def errornorm(u, uh, norm_type="l2", degree_rise=3, mesh=None):
"""
Compute and return the error :math:`e = u - u_h` in the given norm.
*Arguments*
u, uh
:py:class:`Functions <dolfin.functions.function.Function>`
norm_type
Type of norm. The :math:`L^2` -norm is default.
For other norms, see :py:func:`norm <dolfin.fem.norms.norm>`.
degree_rise
The number of degrees above that of u_h used in the
interpolation; i.e. the degree of piecewise polynomials used
to approximate :math:`u` and :math:`u_h` will be the degree
of :math:`u_h` + degree_raise.
mesh
Optional :py:class:`Mesh <dolfin.cpp.Mesh>` on
which to compute the error norm.
In simple cases, one may just define
.. code-block:: python
e = u - uh
and evalute for example the square of the error in the :math:`L^2` -norm by
.. code-block:: python
assemble(e**2*dx(mesh))
However, this is not stable w.r.t. round-off errors considering that
the form compiler may expand(#) the expression above to::
e**2*dx = u**2*dx - 2*u*uh*dx + uh**2*dx
and this might get further expanded into thousands of terms for
higher order elements. Thus, the error will be evaluated by adding
a large number of terms which should sum up to something close to
zero (if the error is small).
This module computes the error by first interpolating both
:math:`u` and :math:`u_h` to a common space (of high accuracy),
then subtracting the two fields (which is easy since they are
expressed in the same basis) and then evaluating the integral.
(#) If using the tensor representation optimizations.
The quadrature represenation does not suffer from this problem.
"""
# Check argument
# if not isinstance(u, cpp.function.GenericFunction):
# cpp.dolfin_error("norms.py",
# "compute error norm",
# "Expecting a Function or Expression for u")
# if not isinstance(uh, cpp.function.Function):
# cpp.dolfin_error("norms.py",
# "compute error norm",
# "Expecting a Function for uh")
# Get mesh
if isinstance(u, cpp.function.Function) and mesh is None:
mesh = u.function_space().mesh()
if isinstance(uh, cpp.function.Function) and mesh is None:
mesh = uh.function_space().mesh()
if hasattr(uh, "_cpp_object") and mesh is None:
mesh = uh._cpp_object.function_space().mesh()
if hasattr(u, "_cpp_object") and mesh is None:
mesh = u._cpp_object.function_space().mesh()
if mesh is None:
cpp.dolfin_error("norms.py",
"compute error norm",
"Missing mesh")
# Get rank
if not u.ufl_shape == uh.ufl_shape:
cpp.dolfin_error("norms.py",
"compute error norm",
"Value shapes don't match")
shape = u.ufl_shape
rank = len(shape)
# Check that uh is associated with a finite element
if uh.ufl_element().degree() is None:
cpp.dolfin_error("norms.py",
"compute error norm",
"Function uh must have a finite element")
# Degree for interpolation space. Raise degree with respect to uh.
degree = uh.ufl_element().degree() + degree_rise
# Check degree of 'exact' solution u
degree_u = u.ufl_element().degree()
if degree_u is not None and degree_u < degree:
cpp.warning("Degree of exact solution may be inadequate for accurate result in errornorm.")
# Create function space
if rank == 0:
V = FunctionSpace(mesh, "Discontinuous Lagrange", degree)
elif rank == 1:
V = VectorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
dim=shape[0])
elif rank > 1:
V = TensorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
shape=shape)
# Interpolate functions into finite element space
pi_u = interpolate(u, V)
pi_uh = interpolate(uh, V)
# Compute the difference
e = Function(V)
e.assign(pi_u)
e.vector().axpy(-1.0, pi_uh.vector())
# Compute norm
return norm(e, norm_type=norm_type, mesh=mesh)
def r2_errornorm(u, uh, norm_type="l2", degree_rise=3, mesh=None ):
"""
This function is a modification of FEniCS's built-in errornorm function that adopts the :math:`r^2dr`
measure as opposed to the standard Cartesian :math:`dx` measure.
For documentation and usage, see the
original module <https://bitbucket.org/fenics-project/dolfin/src/master/python/dolfin/fem/norms.py>_.
"""
# Get mesh
if isinstance(u, cpp.function.Function) and mesh is None:
mesh = u.function_space().mesh()
if isinstance(uh, cpp.function.Function) and mesh is None:
mesh = uh.function_space().mesh()
# if isinstance(uh, MultiMeshFunction) and mesh is None:
# mesh = uh.function_space().multimesh()
if hasattr(uh, "_cpp_object") and mesh is None:
mesh = uh._cpp_object.function_space().mesh()
if hasattr(u, "_cpp_object") and mesh is None:
mesh = u._cpp_object.function_space().mesh()
if mesh is None:
raise RuntimeError("Cannot compute error norm. Missing mesh.")
# Get rank
if not u.ufl_shape == uh.ufl_shape:
raise RuntimeError("Cannot compute error norm. Value shapes do not match.")
shape = u.ufl_shape
rank = len(shape)
# Check that uh is associated with a finite element
if uh.ufl_element().degree() is None:
raise RuntimeError("Cannot compute error norm. Function uh must have a finite element.")
# Degree for interpolation space. Raise degree with respect to uh.
degree = uh.ufl_element().degree() + degree_rise
# Check degree of 'exact' solution u
degree_u = u.ufl_element().degree()
if degree_u is not None and degree_u < degree:
cpp.warning("Degree of exact solution may be inadequate for accurate result in errornorm.")
# Create function space
if rank == 0:
V = FunctionSpace(mesh, "Discontinuous Lagrange", degree)
elif rank == 1:
V = VectorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
dim=shape[0])
elif rank > 1:
V = TensorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
shape=shape)
# Interpolate functions into finite element space
pi_u = interpolate(u, V)
pi_uh = interpolate(uh, V)
# Compute the difference
e = Function(V)
e.assign(pi_u)
e.vector().axpy(-1.0, pi_uh.vector())
# Compute norm
return r2_norm(e, func_degree=degree, norm_type=norm_type, mesh=mesh )