def solve(*args, **kwargs):
"""Solve linear system Ax = b or variational problem a == L or F == 0.
The DOLFIN solve() function can be used to solve either linear
systems or variational problems. The following list explains the
various ways in which the solve() function can be used.
*1. Solving linear systems*
A linear system Ax = b may be solved by calling solve(A, x, b),
where A is a matrix and x and b are vectors. Optional arguments
may be passed to specify the solver method and preconditioner.
Some examples are given below:
.. code-block:: python
solve(A, x, b)
solve(A, x, b, "lu")
solve(A, x, b, "gmres", "ilu")
solve(A, x, b, "cg", "hypre_amg")
Possible values for the solver method and preconditioner depend
on which linear algebra backend is used and how that has been
configured.
To list all available LU methods, run the following command:
.. code-block:: python
list_lu_methods()
To list all available Krylov methods, run the following command:
.. code-block:: python
list_krylov_methods()
To list all available preconditioners, run the following command:
.. code-block:: python
list_preconditioners()
To list all available solver methods, including LU methods, Krylov
methods and, possibly, other methods, run the following command:
.. code-block:: python
list_solver_methods()
*2. Solving linear variational problems*
A linear variational problem a(u, v) = L(v) for all v may be
solved by calling solve(a == L, u, ...), where a is a bilinear
form, L is a linear form, u is a Function (the solution). Optional
arguments may be supplied to specify boundary conditions or solver
parameters. Some examples are given below:
.. code-block:: python
solve(a == L, u)
solve(a == L, u, bcs=bc)
solve(a == L, u, bcs=[bc1, bc2])
solve(a == L, u, bcs=bcs,
solver_parameters={"linear_solver": "lu"},
form_compiler_parameters={"optimize": True})
For available choices for the 'solver_parameters' kwarg, look at:
.. code-block:: python
info(LinearVariationalSolver.default_parameters(), 1)
*3. Solving nonlinear variational problems*
A nonlinear variational problem F(u; v) = 0 for all v may be
solved by calling solve(F == 0, u, ...), where the residual F is a
linear form (linear in the test function v but possibly nonlinear
in the unknown u) and u is a Function (the solution). Optional
arguments may be supplied to specify boundary conditions, the
Jacobian form or solver parameters. If the Jacobian is not
supplied, it will be computed by automatic differentiation of the
residual form. Some examples are given below:
.. code-block:: python
solve(F == 0, u)
solve(F == 0, u, bcs=bc)
solve(F == 0, u, bcs=[bc1, bc2])
solve(F == 0, u, bcs, J=J,
solver_parameters={"linear_solver": "lu"},
form_compiler_parameters={"optimize": True})
For available choices for the 'solver_parameters' kwarg, look at:
.. code-block:: python
info(NonlinearVariationalSolver.default_parameters(), 1)
*4. Solving linear/nonlinear variational problems adaptively
Linear and nonlinear variational problems maybe solved adaptively,
with automated goal-oriented error control. The automated error
control algorithm is based on adaptive mesh refinement in
combination with automated generation of dual-weighted
residual-based error estimates and error indicators.
An adaptive solve may be invoked by giving two additional
arguments to the solve call, a numerical error tolerance and a
goal functional (a Form).
.. code-block:: python
M = u*dx()
tol = 1.e-6
# Linear variational problem
solve(a == L, u, bcs=bc, tol=tol, M=M)
# Nonlinear problem:
solve(F == 0, u, bcs=bc, tol=tol, M=M)
"""
assert(len(args) > 0)
# Call adaptive solve if we get a tolerance
if "tol" in kwargs:
_solve_varproblem_adaptive(*args, **kwargs)
# Call variational problem solver if we get an equation (but not a tolerance)
elif isinstance(args[0], ufl.classes.Equation):
_solve_varproblem(*args, **kwargs)
# Default case, just call the wrapped C++ solve function
else:
if kwargs:
cpp.dolfin_error("solving.py",
"solve linear algebra problem",
"Not expecting keyword arguments when solving "\
"linear algebra problem")
return cpp.la_solve(*args)
def solve(*args, **kwargs):
"""Solve linear system Ax = b or variational problem a == L or F == 0.
The DOLFIN solve() function can be used to solve either linear
systems or variational problems. The following list explains the
various ways in which the solve() function can be used.
*1. Solving linear systems*
A linear system Ax = b may be solved by calling solve(A, x, b),
where A is a matrix and x and b are vectors. Optional arguments
may be passed to specify the solver method and preconditioner.
Some examples are given below:
.. code-block:: python
solve(A, x, b)
solve(A, x, b, "lu")
solve(A, x, b, "gmres", "ilu")
solve(A, x, b, "cg", "hypre_amg")
Possible values for the solver method and preconditioner depend
on which linear algebra backend is used and how that has been
configured.
To list all available LU methods, run the following command:
.. code-block:: python
list_lu_solver_methods()
To list all available Krylov methods, run the following command:
.. code-block:: python
list_krylov_solver_methods()
To list all available preconditioners, run the following command:
.. code-block:: python
list_krylov_solver_preconditioners()
To list all available solver methods, including LU methods, Krylov
methods and, possibly, other methods, run the following command:
.. code-block:: python
list_linear_solver_methods()
*2. Solving linear variational problems*
A linear variational problem a(u, v) = L(v) for all v may be
solved by calling solve(a == L, u, ...), where a is a bilinear
form, L is a linear form, u is a Function (the solution). Optional
arguments may be supplied to specify boundary conditions or solver
parameters. Some examples are given below:
.. code-block:: python
solve(a == L, u)
solve(a == L, u, bcs=bc)
solve(a == L, u, bcs=[bc1, bc2])
solve(a == L, u, bcs=bcs,
solver_parameters={"linear_solver": "lu"},
form_compiler_parameters={"optimize": True})
For available choices for the 'solver_parameters' kwarg, look at:
.. code-block:: python
info(LinearVariationalSolver.default_parameters(), True)
*3. Solving nonlinear variational problems*
A nonlinear variational problem F(u; v) = 0 for all v may be
solved by calling solve(F == 0, u, ...), where the residual F is a
linear form (linear in the test function v but possibly nonlinear
in the unknown u) and u is a Function (the solution). Optional
arguments may be supplied to specify boundary conditions, the
Jacobian form or solver parameters. If the Jacobian is not
supplied, it will be computed by automatic differentiation of the
residual form. Some examples are given below:
.. code-block:: python
solve(F == 0, u)
solve(F == 0, u, bcs=bc)
solve(F == 0, u, bcs=[bc1, bc2])
solve(F == 0, u, bcs, J=J,
solver_parameters={"linear_solver": "lu"},
form_compiler_parameters={"optimize": True})
For available choices for the 'solver_parameters' kwarg, look at:
.. code-block:: python
info(NonlinearVariationalSolver.default_parameters(), True)
*4. Solving linear/nonlinear variational problems adaptively
Linear and nonlinear variational problems maybe solved adaptively,
with automated goal-oriented error control. The automated error
control algorithm is based on adaptive mesh refinement in
combination with automated generation of dual-weighted
residual-based error estimates and error indicators.
An adaptive solve may be invoked by giving two additional
arguments to the solve call, a numerical error tolerance and a
goal functional (a Form).
.. code-block:: python
M = u*dx()
tol = 1.e-6
# Linear variational problem
solve(a == L, u, bcs=bc, tol=tol, M=M)
# Nonlinear problem:
solve(F == 0, u, bcs=bc, tol=tol, M=M)
"""
assert (len(args) > 0)
# Call adaptive solve if we get a tolerance
if "tol" in kwargs:
_solve_varproblem_adaptive(*args, **kwargs)
# Call variational problem solver if we get an equation (but not a
# tolerance)
elif isinstance(args[0], ufl.classes.Equation):
_solve_varproblem(*args, **kwargs)
# Default case, just call the wrapped C++ solve function
else:
if kwargs:
cpp.dolfin_error(
"solving.py", "solve linear algebra problem",
"Not expecting keyword arguments when solving "
"linear algebra problem")
return cpp.la_solve(*args)
def project(v, V=None, bcs=None, mesh=None,
solver_type="cg",
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 BoundaryCondition
<dolfin.fem.bcs.BoundaryCondition>`
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).
"""
# If trying to project an Expression
if V is None and isinstance(v, Expression):
if mesh is not None and isinstance(mesh, cpp.Mesh):
V = FunctionSpaceBase(mesh, v.ufl_element())
else:
raise TypeError, "expected a mesh when projecting an Expression"
# Try extracting function space if not specified
if V is None:
V = _extract_function_space(v, mesh)
# Check arguments
if not isinstance(V, FunctionSpaceBase):
cpp.dolfin_error("projection.py",
"compute projection",
"Illegal function space for projection, not a FunctionSpace: " + str(v))
# Define variational problem for projection
w = TestFunction(V)
Pv = TrialFunction(V)
a = ufl.inner(w, Pv)*ufl.dx()
L = ufl.inner(w, v)*ufl.dx()
# Assemble linear system
A, b = assemble_system(a, L, bcs=bcs,
form_compiler_parameters=form_compiler_parameters)
# Solve linear system for projection
Pv = Function(V)
cpp.la_solve(A, Pv.vector(), b, solver_type, preconditioner_type)
return Pv
def project(v,
V=None,
bcs=None,
mesh=None,
function=None,
solver_type="cg",
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, Expression):
if mesh is not None and isinstance(mesh, cpp.Mesh):
V = FunctionSpaceBase(mesh, v.ufl_element())
else:
raise TypeError(
"expected a mesh when projecting an Expression")
else:
# Otherwise try extracting function space from expression
V = _extract_function_space(v, mesh)
# Check arguments
if not isinstance(V, FunctionSpaceBase):
cpp.dolfin_error(
"projection.py", "compute projection",
"Illegal function space for projection, not a FunctionSpace: " +
str(v))
# 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,
D=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 to adopt the :math:`r^2dr`
measure as opposed to the standard Cartesian :math:`dx` one when the :math:`L_2` projection is
computed. I have removed the case for a multimesh function space as we don't use it.
.. 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.
.. note:: I am manually implementing this bug fix that is not yet in the current release:
<https://bitbucket.org/fenics-project/dolfin/commits/c438724fa5d7f19504d4e0c48695e17b774b2c2d>_
"""
# 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):
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
if not isinstance(V, (FunctionSpace)):
cpp.dolfin_error(
"projection.py", "compute projection",
"Illegal function space for projection, not a FunctionSpace: " +
str(v))
# 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
rD = Expression('pow(x[0],D-1)', D=D, degree=func_degree)
w = TestFunction(V)
Pv = TrialFunction(V)
a = ufl.inner(w, Pv) * rD * dx
L = ufl.inner(w, v) * rD * 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