diff --git a/firedrake/bcs.py b/firedrake/bcs.py
index 6cd6f3d5d7..0d16b0f7cb 100644
--- a/firedrake/bcs.py
+++ b/firedrake/bcs.py
@@ -319,6 +319,8 @@ def reconstruct(self, field=None, V=None, g=None, sub_domain=None, use_split=Fal
             V = V.sub(index)
         if g is None:
             g = self._original_arg
+            if isinstance(g, firedrake.Function) and g.function_space() != V:
+                g = firedrake.Function(V).interpolate(g)
         if sub_domain is None:
             sub_domain = self.sub_domain
         if field is not None:
@@ -739,11 +741,11 @@ def restricted_function_space(V, ids):
         return V
 
     assert len(ids) == len(V)
-    spaces = [Vsub if len(boundary_set) == 0 else
-              firedrake.RestrictedFunctionSpace(Vsub, boundary_set=boundary_set)
-              for Vsub, boundary_set in zip(V, ids)]
+    spaces = [V_ if len(boundary_set) == 0 else
+              firedrake.RestrictedFunctionSpace(V_, boundary_set=boundary_set, name=V_.name)
+              for V_, boundary_set in zip(V, ids)]
 
     if len(spaces) == 1:
         return spaces[0]
     else:
-        return firedrake.MixedFunctionSpace(spaces)
+        return firedrake.MixedFunctionSpace(spaces, name=V.name)
diff --git a/firedrake/dmhooks.py b/firedrake/dmhooks.py
index c1cca65b56..07b334b111 100644
--- a/firedrake/dmhooks.py
+++ b/firedrake/dmhooks.py
@@ -53,11 +53,13 @@ def get_function_space(dm):
     :raises RuntimeError: if no function space was found.
     """
     info = dm.getAttr("__fs_info__")
-    meshref, element, indices, (name, names) = info
+    meshref, element, indices, (name, names), boundary_sets = info
     mesh = meshref()
     if mesh is None:
         raise RuntimeError("Somehow your mesh was collected, this should never happen")
     V = firedrake.FunctionSpace(mesh, element, name=name)
+    if any(boundary_sets):
+        V = firedrake.bcs.restricted_function_space(V, boundary_sets)
     if len(V) > 1:
         for V_, name in zip(V, names):
             V_.topological.name = name
@@ -93,8 +95,8 @@ def set_function_space(dm, V):
     if len(V) > 1:
         names = tuple(V_.name for V_ in V)
     element = V.ufl_element()
-
-    info = (weakref.ref(mesh), element, tuple(reversed(indices)), (V.name, names))
+    boundary_sets = tuple(V_.boundary_set for V_ in V)
+    info = (weakref.ref(mesh), element, tuple(reversed(indices)), (V.name, names), boundary_sets)
     dm.setAttr("__fs_info__", info)
 
 
@@ -457,7 +459,7 @@ def refine(dm, comm):
     if hasattr(V, "_fine"):
         fdm = V._fine.dm
     else:
-        V._fine = firedrake.FunctionSpace(hierarchy[level + 1], V.ufl_element())
+        V._fine = V.reconstruct(mesh=hierarchy[level + 1])
         fdm = V._fine.dm
     V._fine._coarse = V
     return fdm
diff --git a/firedrake/functionspaceimpl.py b/firedrake/functionspaceimpl.py
index cec9dd860c..60d518f884 100644
--- a/firedrake/functionspaceimpl.py
+++ b/firedrake/functionspaceimpl.py
@@ -378,10 +378,13 @@ def make_function_space(cls, mesh, element, name=None):
             new = cls.create(new, mesh)
         return new
 
-    def reconstruct(self, mesh=None, name=None, **kwargs):
+    def reconstruct(self, mesh=None, element=None, boundary_set=None, name=None, **kwargs):
         r"""Reconstruct this :class:`.WithGeometryBase` .
 
         :kwarg mesh: the new :func:`~.Mesh` (defaults to same mesh)
+        :kwarg element: the new :class:`finat.ufl.FiniteElement` (defaults to same element)
+        :kwarg boundary_set: boundary subdomain labels defining a new
+                             :func:`~.RestrictedFunctionSpace` (defaults to same boundary_set)
         :kwarg name: the new name (defaults to None)
         :returns: the new function space of the same class as ``self``.
 
@@ -389,6 +392,9 @@ def reconstruct(self, mesh=None, name=None, **kwargs):
         For details see :meth:`finat.ufl.finiteelement.FiniteElement.reconstruct`.
         """
         V_parent = self
+        if boundary_set is None:
+            boundary_set = V_parent.boundary_set
+
         # Deal with ProxyFunctionSpace
         indices = []
         while True:
@@ -403,8 +409,9 @@ def reconstruct(self, mesh=None, name=None, **kwargs):
 
         if mesh is None:
             mesh = V_parent.mesh()
+        if element is None:
+            element = V_parent.ufl_element()
 
-        element = V_parent.ufl_element()
         cell = mesh.topology.ufl_cell()
         if len(kwargs) > 0 or element.cell != cell:
             element = element.reconstruct(cell=cell, **kwargs)
@@ -412,6 +419,9 @@ def reconstruct(self, mesh=None, name=None, **kwargs):
         V = type(self).make_function_space(mesh, element, name=name)
         for i in reversed(indices):
             V = V.sub(i)
+
+        if boundary_set:
+            V = RestrictedFunctionSpace(V, boundary_set=boundary_set, name=V.name)
         return V
 
 
@@ -876,6 +886,14 @@ class RestrictedFunctionSpace(FunctionSpace):
     If using this class to solve or similar, a list of DirichletBCs will still
     need to be specified on this space and passed into the function.
     """
+    def __new__(cls, function_space, boundary_set=frozenset(), name=None):
+        mesh = function_space.mesh()
+        if mesh is not mesh.topology:
+            V = RestrictedFunctionSpace(function_space.topological,
+                                        boundary_set=boundary_set, name=name)
+            return type(function_space).create(V, mesh)
+        return FunctionSpace.__new__(cls)
+
     def __init__(self, function_space, boundary_set=frozenset(), name=None):
         label = ""
         boundary_set_ = []
@@ -901,8 +919,7 @@ def __init__(self, function_space, boundary_set=frozenset(), name=None):
                                                      function_space.ufl_element(),
                                                      label=self._label)
         self.function_space = function_space
-        self.name = name or (function_space.name or "Restricted" + "_"
-                             + "_".join(sorted(map(str, self.boundary_set))))
+        self.name = name or function_space.name
 
     def set_shared_data(self):
         sdata = get_shared_data(self._mesh, self.ufl_element(), self.boundary_set)
@@ -1200,7 +1217,7 @@ class ProxyFunctionSpace(FunctionSpace):
     """
     def __new__(cls, mesh, element, name=None):
         topology = mesh.topology
-        self = super(ProxyFunctionSpace, cls).__new__(cls)
+        self = FunctionSpace.__new__(cls)
         if mesh is not topology:
             return WithGeometry.create(self, mesh)
         else:
@@ -1255,7 +1272,7 @@ class ProxyRestrictedFunctionSpace(RestrictedFunctionSpace):
     """
     def __new__(cls, function_space, boundary_set=frozenset(), name=None):
         topology = function_space._mesh.topology
-        self = super(ProxyRestrictedFunctionSpace, cls).__new__(cls)
+        self = FunctionSpace.__new__(cls)
         if function_space._mesh is not topology:
             return WithGeometry.create(self, function_space._mesh)
         else:
diff --git a/firedrake/mg/embedded.py b/firedrake/mg/embedded.py
index 551c09a7e2..17b8af37eb 100644
--- a/firedrake/mg/embedded.py
+++ b/firedrake/mg/embedded.py
@@ -90,6 +90,9 @@ def cache(self, V):
         except KeyError:
             return self.caches.setdefault(key, TransferManager.Cache(*key))
 
+    def cache_key(self, V):
+        return (V.dim(), tuple(V.boundary_set))
+
     def V_dof_weights(self, V):
         """Dof weights for averaging projection.
 
@@ -97,7 +100,7 @@ def V_dof_weights(self, V):
         :returns: A PETSc Vec.
         """
         cache = self.cache(V)
-        key = V.dim()
+        key = self.cache_key(V)
         try:
             return cache._V_dof_weights[key]
         except KeyError:
@@ -122,7 +125,7 @@ def V_DG_mass(self, V, DG):
         :returns: A PETSc Mat mapping from V -> DG
         """
         cache = self.cache(V)
-        key = V.dim()
+        key = self.cache_key(V)
         try:
             return cache._V_DG_mass[key]
         except KeyError:
@@ -153,7 +156,7 @@ def V_approx_inv_mass(self, V, DG):
         :returns: A PETSc Mat mapping from V -> DG.
         """
         cache = self.cache(V)
-        key = V.dim()
+        key = self.cache_key(V)
         try:
             return cache._V_approx_inv_mass[key]
         except KeyError:
@@ -171,7 +174,7 @@ def V_inv_mass_ksp(self, V):
         :returns: A PETSc KSP for inverting (V, V).
         """
         cache = self.cache(V)
-        key = V.dim()
+        key = self.cache_key(V)
         try:
             return cache._V_inv_mass_ksp[key]
         except KeyError:
@@ -193,7 +196,7 @@ def DG_work(self, V):
         """
         needs_dual = ufl.duals.is_dual(V)
         cache = self.cache(V)
-        key = (V.dim(), needs_dual)
+        key = self.cache_key(V) + (needs_dual,)
         try:
             return cache._DG_work[key]
         except KeyError:
@@ -210,7 +213,7 @@ def work_vec(self, V):
         :returns: A PETSc Vec for V.
         """
         cache = self.cache(V)
-        key = V.dim()
+        key = self.cache_key(V)
         try:
             return cache._work_vec[key]
         except KeyError:
diff --git a/firedrake/mg/interface.py b/firedrake/mg/interface.py
index f962253db3..22faeb3e8d 100644
--- a/firedrake/mg/interface.py
+++ b/firedrake/mg/interface.py
@@ -58,7 +58,6 @@ def prolong(coarse, fine):
     repeat = (fine_level - coarse_level)*refinements_per_level
     next_level = coarse_level * refinements_per_level
 
-    element = Vc.ufl_element()
     meshes = hierarchy._meshes
     for j in range(repeat):
         next_level += 1
@@ -66,10 +65,10 @@ def prolong(coarse, fine):
             next = fine
             Vf = fine.function_space()
         else:
-            Vf = firedrake.FunctionSpace(meshes[next_level], element)
+            Vf = Vc.reconstruct(mesh=meshes[next_level])
             next = firedrake.Function(Vf)
 
-        coarse_coords = Vc.mesh().coordinates
+        coarse_coords = get_coordinates(Vc)
         fine_to_coarse = utils.fine_node_to_coarse_node_map(Vf, Vc)
         fine_to_coarse_coords = utils.fine_node_to_coarse_node_map(Vf, coarse_coords.function_space())
         kernel = kernels.prolong_kernel(coarse)
@@ -119,7 +118,6 @@ def restrict(fine_dual, coarse_dual):
     repeat = (fine_level - coarse_level)*refinements_per_level
     next_level = fine_level * refinements_per_level
 
-    element = Vc.ufl_element()
     meshes = hierarchy._meshes
 
     for j in range(repeat):
@@ -128,15 +126,15 @@ def restrict(fine_dual, coarse_dual):
             coarse_dual.dat.zero()
             next = coarse_dual
         else:
-            Vc = firedrake.FunctionSpace(meshes[next_level], element)
-            next = firedrake.Cofunction(Vc.dual())
+            Vc = Vf.reconstruct(mesh=meshes[next_level])
+            next = firedrake.Cofunction(Vc)
         Vc = next.function_space()
         # XXX: Should be able to figure out locations by pushing forward
         # reference cell node locations to physical space.
         # x = \sum_i c_i \phi_i(x_hat)
-        node_locations = utils.physical_node_locations(Vf)
+        node_locations = utils.physical_node_locations(Vf.dual())
 
-        coarse_coords = Vc.mesh().coordinates
+        coarse_coords = get_coordinates(Vc.dual())
         fine_to_coarse = utils.fine_node_to_coarse_node_map(Vf, Vc)
         fine_to_coarse_coords = utils.fine_node_to_coarse_node_map(Vf, coarse_coords.function_space())
         # Have to do this, because the node set core size is not right for
@@ -195,7 +193,6 @@ def inject(fine, coarse):
     repeat = (fine_level - coarse_level)*refinements_per_level
     next_level = fine_level * refinements_per_level
 
-    element = Vc.ufl_element()
     meshes = hierarchy._meshes
 
     for j in range(repeat):
@@ -205,12 +202,12 @@ def inject(fine, coarse):
             next = coarse
             Vc = next.function_space()
         else:
-            Vc = firedrake.FunctionSpace(meshes[next_level], element)
+            Vc = Vf.reconstruct(mesh=meshes[next_level])
             next = firedrake.Function(Vc)
         if not dg:
             node_locations = utils.physical_node_locations(Vc)
 
-            fine_coords = Vf.mesh().coordinates
+            fine_coords = get_coordinates(Vf)
             coarse_node_to_fine_nodes = utils.coarse_node_to_fine_node_map(Vc, Vf)
             coarse_node_to_fine_coords = utils.coarse_node_to_fine_node_map(Vc, fine_coords.function_space())
 
@@ -242,3 +239,11 @@ def inject(fine, coarse):
         fine = next
         Vf = Vc
     return coarse
+
+
+def get_coordinates(V):
+    coords = V.mesh().coordinates
+    if V.boundary_set:
+        W = V.reconstruct(element=coords.function_space().ufl_element())
+        coords = firedrake.Function(W).interpolate(coords)
+    return coords
diff --git a/firedrake/mg/ufl_utils.py b/firedrake/mg/ufl_utils.py
index 9e92159d25..6786ba1f43 100644
--- a/firedrake/mg/ufl_utils.py
+++ b/firedrake/mg/ufl_utils.py
@@ -186,7 +186,7 @@ def inject_on_restrict(fine, restriction, rscale, injection, coarse):
                 if isinstance(g, firedrake.Function) and hasattr(g, "_child"):
                     manager.inject(g, g._child)
 
-    V = problem.u.function_space()
+    V = problem.u_restrict.function_space()
     if not hasattr(V, "_coarse"):
         # The hook is persistent and cumulative, but also problem-independent.
         # Therefore, we are only adding it once.
@@ -201,7 +201,7 @@ def inject_on_restrict(fine, restriction, rscale, injection, coarse):
     for c in coefficients:
         coefficient_mapping[c] = self(c, self, coefficient_mapping=coefficient_mapping)
 
-    u = coefficient_mapping[problem.u]
+    u = coefficient_mapping[problem.u_restrict]
 
     bcs = [self(bc, self) for bc in problem.bcs]
     J = self(problem.J, self, coefficient_mapping=coefficient_mapping)
@@ -277,7 +277,7 @@ def coarsen_snescontext(context, self, coefficient_mapping=None):
         if isinstance(val, (firedrake.Function, firedrake.Cofunction)):
             V = val.function_space()
             coarseneddm = V.dm
-            parentdm = get_parent(context._problem.u.function_space().dm)
+            parentdm = get_parent(context._problem.u_restrict.function_space().dm)
 
             # Now attach the hook to the parent DM
             if get_appctx(coarseneddm) is None:
@@ -369,8 +369,8 @@ def create_interpolation(dmc, dmf):
 
     manager = get_transfer_manager(dmf)
 
-    V_c = cctx._problem.u.function_space()
-    V_f = fctx._problem.u.function_space()
+    V_c = cctx._problem.u_restrict.function_space()
+    V_f = fctx._problem.u_restrict.function_space()
 
     row_size = V_f.dof_dset.layout_vec.getSizes()
     col_size = V_c.dof_dset.layout_vec.getSizes()
@@ -395,8 +395,8 @@ def create_injection(dmc, dmf):
 
     manager = get_transfer_manager(dmf)
 
-    V_c = cctx._problem.u.function_space()
-    V_f = fctx._problem.u.function_space()
+    V_c = cctx._problem.u_restrict.function_space()
+    V_f = fctx._problem.u_restrict.function_space()
 
     row_size = V_f.dof_dset.layout_vec.getSizes()
     col_size = V_c.dof_dset.layout_vec.getSizes()
diff --git a/firedrake/mg/utils.py b/firedrake/mg/utils.py
index 80e5e820f6..025e5e462f 100644
--- a/firedrake/mg/utils.py
+++ b/firedrake/mg/utils.py
@@ -11,7 +11,7 @@
 def fine_node_to_coarse_node_map(Vf, Vc):
     if len(Vf) > 1:
         assert len(Vf) == len(Vc)
-        return op2.MixedMap(fine_node_to_coarse_node_map(f, c) for f, c in zip(Vf, Vc))
+        return op2.MixedMap(map(fine_node_to_coarse_node_map, Vf, Vc))
     mesh = Vf.mesh()
     assert hasattr(mesh, "_shared_data_cache")
     hierarchyf, levelf = get_level(Vf.mesh())
@@ -25,10 +25,7 @@ def fine_node_to_coarse_node_map(Vf, Vc):
     if levelc + increment != levelf:
         raise ValueError("Can't map between level %s and level %s" % (levelc, levelf))
 
-    key = (entity_dofs_key(Vc.finat_element.entity_dofs())
-           + entity_dofs_key(Vf.finat_element.entity_dofs())
-           + (levelc, levelf))
-
+    key = _cache_key(Vc, Vf)
     cache = mesh._shared_data_cache["hierarchy_fine_node_to_coarse_node_map"]
     try:
         return cache[key]
@@ -49,7 +46,7 @@ def fine_node_to_coarse_node_map(Vf, Vc):
 def coarse_node_to_fine_node_map(Vc, Vf):
     if len(Vf) > 1:
         assert len(Vf) == len(Vc)
-        return op2.MixedMap(coarse_node_to_fine_node_map(f, c) for f, c in zip(Vf, Vc))
+        return op2.MixedMap(map(coarse_node_to_fine_node_map, Vf, Vc))
     mesh = Vc.mesh()
     assert hasattr(mesh, "_shared_data_cache")
     hierarchyf, levelf = get_level(Vf.mesh())
@@ -63,10 +60,7 @@ def coarse_node_to_fine_node_map(Vc, Vf):
     if levelc + increment != levelf:
         raise ValueError("Can't map between level %s and level %s" % (levelc, levelf))
 
-    key = (entity_dofs_key(Vc.finat_element.entity_dofs())
-           + entity_dofs_key(Vf.finat_element.entity_dofs())
-           + (levelc, levelf))
-
+    key = _cache_key(Vc, Vf)
     cache = mesh._shared_data_cache["hierarchy_coarse_node_to_fine_node_map"]
     try:
         return cache[key]
@@ -101,7 +95,7 @@ def coarse_cell_to_fine_node_map(Vc, Vf):
     if levelc + increment != levelf:
         raise ValueError("Can't map between level %s and level %s" % (levelc, levelf))
 
-    key = (entity_dofs_key(Vf.finat_element.entity_dofs()) + (levelc, levelf))
+    key = _cache_key(Vc, Vf)
     cache = mesh._shared_data_cache["hierarchy_coarse_cell_to_fine_node_map"]
     try:
         return cache[key]
@@ -142,11 +136,12 @@ def physical_node_locations(V):
     # This is a defaultdict, so the first time we access the key we
     # get a fresh dict for the cache.
     cache = mesh._geometric_shared_data_cache["hierarchy_physical_node_locations"]
-    key = element
+    key = (element, tuple(V.boundary_set))
     try:
         return cache[key]
     except KeyError:
-        Vc = firedrake.VectorFunctionSpace(mesh, element)
+        Vc = V.collapse().reconstruct(element=finat.ufl.VectorElement(element, dim=mesh.geometric_dimension()))
+
         # FIXME: This is unsafe for DG coordinates and CG target spaces.
         locations = firedrake.assemble(firedrake.Interpolate(firedrake.SpatialCoordinate(mesh), Vc))
         return cache.setdefault(key, locations)
@@ -171,3 +166,10 @@ def get_level(obj):
 def has_level(obj):
     """Does the provided object have level info?"""
     return hasattr(obj.topological, "__level_info__")
+
+
+def _cache_key(Vc, Vf):
+    return (entity_dofs_key(Vc.finat_element.entity_dofs())
+            + entity_dofs_key(Vf.finat_element.entity_dofs())
+            + (Vc.dim(), Vf.dim())
+            + (tuple(Vc.boundary_set), tuple(Vf.boundary_set)))
diff --git a/firedrake/preconditioners/bddc.py b/firedrake/preconditioners/bddc.py
index 5fb1fca51a..a39160ac45 100644
--- a/firedrake/preconditioners/bddc.py
+++ b/firedrake/preconditioners/bddc.py
@@ -4,7 +4,6 @@
 from firedrake.petsc import PETSc
 from firedrake.dmhooks import get_function_space, get_appctx
 from firedrake.ufl_expr import TestFunction, TrialFunction
-from firedrake.functionspace import FunctionSpace, VectorFunctionSpace, TensorFunctionSpace
 from ufl import curl, div, HCurl, HDiv, inner, dx
 from pyop2.utils import as_tuple
 import numpy
@@ -88,13 +87,7 @@ def initialize(self, pc):
             if B is None:
                 from firedrake.assemble import assemble
                 d = {HCurl: curl, HDiv: div}[sobolev_space]
-                if V.shape == ():
-                    make_function_space = FunctionSpace
-                elif len(V.shape) == 1:
-                    make_function_space = VectorFunctionSpace
-                else:
-                    make_function_space = TensorFunctionSpace
-                Q = make_function_space(V.mesh(), "DG", degree-1)
+                Q = V.reconstruct(family="DG", degree=degree-1)
                 b = inner(d(TrialFunction(V)), TestFunction(Q)) * dx(degree=2*(degree-1))
                 B = assemble(b, mat_type="matfree")
             bddcpc.setBDDCDivergenceMat(B.petscmat)
@@ -103,7 +96,7 @@ def initialize(self, pc):
             if gradient is None:
                 from firedrake.preconditioners.fdm import tabulate_exterior_derivative
                 from firedrake.preconditioners.hiptmair import curl_to_grad
-                Q = FunctionSpace(V.mesh(), curl_to_grad(V.ufl_element()))
+                Q = V.reconstruct(element=curl_to_grad(V.ufl_element()))
                 gradient = tabulate_exterior_derivative(Q, V)
                 corners = get_vertex_dofs(Q)
                 gradient.compose('_elements_corners', corners)
@@ -135,7 +128,7 @@ def applyTranspose(self, pc, x, y):
 
 
 def get_vertex_dofs(V):
-    W = FunctionSpace(V.mesh(), restrict(V.ufl_element(), "vertex"))
+    W = V.reconstruct(element=restrict(V.ufl_element(), "vertex"))
     indices = get_restriction_indices(V, W)
     V.dof_dset.lgmap.apply(indices, result=indices)
     vertex_dofs = PETSc.IS().createGeneral(indices, comm=V.comm)
diff --git a/firedrake/preconditioners/facet_split.py b/firedrake/preconditioners/facet_split.py
index 4c8e654ce4..5848e0179b 100644
--- a/firedrake/preconditioners/facet_split.py
+++ b/firedrake/preconditioners/facet_split.py
@@ -4,6 +4,7 @@
 from finat.ufl import RestrictedElement, MixedElement, TensorElement, VectorElement
 from firedrake.petsc import PETSc
 from firedrake.preconditioners.base import PCBase
+from firedrake.bcs import restricted_function_space
 import firedrake.dmhooks as dmhooks
 import numpy
 
@@ -58,6 +59,8 @@ def initialize(self, pc):
         element = V.ufl_element()
         elements = [restrict(element, domain) for domain in domains]
         W = FunctionSpace(V.mesh(), elements[0] if len(elements) == 1 else MixedElement(elements))
+        if V.boundary_set:
+            W = restricted_function_space(W, [V.boundary_set]*len(W))
 
         args = (TestFunction(W), TrialFunction(W))
         if len(W) > 1:
diff --git a/firedrake/preconditioners/fdm.py b/firedrake/preconditioners/fdm.py
index 1b6aa9c9aa..a52fba5aae 100644
--- a/firedrake/preconditioners/fdm.py
+++ b/firedrake/preconditioners/fdm.py
@@ -246,7 +246,7 @@ def allocate_matrix(self, Amat, V, J, bcs, fcp, pmat_type, use_static_condensati
         elif len(ifacet) == 1:
             Vfacet = V[ifacet[0]]
             ebig, = set(unrestrict_element(Vsub.ufl_element()) for Vsub in V)
-            Vbig = FunctionSpace(V.mesh(), ebig)
+            Vbig = V.reconstruct(element=ebig)
             if len(V) > 1:
                 dims = [Vsub.finat_element.space_dimension() for Vsub in V]
                 assert sum(dims) == Vbig.finat_element.space_dimension()
@@ -436,9 +436,9 @@ def condense(self, A, J, bcs, fcp, pc_type="icc"):
         V0 = next((Vi for Vi in V if is_restricted(Vi.finat_element)[0]), None)
         V1 = next((Vi for Vi in V if is_restricted(Vi.finat_element)[1]), None)
         if V0 is None:
-            V0 = FunctionSpace(V.mesh(), restrict_element(self.embedding_element, "interior"))
+            V0 = V.reconstruct(element=restrict_element(self.embedding_element, "interior"))
         if V1 is None:
-            V1 = FunctionSpace(V.mesh(), restrict_element(self.embedding_element, "facet"))
+            V1 = V.reconstruct(element=restrict_element(self.embedding_element, "facet"))
         if len(V) == 1:
             J00 = J(*(t.reconstruct(function_space=V0) for t in J.arguments()))
         elif len(V) == 2:
@@ -446,7 +446,7 @@ def condense(self, A, J, bcs, fcp, pc_type="icc"):
             ises = V.dof_dset.field_ises
             Smats[V[0], V[1]] = A.createSubMatrix(ises[0], ises[1])
             Smats[V[1], V[0]] = A.createSubMatrix(ises[1], ises[0])
-            unindexed = {Vsub: FunctionSpace(Vsub.mesh(), Vsub.ufl_element()) for Vsub in V}
+            unindexed = {Vsub: Vsub.collapse() for Vsub in V}
             bcs = tuple(bc.reconstruct(V=unindexed[bc.function_space()], g=0) for bc in bcs)
         else:
             raise ValueError("Expecting at most 2 components")
@@ -708,7 +708,7 @@ def _element_kernels(self):
             element_kernel = TripleProductKernel(R1, M, C1)
             schur_kernel = self.schur_kernel.get(Vrow) if Vrow == Vcol else None
             if schur_kernel is not None:
-                V0 = FunctionSpace(Vrow.mesh(), restrict_element(self.embedding_element, "interior"))
+                V0 = Vrow.collapse().reconstruct(element=restrict_element(self.embedding_element, "interior"))
                 C0 = self.assemble_reference_tensor(V0, sort_interior=True)
                 R0 = self.assemble_reference_tensor(V0, sort_interior=True, transpose=True)
                 element_kernel = schur_kernel(element_kernel,
@@ -826,7 +826,7 @@ def set_values(self, A, Vrow, Vcol, mat_type=None):
             element_kernel = TripleProductKernel(R1, M, C1)
             schur_kernel = self.schur_kernel.get(Vrow) if on_diag else None
             if schur_kernel is not None:
-                V0 = FunctionSpace(Vrow.mesh(), restrict_element(self.embedding_element, "interior"))
+                V0 = Vrow.collapse().reconstruct(element=restrict_element(self.embedding_element, "interior"))
                 C0 = self.assemble_reference_tensor(V0, sort_interior=True)
                 R0 = self.assemble_reference_tensor(V0, sort_interior=True, transpose=True)
                 element_kernel = schur_kernel(element_kernel,
@@ -1462,9 +1462,9 @@ def __init__(self, kernel, name=None):
         fcp = self.child.fcp
         args = form.arguments()
         Q = args[0].function_space()
-        V = FunctionSpace(Q.mesh(), unrestrict_element(Q.ufl_element()))
-        V0 = FunctionSpace(Q.mesh(), restrict_element(V.ufl_element(), "interior"))
-        V1 = FunctionSpace(Q.mesh(), restrict_element(V.ufl_element(), "facet"))
+        V = Q.reconstruct(element=unrestrict_element(Q.ufl_element()))
+        V0 = Q.reconstruct(element=restrict_element(V.ufl_element(), "interior"))
+        V1 = Q.reconstruct(element=restrict_element(V.ufl_element(), "facet"))
         idofs = PETSc.IS().createBlock(V.block_size, restricted_dofs(V0.finat_element, V.finat_element), comm=comm)
         fdofs = PETSc.IS().createBlock(V.block_size, restricted_dofs(V1.finat_element, V.finat_element), comm=comm)
         size = idofs.size + fdofs.size
diff --git a/firedrake/preconditioners/hiptmair.py b/firedrake/preconditioners/hiptmair.py
index 858a07d0d5..0b9d55b06f 100644
--- a/firedrake/preconditioners/hiptmair.py
+++ b/firedrake/preconditioners/hiptmair.py
@@ -5,7 +5,6 @@
 from firedrake.bcs import DirichletBC
 from firedrake.petsc import PETSc
 from firedrake.preconditioners.base import PCBase
-from firedrake.functionspace import FunctionSpace
 from firedrake.ufl_expr import TestFunction, TrialFunction
 from firedrake.preconditioners.hypre_ams import chop
 from firedrake.preconditioners.facet_split import restrict
@@ -146,7 +145,6 @@ def coarsen(self, pc):
 
         a, bcs = self.form(pc)
         V = a.arguments()[-1].function_space()
-        mesh = V.mesh()
         element = V.ufl_element()
         formdegree = V.finat_element.formdegree
         if formdegree == 1:
@@ -163,7 +161,7 @@ def coarsen(self, pc):
         if domain:
             celement = restrict(celement, domain)
 
-        coarse_space = FunctionSpace(mesh, celement)
+        coarse_space = V.reconstruct(element=celement)
         assert coarse_space.finat_element.formdegree + 1 == formdegree
         coarse_space_bcs = [bc.reconstruct(V=coarse_space, g=0) for bc in bcs]
 
diff --git a/firedrake/preconditioners/hypre_ads.py b/firedrake/preconditioners/hypre_ads.py
index 5affcf8c1f..bf9fbe4b85 100644
--- a/firedrake/preconditioners/hypre_ads.py
+++ b/firedrake/preconditioners/hypre_ads.py
@@ -1,11 +1,11 @@
 import firedrake.assemble as assemble
 from firedrake.preconditioners.base import PCBase
 from firedrake.petsc import PETSc
-from firedrake.functionspace import FunctionSpace, VectorFunctionSpace
 from firedrake.ufl_expr import TestFunction
 from firedrake.dmhooks import get_function_space
 from firedrake.preconditioners.hypre_ams import chop
 from firedrake.__future__ import interpolate
+from finat.ufl import VectorElement
 from ufl import grad, curl, SpatialCoordinate
 from pyop2.utils import as_tuple
 
@@ -26,8 +26,8 @@ def initialize(self, obj):
         if formdegree != 2 or degree != 1:
             raise ValueError("Hypre ADS requires lowest order RT elements! (not %s of degree %d)" % (family, degree))
 
-        P1 = FunctionSpace(mesh, "Lagrange", 1)
-        NC1 = FunctionSpace(mesh, "N1curl" if mesh.ufl_cell().is_simplex() else "NCE", 1)
+        P1 = V.reconstruct(family="Lagrange", degree=1)
+        NC1 = V.reconstruct(family="N1curl" if mesh.ufl_cell().is_simplex() else "NCE", degree=1)
         G_callback = appctx.get("get_gradient", None)
         if G_callback is None:
             G = chop(assemble.assemble(interpolate(grad(TestFunction(P1)), NC1)).petscmat)
@@ -49,7 +49,7 @@ def initialize(self, obj):
         pc.setHYPREDiscreteGradient(G)
         pc.setHYPREDiscreteCurl(C)
 
-        VectorP1 = VectorFunctionSpace(mesh, "Lagrange", 1)
+        VectorP1 = P1.reconstruct(element=VectorElement(P1.ufl_element()))
         interp = interpolate(SpatialCoordinate(mesh), VectorP1)
         pc.setCoordinates(assemble.assemble(interp).dat.data_ro.copy())
 
diff --git a/firedrake/preconditioners/hypre_ams.py b/firedrake/preconditioners/hypre_ams.py
index dc610e005d..8445d429aa 100644
--- a/firedrake/preconditioners/hypre_ams.py
+++ b/firedrake/preconditioners/hypre_ams.py
@@ -1,13 +1,13 @@
 import firedrake.assemble as assemble
 from firedrake.preconditioners.base import PCBase
 from firedrake.petsc import PETSc
-from firedrake.functionspace import FunctionSpace, VectorFunctionSpace
 from firedrake.ufl_expr import TestFunction
 from firedrake.dmhooks import get_function_space
 from firedrake.utils import complex_mode
 from firedrake_citations import Citations
 from firedrake import SpatialCoordinate
 from firedrake.__future__ import interpolate
+from finat.ufl import VectorElement
 from ufl import grad
 from pyop2.utils import as_tuple
 
@@ -47,7 +47,7 @@ def initialize(self, obj):
         if formdegree != 1 or degree != 1:
             raise ValueError("Hypre AMS requires lowest order Nedelec elements! (not %s of degree %d)" % (family, degree))
 
-        P1 = FunctionSpace(mesh, "Lagrange", 1)
+        P1 = V.reconstruct(family="Lagrange", degree=1)
         G_callback = appctx.get("get_gradient", None)
         if G_callback is None:
             G = chop(assemble.assemble(interpolate(grad(TestFunction(P1)), V)).petscmat)
@@ -67,7 +67,7 @@ def initialize(self, obj):
         if zero_beta:
             pc.setHYPRESetBetaPoissonMatrix(None)
 
-        VectorP1 = VectorFunctionSpace(mesh, "Lagrange", 1)
+        VectorP1 = P1.reconstruct(element=VectorElement(P1.ufl_element()))
         interp = interpolate(SpatialCoordinate(mesh), VectorP1)
         pc.setCoordinates(assemble.assemble(interp).dat.data_ro.copy())
         pc.setFromOptions()
diff --git a/firedrake/preconditioners/patch.py b/firedrake/preconditioners/patch.py
index 6d2bced79f..cdcca9c2f3 100644
--- a/firedrake/preconditioners/patch.py
+++ b/firedrake/preconditioners/patch.py
@@ -4,7 +4,6 @@
 from firedrake.solving_utils import _SNESContext
 from firedrake.utils import cached_property, complex_mode, IntType
 from firedrake.dmhooks import get_appctx, push_appctx, pop_appctx
-from firedrake.functionspace import FunctionSpace
 from firedrake.interpolation import Interpolate
 
 from collections import namedtuple
@@ -654,22 +653,15 @@ def sort_entities(self, dm, axis, dir, ndiv=None, divisions=None):
             raise RuntimeError("Must either set ndiv or divisions for PlaneSmoother!")
 
         mesh = dm.getAttr("__firedrake_mesh__")
-        ele = mesh.coordinates.function_space().ufl_element()
-        V = mesh.coordinates.function_space()
-        if V.finat_element.entity_dofs() == V.finat_element.entity_closure_dofs():
+        coordinates = mesh.coordinates
+        V = coordinates.function_space()
+        if V.finat_element.is_dg():
             # We're using DG or DQ for our coordinates, so we got
             # a periodic mesh. We need to interpolate to CGk
             # with access descriptor MAX to define a consistent opinion
             # about where the vertices are.
-            CGkele = ele.reconstruct(family="Lagrange")
-            # Need to supply the actual mesh to the FunctionSpace constructor,
-            # not its weakref proxy (the variable `mesh`)
-            # as interpolation fails because they are not hashable
-            CGk = FunctionSpace(mesh.coordinates.function_space().mesh(), CGkele)
-            coordinates = Interpolate(mesh.coordinates, CGk, access=op2.MAX)
-            coordinates = assemble(coordinates)
-        else:
-            coordinates = mesh.coordinates
+            CGk = V.reconstruct(family="Lagrange")
+            coordinates = assemble(Interpolate(coordinates, CGk, access=op2.MAX))
 
         select = partial(select_entity, dm=dm, exclude="pyop2_ghost")
         entities = [(p, self.coords(dm, p, coordinates)) for p in
diff --git a/firedrake/preconditioners/pmg.py b/firedrake/preconditioners/pmg.py
index 48df37d615..04797e95aa 100644
--- a/firedrake/preconditioners/pmg.py
+++ b/firedrake/preconditioners/pmg.py
@@ -181,21 +181,21 @@ def coarsen(self, fdm, comm):
         assert parent is not None
 
         test, trial = fctx.J.arguments()
-        fV = test.function_space()
+        fV = trial.function_space()
         cele = self.coarsen_element(fV.ufl_element())
 
         # Have we already done this?
         cctx = fctx._coarse
         if cctx is not None:
-            cV = cctx.J.arguments()[0].function_space()
-            if (cV.ufl_element() == cele) and (cV.mesh() == fV.mesh()):
+            cV = cctx.J.arguments()[1].function_space()
+            if (cV.ufl_element() == cele) and (cV.mesh() == fV.mesh()) and all(cV_.boundary_set == fV_.boundary_set for cV_, fV_ in zip(cV, fV)):
                 return cV.dm
 
-        cV = firedrake.FunctionSpace(fV.mesh(), cele)
+        cV = fV.reconstruct(element=cele)
         cdm = cV.dm
 
         fproblem = fctx._problem
-        fu = fproblem.u
+        fu = fproblem.u_restrict
         cu = firedrake.Function(cV)
 
         fdeg = PMGBase.max_degree(fV.ufl_element())
@@ -370,8 +370,8 @@ def create_transfer(self, mat_type, cctx, fctx, cbcs, fbcs):
                 construct_mat = prolongation_matrix_aij
             else:
                 raise ValueError("Unknown matrix type")
-            cV = cctx.J.arguments()[0].function_space()
-            fV = fctx.J.arguments()[0].function_space()
+            cV = cctx._problem.u_restrict.function_space()
+            fV = fctx._problem.u_restrict.function_space()
             cbcs = tuple(cctx._problem.bcs) if cbcs else tuple()
             fbcs = tuple(fctx._problem.bcs) if fbcs else tuple()
             return cache.setdefault(key, construct_mat(cV, fV, cbcs, fbcs))
@@ -1179,7 +1179,7 @@ def make_permutation_code(V, vshape, pshape, t_in, t_out, array_name):
 
 def reference_value_space(V):
     element = finat.ufl.WithMapping(V.ufl_element(), mapping="identity")
-    return firedrake.FunctionSpace(V.mesh(), element)
+    return V.collapse().reconstruct(element=element)
 
 
 class StandaloneInterpolationMatrix(object):
@@ -1206,13 +1206,13 @@ def __init__(self, Vc, Vf, Vc_bcs, Vf_bcs):
             self.Vf = reference_value_space(self.Vf)
             self.uc = firedrake.Function(self.Vc, val=self.uc.dat)
             self.uf = firedrake.Function(self.Vf, val=self.uf.dat)
-            self.Vc_bcs = [bc.reconstruct(V=self.Vc) for bc in self.Vc_bcs]
-            self.Vf_bcs = [bc.reconstruct(V=self.Vf) for bc in self.Vf_bcs]
+            self.Vc_bcs = [bc.reconstruct(V=self.Vc, g=0) for bc in self.Vc_bcs]
+            self.Vf_bcs = [bc.reconstruct(V=self.Vf, g=0) for bc in self.Vf_bcs]
 
     def work_function(self, V):
         if isinstance(V, firedrake.Function):
             return V
-        key = (V.ufl_element(), V.mesh())
+        key = (V.ufl_element(), V.mesh(), tuple(V.boundary_set))
         try:
             return self._cache_work[key]
         except KeyError:
@@ -1337,17 +1337,14 @@ def make_blas_kernels(self, Vf, Vc):
             restrict = [""]*5
             # get embedding element for Vf with identity mapping and collocated vector component DOFs
             try:
-                qelem = felem
-                if qelem.mapping() != "identity":
-                    qelem = qelem.reconstruct(mapping="identity")
-                Qf = Vf if qelem == felem else firedrake.FunctionSpace(Vf.mesh(), qelem)
+                Qf = Vf if felem.mapping() == "identity" else Vf.reconstruct(mapping="identity")
                 mapping_output = make_mapping_code(Qf, cmapping, fmapping, "t0", "t1")
                 in_place_mapping = True
             except Exception:
                 qelem = finat.ufl.FiniteElement("DQ", cell=felem.cell, degree=PMGBase.max_degree(felem))
                 if Vf.value_shape:
                     qelem = finat.ufl.TensorElement(qelem, shape=Vf.value_shape, symmetry=felem.symmetry())
-                Qf = firedrake.FunctionSpace(Vf.mesh(), qelem)
+                Qf = Vf.reconstruct(element=qelem)
                 mapping_output = make_mapping_code(Qf, cmapping, fmapping, "t0", "t1")
 
             qshape = (Qf.block_size, Qf.finat_element.space_dimension())
diff --git a/pyop2/types/dat.py b/pyop2/types/dat.py
index 45900effef..7dcd08e589 100644
--- a/pyop2/types/dat.py
+++ b/pyop2/types/dat.py
@@ -105,7 +105,6 @@ def _wrapper_cache_key_(self):
     @utils.validate_in(('access', _modes, ex.ModeValueError))
     def __call__(self, access, path=None):
         from pyop2.parloop import DatLegacyArg
-
         if conf.configuration["type_check"] and path and path.toset != self.dataset.set:
             raise ex.MapValueError("To Set of Map does not match Set of Dat.")
         return DatLegacyArg(self, path, access)
diff --git a/tests/firedrake/multigrid/test_p_multigrid.py b/tests/firedrake/multigrid/test_p_multigrid.py
index 39437b0204..ea26f1c44a 100644
--- a/tests/firedrake/multigrid/test_p_multigrid.py
+++ b/tests/firedrake/multigrid/test_p_multigrid.py
@@ -137,7 +137,8 @@ def mat_type(request):
     return request.param
 
 
-def test_p_multigrid_scalar(mesh, mat_type):
+@pytest.mark.parametrize("restrict", [False, True], ids=("unrestrict", "restrict"))
+def test_p_multigrid_scalar(mesh, mat_type, restrict):
     V = FunctionSpace(mesh, "CG", 4)
 
     u = Function(V)
@@ -175,7 +176,7 @@ def test_p_multigrid_scalar(mesh, mat_type):
           "pmg_mg_coarse_mg_coarse_ksp_monitor": None,
           "pmg_mg_coarse_mg_coarse_pc_type": "gamg",
           "pmg_mg_coarse_mg_coarse_pc_gamg_threshold": 0}
-    problem = NonlinearVariationalProblem(F, u, bcs)
+    problem = NonlinearVariationalProblem(F, u, bcs, restrict=restrict)
     solver = NonlinearVariationalSolver(problem, solver_parameters=sp)
     solver.solve()
 
diff --git a/tests/firedrake/regression/test_restricted_function_space.py b/tests/firedrake/regression/test_restricted_function_space.py
index 8d8a19a56d..ddbc876ddc 100644
--- a/tests/firedrake/regression/test_restricted_function_space.py
+++ b/tests/firedrake/regression/test_restricted_function_space.py
@@ -362,3 +362,97 @@ def test_restricted_function_space_extrusion_stokes(ncells):
     # -- Actually, the ordering is the same.
     assert np.allclose(sol_res.subfunctions[0].dat.data_ro_with_halos, sol.subfunctions[0].dat.data_ro_with_halos)
     assert np.allclose(sol_res.subfunctions[1].dat.data_ro_with_halos, sol.subfunctions[1].dat.data_ro_with_halos)
+
+
+@pytest.mark.parametrize("names", [(None, None), (None, "name1"), ("name0", "name1")])
+def test_restrict_fieldsplit(names):
+    mesh = UnitSquareMesh(2, 2)
+    V = FunctionSpace(mesh, "CG", 1, name=names[0])
+    Q = FunctionSpace(mesh, "CG", 2, name=names[1])
+    Z = V * Q
+
+    z = Function(Z)
+    test = TestFunction(Z)
+    z_exact = Constant([1, -1])
+
+    F = inner(z - z_exact, test) * dx
+    bcs = [DirichletBC(Z.sub(i), z_exact[i], (i+1, i+3)) for i in range(len(Z))]
+
+    problem = NonlinearVariationalProblem(F, z, bcs=bcs, restrict=True)
+    solver = NonlinearVariationalSolver(problem, solver_parameters={
+        "snes_type": "ksponly",
+        "ksp_type": "preonly",
+        "pc_type": "fieldsplit",
+        "pc_fieldsplit_type": "additive",
+        f"fieldsplit_{names[0] or 0}_pc_type": "lu",
+        f"fieldsplit_{names[1] or 1}_pc_type": "lu"},
+        options_prefix="")
+    solver.solve()
+
+    # Test prefixes for the restricted spaces
+    pc = solver.snes.ksp.pc
+    for field, ksp in enumerate(pc.getFieldSplitSubKSP()):
+        name = Z[field].name or field
+        assert ksp.getOptionsPrefix() == f"fieldsplit_{name}_"
+
+    assert errornorm(z_exact[0], z.subfunctions[0]) < 1E-10
+    assert errornorm(z_exact[1], z.subfunctions[1]) < 1E-10
+
+
+def test_restrict_python_pc():
+    mesh = UnitSquareMesh(2, 2)
+    V = FunctionSpace(mesh, "CG", 1)
+    u = Function(V)
+    test = TestFunction(V)
+
+    x, y = SpatialCoordinate(mesh)
+    u_exact = x + y
+    g = Function(V).interpolate(u_exact)
+
+    F = inner(u - u_exact, test) * dx
+    bcs = [DirichletBC(V, g, 1), DirichletBC(V, u_exact, 2)]
+
+    problem = NonlinearVariationalProblem(F, u, bcs=bcs, restrict=True)
+    solver = NonlinearVariationalSolver(problem, solver_parameters={
+        "snes_type": "ksponly",
+        "mat_type": "matfree",
+        "ksp_type": "preonly",
+        "pc_type": "python",
+        "pc_python_type": "firedrake.AssembledPC",
+        "assembled_pc_type": "lu"})
+    solver.solve()
+
+    assert errornorm(u_exact, u) < 1E-10
+
+
+def test_restrict_multigrid():
+    base = UnitSquareMesh(2, 2)
+    refine = 2
+    mh = MeshHierarchy(base, refine)
+    mesh = mh[-1]
+
+    V = FunctionSpace(mesh, "CG", 1)
+    u = Function(V)
+    test = TestFunction(V)
+
+    x, y = SpatialCoordinate(mesh)
+    u_exact = x + y
+    g = Function(V).interpolate(u_exact)
+
+    F = inner(grad(u - u_exact), grad(test)) * dx
+    bcs = [DirichletBC(V, g, 1), DirichletBC(V, u_exact, 2)]
+
+    problem = NonlinearVariationalProblem(F, u, bcs=bcs, restrict=True)
+    solver = NonlinearVariationalSolver(problem, solver_parameters={
+        "snes_type": "ksponly",
+        "ksp_type": "cg",
+        "ksp_rtol": 1E-10,
+        "ksp_max_it": 10,
+        "ksp_monitor": None,
+        "pc_type": "mg",
+        "mg_levels_ksp_type": "chebyshev",
+        "mg_levels_pc_type": "jacobi",
+        "mg_coarse_pc_type": "lu"})
+    solver.solve()
+
+    assert errornorm(u_exact, u) < 1E-10