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 project(v,
V=None,
bcs=None,
mesh=None,
function=None,
solver_type="lu",
preconditioner_type="default",
form_compiler_parameters=None):
"""Return projection of given expression *v* onto the finite element
space *V*.
*Arguments*
v
a :py:class:`Function <dolfin.functions.function.Function>` or
an :py:class:`Expression <dolfin.functions.expression.Expression>`
bcs
Optional argument :py:class:`list of DirichletBC
<dolfin.fem.bcs.DirichletBC>`
V
Optional argument :py:class:`FunctionSpace
<dolfin.functions.functionspace.FunctionSpace>`
mesh
Optional argument :py:class:`mesh <dolfin.cpp.Mesh>`.
solver_type
see :py:func:`solve <dolfin.fem.solving.solve>` for options.
preconditioner_type
see :py:func:`solve <dolfin.fem.solving.solve>` for options.
form_compiler_parameters
see :py:class:`Parameters <dolfin.cpp.Parameters>` for more
information.
*Example of usage*
.. code-block:: python
v = Expression("sin(pi*x[0])")
V = FunctionSpace(mesh, "Lagrange", 1)
Pv = project(v, V)
This is useful for post-processing functions or expressions
which are not readily handled by visualization tools (such as
for example discontinuous functions).
"""
# Try figuring out a function space if not specified
if V is None:
# Create function space based on Expression element if trying
# to project an Expression
if isinstance(v, dolfin.function.expression.Expression):
if mesh is not None and isinstance(mesh, cpp.mesh.Mesh):
V = FunctionSpace(mesh, v.ufl_element())
# else:
# cpp.dolfin_error("projection.py",
# "perform projection",
# "Expected a mesh when projecting an Expression")
else:
# Otherwise try extracting function space from expression
V = _extract_function_space(v, mesh)
# Check arguments
# Ensure we have a mesh and attach to measure
if mesh is None:
mesh = V.mesh()
dx = ufl.dx(mesh)
# Define variational problem for projection
w = TestFunction(V)
Pv = TrialFunction(V)
a = ufl.inner(w, Pv) * dx
L = ufl.inner(w, v) * dx
# Assemble linear system
A, b = assemble_system(
a, L, bcs=bcs, form_compiler_parameters=form_compiler_parameters)
# Solve linear system for projection
if function is None:
function = Function(V)
cpp.la.solve(A, function.vector(), b, solver_type, preconditioner_type)
return function
def project(v, V=None, func_degree=None, bcs=None, mesh=None,
function=None,
solver_type="lu",
preconditioner_type="default",
form_compiler_parameters=None):
"""
This function is a modification of FEniCS's built-in project 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/projection.py>`_.
.. note:: Note the extra argument func_degree: this is used to interpolate the :math:`r^2`
Expression to the same degree as used in the definition of the Trial and Test function
spaces.
"""
# Try figuring out a function space if not specified
if V is None:
# Create function space based on Expression element if trying
# to project an Expression
if isinstance(v, Expression):
# FIXME: Add handling of cpp.MultiMesh
if mesh is not None and isinstance(mesh, cpp.mesh.Mesh):
V = FunctionSpace(mesh, v.ufl_element())
# else:
# cpp.dolfin_error("projection.py",
# "perform projection",
# "Expected a mesh when projecting an Expression")
else:
# Otherwise try extracting function space from expression
V = _extract_function_space(v, mesh)
# Ensure we have a mesh and attach to measure
if mesh is None:
mesh = V.mesh()
dx = ufl.dx(mesh)
# Define variational problem for projection
# DS: HERE IS WHERE I MODIFY
r2 = Expression('pow(x[0],2)', degree=func_degree)
w = TestFunction(V)
Pv = TrialFunction(V)
a = ufl.inner(w, Pv) * r2 * dx
L = ufl.inner(w, v) * r2 * dx
# Assemble linear system
A, b = assemble_system(a, L, bcs=bcs,
form_compiler_parameters=form_compiler_parameters)
# Solve linear system for projection
if function is None:
function = Function(V)
cpp.la.solve(A, function.vector(), b, solver_type, preconditioner_type)
return function
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 )
class DOLFINErrorControlGenerator(ErrorControlGenerator):
"""
This class provides a realization of
ffc.errorcontrol.errorcontrolgenerators.ErrorControlGenerator for
use with UFL forms defined over DOLFIN objects
"""
def __init__(self, F, M, u):
"""
*Arguments*
F (tuple or Form)
tuple of (bilinear, linear) forms or linear form
M (Form)
functional or linear form
u (Coefficient)
The coefficient considered as the unknown.
"""
ErrorControlGenerator.__init__(self, __import__("dolfin"), F, M, u)
def initialize_data(self):
"""
Extract required objects for defining error control
forms. This will be stored and reused.
"""
# Developer's note: The UFL-FFC-DOLFIN--PyDOLFIN toolchain for
# error control is quite fine-tuned. In particular, the order
# of coefficients in forms is (and almost must be) used for
# their assignment. This means that the order in which these
# coefficients are defined matters and should be considered
# fixed.
# Primal trial element space
self._V = self.u.function_space()
# Primal test space == Dual trial space
Vhat = self.weak_residual.arguments()[0].function_space()
# Discontinuous version of primal trial element space
self._dV = tear(self._V)
# Extract cell and geometric dimension
mesh = self._V.mesh()
dim = mesh.topology().dim()
# Function representing improved dual
E = increase_order(Vhat)
self._Ez_h = Function(E)
# Function representing cell bubble function
B = FunctionSpace(mesh, "B", dim + 1)
self._b_T = Function(B)
self._b_T.vector()[:] = 1.0
# Function representing strong cell residual
self._R_T = Function(self._dV)
# Function representing cell cone function
C = FunctionSpace(mesh, "DG", dim)
self._b_e = Function(C)
# Function representing strong facet residual
self._R_dT = Function(self._dV)
# Function for discrete dual on primal test space
self._z_h = Function(Vhat)
# Piecewise constants for assembling indicators
self._DG0 = FunctionSpace(mesh, "DG", 0)
class DOLFINErrorControlGenerator(ErrorControlGenerator):
"""
This class provides a realization of
ffc.errorcontrol.errorcontrolgenerators.ErrorControlGenerator for
use with UFL forms defined over DOLFIN objects
"""
def __init__(self, F, M, u):
"""
*Arguments*
F (tuple or Form)
tuple of (bilinear, linear) forms or linear form
M (Form)
functional or linear form
u (Coefficient)
The coefficient considered as the unknown.
"""
ErrorControlGenerator.__init__(self, __import__("dolfin"), F, M, u)
def initialize_data(self):
"""
Extract required objects for defining error control
forms. This will be stored and reused.
"""
# Developer's note: The UFL-FFC-DOLFIN--PyDOLFIN toolchain for
# error control is quite fine-tuned. In particular, the order
# of coefficients in forms is (and almost must be) used for
# their assignment. This means that the order in which these
# coefficients are defined matters and should be considered
# fixed.
# Primal trial element space
self._V = self.u.function_space()
# Primal test space == Dual trial space
Vhat = self.weak_residual.arguments()[0].function_space()
# Discontinuous version of primal trial element space
self._dV = tear(self._V)
# Extract cell and geometric dimension
mesh = self._V.mesh()
dim = mesh.topology().dim()
# Function representing improved dual
E = increase_order(Vhat)
self._Ez_h = Function(E)
# Function representing cell bubble function
B = FunctionSpace(mesh, "B", dim + 1)
self._b_T = Function(B)
self._b_T.vector()[:] = 1.0
# Function representing strong cell residual
self._R_T = Function(self._dV)
# Function representing cell cone function
C = FunctionSpace(mesh, "DG", dim)
self._b_e = Function(C)
# Function representing strong facet residual
self._R_dT = Function(self._dV)
# Function for discrete dual on primal test space
self._z_h = Function(Vhat)
# Piecewise constants for assembling indicators
self._DG0 = FunctionSpace(mesh, "DG", 0)