diff --git a/test/test_change_to_local.py b/test/test_change_to_local.py
index 973195bc7..807bf1645 100755
--- a/test/test_change_to_local.py
+++ b/test/test_change_to_local.py
@@ -12,8 +12,8 @@
 def test_change_to_reference_grad():
     cell = triangle
     domain = Mesh(FiniteElement("Lagrange", cell, 1, (2,), identity_pullback, H1))
-    U = FunctionSpace(domain, FiniteElement("Lagrange", cell, 1, (), identity_pullback, H1))
-    V = FunctionSpace(domain, FiniteElement("Lagrange", cell, 1, (2,), identity_pullback, H1))
+    U = FunctionSpace(domain, FiniteElement("Lagrange", cell, 3, (), identity_pullback, H1))
+    V = FunctionSpace(domain, FiniteElement("Lagrange", cell, 3, (2,), identity_pullback, H1))
     u = Coefficient(U)
     v = Coefficient(V)
     Jinv = JacobianInverse(domain)
diff --git a/ufl/checks.py b/ufl/checks.py
index 09ae4453b..31dfe18b5 100644
--- a/ufl/checks.py
+++ b/ufl/checks.py
@@ -11,8 +11,9 @@
 from ufl.core.expr import Expr
 from ufl.core.terminal import FormArgument
 from ufl.corealg.traversal import traverse_unique_terminals
-from ufl.geometry import GeometricQuantity
+from ufl.geometry import GeometricQuantity, SpatialCoordinate
 from ufl.sobolevspace import H1
+from ufl.domain import extract_unique_domain
 
 
 def is_python_scalar(expression):
@@ -34,7 +35,20 @@ def is_true_ufl_scalar(expression):
 
 def is_cellwise_constant(expr):
     """Return whether expression is constant over a single cell."""
-    # TODO: Implement more accurately considering e.g. derivatives?
+    from ufl.coefficient import Coefficient
+    from ufl.differentiation import ReferenceGrad
+
+    if isinstance(expr, ReferenceGrad):
+        (expr,) = expr.ufl_operands
+        if is_cellwise_constant(expr):
+            return True
+        elif isinstance(expr, SpatialCoordinate):
+            domain = extract_unique_domain(expr)
+            return domain.is_piecewise_linear_simplex_domain()
+        elif isinstance(expr, Coefficient):
+            element = expr.ufl_element()
+            return element.embedded_superdegree <= 1
+
     return all(e.is_cellwise_constant() for e in traverse_unique_terminals(expr))
 
 
diff --git a/ufl/core/expr.py b/ufl/core/expr.py
index 41b6e55a6..368c6cba3 100644
--- a/ufl/core/expr.py
+++ b/ufl/core/expr.py
@@ -266,7 +266,6 @@ def ufl_domain(self):
             DeprecationWarning,
         )
         from ufl.domain import extract_unique_domain
-
         return extract_unique_domain(self)
 
     # --- Functions for float evaluation ---