diff --git a/firedrake/assemble.py b/firedrake/assemble.py
index 607661831f..c93594ce52 100644
--- a/firedrake/assemble.py
+++ b/firedrake/assemble.py
@@ -128,13 +128,87 @@ def get_form_assembler(form, tensor, *args, **kwargs):
             raise ValueError('Expecting a 0-, 1-, or 2-form: got %s' % (form))
     elif isinstance(form, ufl.core.expr.Expr) and not isinstance(form, ufl.core.base_form_operator.BaseFormOperator):
         # BaseForm preprocessing can turn BaseForm into an Expr (cf. case (6) in `restructure_base_form`)
-        return functools.partial(_assemble_expr, form)
+        return functools.partial(ExprAssembler(form).assemble, tensor=tensor)
     elif isinstance(form, ufl.form.BaseForm):
         return functools.partial(BaseFormAssembler(form, *args, **kwargs).assemble, tensor=tensor)
     else:
         raise ValueError(f'Expecting a BaseForm, slate.TensorBase, or Expr object: got {form}')
 
 
+class ExprAssembler:
+    """Expression assembler.
+
+    Parameters
+    ----------
+    expr : ufl.core.expr.Expr
+        Expression.
+
+    """
+
+    def __init__(self, expr):
+        self._expr = expr
+
+    def assemble(self, tensor=None):
+        """Assemble the pointwise expression.
+
+        Parameters
+        ----------
+        tensor : firedrake.function.Function or firedrake.cofunction.Cofunction or matrix.MatrixBase
+            Output tensor.
+
+        Returns
+        -------
+        float or firedrake.cofunction.Cofunction or firedrake.function.Function or matrix.MatrixBase
+            Result of evaluation: `float` for 0-forms, `firedrake.cofunction.Cofunction` or `firedrake.function.Function` for 1-forms, and `matrix.MatrixBase` for 2-forms.
+
+        """
+        from ufl.algorithms.analysis import extract_base_form_operators
+        from ufl.checks import is_scalar_constant_expression
+
+        assert tensor is None
+        expr = self._expr
+        # Get BaseFormOperators (e.g. `Interpolate` or `ExternalOperator`)
+        base_form_operators = extract_base_form_operators(expr)
+
+        # -- Linear combination involving 2-form BaseFormOperators -- #
+        # Example: a * dNdu1(u1, u2; v1, v*) + b * dNdu2(u1, u2; v2, v*)
+        # with u1, u2 Functions, v1, v2, v* BaseArguments, dNdu1, dNdu2 BaseFormOperators, and a, b scalars.
+        if len(base_form_operators) and any(len(e.arguments()) > 1 for e in base_form_operators):
+            if isinstance(expr, ufl.algebra.Sum):
+                a, b = [assemble(e) for e in expr.ufl_operands]
+                # Only Expr resulting in a Matrix if assembled are BaseFormOperator
+                if not all(isinstance(op, matrix.AssembledMatrix) for op in (a, b)):
+                    raise TypeError('Mismatching Sum shapes')
+                return get_form_assembler(ufl.FormSum((a, 1), (b, 1)), None)()
+            elif isinstance(expr, ufl.algebra.Product):
+                a, b = expr.ufl_operands
+                scalar = [e for e in expr.ufl_operands if is_scalar_constant_expression(e)]
+                if scalar:
+                    base_form = a if a is scalar else b
+                    assembled_mat = assemble(base_form)
+                    return get_form_assembler(ufl.FormSum((assembled_mat, scalar[0])), None)()
+                a, b = [assemble(e) for e in (a, b)]
+                return get_form_assembler(ufl.action(a, b), None)()
+        # -- Linear combination of Functions and 1-form BaseFormOperators -- #
+        # Example: a * u1 + b * u2 + c * N(u1; v*) + d * N(u2; v*)
+        # with u1, u2 Functions, N a BaseFormOperator, and a, b, c, d scalars or 0-form BaseFormOperators.
+        else:
+            base_form_operators = extract_base_form_operators(expr)
+            assembled_bfops = [firedrake.assemble(e) for e in base_form_operators]
+            # Substitute base form operators with their output before examining the expression
+            # which avoids conflict when determining function space, for example:
+            # extract_coefficients(Interpolate(u, V2)) with u \in V1 will result in an output function space V1
+            # instead of V2.
+            if base_form_operators:
+                expr = ufl.replace(expr, dict(zip(base_form_operators, assembled_bfops)))
+            try:
+                coefficients = ufl.algorithms.extract_coefficients(expr)
+                V, = set(c.function_space() for c in coefficients) - {None}
+            except ValueError:
+                raise ValueError("Cannot deduce correct target space from pointwise expression")
+            return firedrake.Function(V).assign(expr)
+
+
 class AbstractFormAssembler(abc.ABC):
     """Abstract assembler class for forms.
 
@@ -807,57 +881,6 @@ def allocate_matrix(expr, bcs=None, *, mat_type=None, sub_mat_type=None,
                          options_prefix=options_prefix)
 
 
-def _assemble_expr(expr):
-    """Assemble a pointwise expression.
-
-    :arg expr: The :class:`ufl.core.expr.Expr` to be evaluated.
-    :returns: A :class:`firedrake.Function` containing the result of this evaluation.
-    """
-    from ufl.algorithms.analysis import extract_base_form_operators
-    from ufl.checks import is_scalar_constant_expression
-
-    # Get BaseFormOperators (e.g. `Interpolate` or `ExternalOperator`)
-    base_form_operators = extract_base_form_operators(expr)
-
-    # -- Linear combination involving 2-form BaseFormOperators -- #
-    # Example: a * dNdu1(u1, u2; v1, v*) + b * dNdu2(u1, u2; v2, v*)
-    # with u1, u2 Functions, v1, v2, v* BaseArguments, dNdu1, dNdu2 BaseFormOperators, and a, b scalars.
-    if len(base_form_operators) and any(len(e.arguments()) > 1 for e in base_form_operators):
-        if isinstance(expr, ufl.algebra.Sum):
-            a, b = [assemble(e) for e in expr.ufl_operands]
-            # Only Expr resulting in a Matrix if assembled are BaseFormOperator
-            if not all(isinstance(op, matrix.AssembledMatrix) for op in (a, b)):
-                raise TypeError('Mismatching Sum shapes')
-            return get_form_assembler(ufl.FormSum((a, 1), (b, 1)), None)()
-        elif isinstance(expr, ufl.algebra.Product):
-            a, b = expr.ufl_operands
-            scalar = [e for e in expr.ufl_operands if is_scalar_constant_expression(e)]
-            if scalar:
-                base_form = a if a is scalar else b
-                assembled_mat = assemble(base_form)
-                return get_form_assembler(ufl.FormSum((assembled_mat, scalar[0])), None)()
-            a, b = [assemble(e) for e in (a, b)]
-            return get_form_assembler(ufl.action(a, b), None)()
-    # -- Linear combination of Functions and 1-form BaseFormOperators -- #
-    # Example: a * u1 + b * u2 + c * N(u1; v*) + d * N(u2; v*)
-    # with u1, u2 Functions, N a BaseFormOperator, and a, b, c, d scalars or 0-form BaseFormOperators.
-    else:
-        base_form_operators = extract_base_form_operators(expr)
-        assembled_bfops = [firedrake.assemble(e) for e in base_form_operators]
-        # Substitute base form operators with their output before examining the expression
-        # which avoids conflict when determining function space, for example:
-        # extract_coefficients(Interpolate(u, V2)) with u \in V1 will result in an output function space V1
-        # instead of V2.
-        if base_form_operators:
-            expr = ufl.replace(expr, dict(zip(base_form_operators, assembled_bfops)))
-        try:
-            coefficients = ufl.algorithms.extract_coefficients(expr)
-            V, = set(c.function_space() for c in coefficients) - {None}
-        except ValueError:
-            raise ValueError("Cannot deduce correct target space from pointwise expression")
-        return firedrake.Function(V).assign(expr)
-
-
 class FormAssembler(AbstractFormAssembler):
     """Form assembler.