facet_topology.py
new file mode 100644
index 0000000000..0d1fe8c492
--- /dev/null
+++ b/tests/firedrake/regression/test_cell_facet_topology.py
@@ -0,0 +1,211 @@
+import numpy as np
+import pytest
+
+from firedrake import *
+
+
+@pytest.fixture(params=[
+    # No interior facets
+    pytest.param(lambda: UnitIntervalMesh(1), id="interval-1"),
+    pytest.param(lambda: UnitSquareMesh(1, 1), id="tri-square-1x1"),
+    pytest.param(lambda: UnitSquareMesh(1, 1, quadrilateral=True), id="quad-square-1x1"),
+    pytest.param(lambda: UnitCubeMesh(1, 1, 1), id="tet-cube-1x1x1"),
+    pytest.param(lambda: UnitCubeMesh(1, 1, 1, hexahedral=True), id="hex-cube-1x1x1"),
+
+    # With interior facets
+    pytest.param(lambda: UnitIntervalMesh(2), id="interval-2"),
+    pytest.param(lambda: UnitSquareMesh(2, 1), id="tri-square-2x1"),
+    pytest.param(lambda: UnitSquareMesh(2, 1, quadrilateral=True), id="quad-square-2x1"),
+    pytest.param(lambda: UnitCubeMesh(2, 1, 1), id="tet-cube-2x1x1"),
+    pytest.param(lambda: UnitCubeMesh(2, 1, 1, hexahedral=True), id="hex-cube-2x1x1"),
+])
+def mesh(request):
+    return request.param()
+
+
+@pytest.mark.parallel([1, 3])
+def test_cell_facet_neighbours_are_valid(mesh):
+    topology = mesh.topology
+
+    # Check that every cell has exactly one neighbour entry per local facet
+    owned_neighbours = topology.cell_facet_neighbours.data_ro
+    neighbours_with_halos = topology.cell_facet_neighbours.data_ro_with_halos
+
+    assert owned_neighbours.shape == (
+        topology.cell_set.size,
+        mesh.ufl_cell().num_facets,
+    )
+
+    assert neighbours_with_halos.shape == (
+        topology.cell_set.total_size,
+        mesh.ufl_cell().num_facets,
+    )
+
+    for c, row in enumerate(owned_neighbours):
+        for n in row:
+            if n == -1:
+                continue
+
+            # Check that every non-boundary neighbour has a valid cell number
+            assert 0 <= n < topology.cell_set.total_size
+
+            # Check reciprocity: If c is the neighbour of n across a given facet,
+            # then n also appears as the neighbour of c
+            assert c in neighbours_with_halos[n]
+
+
+@pytest.mark.parallel([1, 3])
+def test_cell_facet_neighbours_match_interior_facets(mesh):
+    topology = mesh.topology
+
+    neighbours = topology.cell_facet_neighbours.data_ro_with_halos
+
+    # Test only with owned facets while allowing either adjacent cell to be halo
+    nowned_facets = topology.interior_facets.set.size
+    facet_cells = topology.interior_facets.facet_cell[:nowned_facets]
+    local_facets = topology.interior_facets.local_facet_dat.data_ro[:nowned_facets]
+
+    # Check that the neighbouring cell is attached to the right facet
+    for (c0, c1), (lf0, lf1) in zip(facet_cells, local_facets):
+        if c0 == -1 or c1 == -1:
+            # Skip interior facets for which the rank doesn't know both adjacent cells
+            continue
+
+        assert 0 <= c0 < topology.cell_set.total_size
+        assert 0 <= c1 < topology.cell_set.total_size
+
+        assert neighbours[c0, lf0] == c1
+        assert neighbours[c1, lf1] == c0
+
+        f0 = topology._cell_facet_point(c0, lf0)
+        f1 = topology._cell_facet_point(c1, lf1)
+
+        assert f0 == f1
+
+
+@pytest.mark.parallel([1, 3])
+def test_cell_facet_exterior_mask(mesh):
+    topology = mesh.topology
+
+    mask = topology.cell_facet_exterior_mask
+
+    expected = np.zeros(
+        (topology.cell_set.total_size, mesh.ufl_cell().num_facets),
+        dtype=bool,
+    )
+
+    facet_cells = np.asarray(topology.exterior_facets.facet_cell).reshape((-1, 1))
+    local_facets = np.asarray(
+        topology.exterior_facets.local_facet_dat.data_ro_with_halos
+    ).reshape((-1, 1))
+
+    for (c,), (lf,) in zip(facet_cells, local_facets):
+        if c == -1:
+            continue
+        assert 0 <= c < topology.cell_set.total_size
+        expected[c, lf] = True
+
+    assert mask.shape == expected.shape
+    assert np.array_equal(mask, expected)
+
+
+@pytest.mark.parallel([1, 3])
+def test_cell_facet_coord_transforms_are_inverse_on_interior_facets(mesh):
+    """Checks that the forward and backward transforms are consistent for two adjacent cells over a fixed shared facet."""
+    topology = mesh.topology
+
+    A_dat, b_dat = topology.cell_facet_coord_transforms
+    A = A_dat.data_ro_with_halos
+    b = b_dat.data_ro_with_halos
+
+    embed_A, embed_b, _ = topology._get_facet_embedding_maps()
+
+    # Restrict to owned interior facets
+    nowned_facets = topology.interior_facets.set.size
+    facet_cells = topology.interior_facets.facet_cell[:nowned_facets]
+    local_facets = topology.interior_facets.local_facet_dat.data_ro[:nowned_facets]
+
+    for (c0, c1), (lf0, lf1) in zip(facet_cells, local_facets):
+        if c0 == -1 or c1 == -1:
+            continue
+
+        assert 0 <= c0 < topology.cell_set.total_size
+        assert 0 <= c1 < topology.cell_set.total_size
+
+        assert np.isfinite(A[c0, lf0]).all()
+        assert np.isfinite(b[c0, lf0]).all()
+        assert np.isfinite(A[c1, lf1]).all()
+        assert np.isfinite(b[c1, lf1]).all()
+
+        facet_dim = embed_A[lf0].shape[1]
+        points = [np.zeros(facet_dim)]
+        points += [np.eye(facet_dim)[i] for i in range(facet_dim)]
+
+        for Xf in points:
+            x = embed_A[lf0] @ Xf + embed_b[lf0]
+            y = A[c0, lf0] @ x + b[c0, lf0]
+            x_back = A[c1, lf1] @ y + b[c1, lf1]
+
+            assert np.allclose(x_back, x)
+
+
+@pytest.mark.parallel([1, 3])
+def test_cell_facet_coord_transforms_map_to_neighbour_facet(mesh):
+    """Checks geometric consistency, that is the linear transform for a cell c on local facet lf
+    sends points on that facet to points on the corresponding facet of the neighbouring cell."""
+    topology = mesh.topology
+
+    A_dat, b_dat = topology.cell_facet_coord_transforms
+    A = A_dat.data_ro_with_halos
+    b = b_dat.data_ro_with_halos
+
+    embed_A, embed_b, _ = topology._get_facet_embedding_maps()
+
+    # Restrict to owned interior facets
+    nowned_facets = topology.interior_facets.set.size
+    facet_cells = topology.interior_facets.facet_cell[:nowned_facets]
+    local_facets = topology.interior_facets.local_facet_dat.data_ro[:nowned_facets]
+
+    for (c0, c1), (lf0, lf1) in zip(facet_cells, local_facets):
+        if c0 == -1 or c1 == -1:
+            continue
+
+        assert 0 <= c0 < topology.cell_set.total_size
+        assert 0 <= c1 < topology.cell_set.total_size
+
+        facet_dim = embed_A[lf0].shape[1]
+        points = [np.zeros(facet_dim)]
+        points += [np.eye(facet_dim)[i] for i in range(facet_dim)]
+
+        for Xf in points:
+            X = embed_A[lf0] @ Xf + embed_b[lf0]
+            Y = A[c0, lf0] @ X + b[c0, lf0]
+
+            # Y should lie on neighbour local facet lf1.
+            Yf = np.linalg.pinv(embed_A[lf1]) @ (Y - embed_b[lf1])
+            assert np.allclose(embed_A[lf1] @ Yf + embed_b[lf1], Y)
+
+        facet_dim = embed_A[lf1].shape[1]
+        points = [np.zeros(facet_dim)]
+        points += [np.eye(facet_dim)[i] for i in range(facet_dim)]
+
+        for Yf in points:
+            Y = embed_A[lf1] @ Yf + embed_b[lf1]
+            X = A[c1, lf1] @ Y + b[c1, lf1]
+
+            Xf = np.linalg.pinv(embed_A[lf0]) @ (X - embed_b[lf0])
+            assert np.allclose(embed_A[lf0] @ Xf + embed_b[lf0], X)
+
+
+@pytest.mark.parallel([1, 3])
+def test_cell_facet_topology_parallel_smoke():
+    mesh = UnitSquareMesh(4, 4)
+    topology = mesh.topology
+
+    topology.cell_facet_neighbours.data_ro_with_halos
+
+    A, b = topology.cell_facet_coord_transforms
+    A.data_ro_with_halos
+    b.data_ro_with_halos
+
+    topology.cell_facet_exterior_mask