pymor.discretizers.builtin.cg
¶
This module provides some operators for continuous finite element discretizations.
Module Contents¶
Classes¶
Linear finite element functional representing the inner product with an L2- |
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Linear finite element functional representing the inner product with an |
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Linear finite element functional for enforcing Dirichlet boundary values. |
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Bilinear finite element functional representing the inner product with an L2- |
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Diffusion |
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Diffusion |
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Linear advection |
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Linear advection |
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Robin boundary |
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Vector-like Lagrange interpolation |
Functions¶
Discretizes a |
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Discretizes an |
Attributes¶
- class pymor.discretizers.builtin.cg.L2ProductFunctionalP1(grid, function, dirichlet_clear_dofs=False, boundary_info=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Linear finite element functional representing the inner product with an L2-
Function
.Parameters
- grid
Grid
for which to assemble the functional.- function
The
Function
with which to take the inner product.- dirichlet_clear_dofs
If
True
, set dirichlet boundary DOFs to zero.- boundary_info
BoundaryInfo
determining the Dirichlet boundaries in casedirichlet_clear_dofs
is set toTrue
.- name
The name of the functional.
- class pymor.discretizers.builtin.cg.BoundaryL2ProductFunctional(grid, function, boundary_type=None, dirichlet_clear_dofs=False, boundary_info=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Linear finite element functional representing the inner product with an L2-
Function
on the boundary.Parameters
- grid
Grid
for which to assemble the functional.- function
The
Function
with which to take the inner product.- boundary_type
The type of domain boundary (e.g. ‘neumann’) on which to assemble the functional. If
None
the functional is assembled over the whole boundary.- dirichlet_clear_dofs
If
True
, set dirichlet boundary DOFs to zero.- boundary_info
If
boundary_type
is specified ordirichlet_clear_dofs
isTrue
, theBoundaryInfo
determining which boundary entity belongs to which physical boundary.- name
The name of the functional.
- class pymor.discretizers.builtin.cg.BoundaryDirichletFunctional(grid, dirichlet_data, boundary_info, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Linear finite element functional for enforcing Dirichlet boundary values.
Parameters
- grid
Grid
for which to assemble the functional.- dirichlet_data
Function
providing the Dirichlet boundary values.- boundary_info
BoundaryInfo
determining the Dirichlet boundaries.- name
The name of the functional.
- class pymor.discretizers.builtin.cg.L2ProductFunctionalQ1(grid, function, dirichlet_clear_dofs=False, boundary_info=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Bilinear finite element functional representing the inner product with an L2-
Function
.Parameters
- grid
Grid
for which to assemble the functional.- function
The
Function
with which to take the inner product.- dirichlet_clear_dofs
If
True
, set dirichlet boundary DOFs to zero.- boundary_info
BoundaryInfo
determining the Dirichlet boundaries in casedirichlet_clear_dofs
is set toTrue
.- name
The name of the functional.
- class pymor.discretizers.builtin.cg.L2ProductP1(grid, boundary_info, dirichlet_clear_rows=True, dirichlet_clear_columns=False, dirichlet_clear_diag=False, coefficient_function=None, solver_options=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Operator
representing the L2-product between linear finite element functions.Parameters
- grid
The
Grid
for which to assemble the product.- boundary_info
BoundaryInfo
for the treatment of Dirichlet boundary conditions.- dirichlet_clear_rows
If
True
, set the rows of the system matrix corresponding to Dirichlet boundary DOFs to zero.- dirichlet_clear_columns
If
True
, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero.- dirichlet_clear_diag
If
True
, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise, if eitherdirichlet_clear_rows
ordirichlet_clear_columns
isTrue
, the diagonal entries are set to one.- coefficient_function
Coefficient
Function
for product withshape_range == ()
. IfNone
, constant one is assumed.- solver_options
The
solver_options
for the operator.- name
The name of the product.
- class pymor.discretizers.builtin.cg.L2ProductQ1(grid, boundary_info, dirichlet_clear_rows=True, dirichlet_clear_columns=False, dirichlet_clear_diag=False, coefficient_function=None, solver_options=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Operator
representing the L2-product between bilinear finite element functions.Parameters
- grid
The
Grid
for which to assemble the product.- boundary_info
BoundaryInfo
for the treatment of Dirichlet boundary conditions.- dirichlet_clear_rows
If
True
, set the rows of the system matrix corresponding to Dirichlet boundary DOFs to zero.- dirichlet_clear_columns
If
True
, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero.- dirichlet_clear_diag
If
True
, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise, if eitherdirichlet_clear_rows
ordirichlet_clear_columns
isTrue
, the diagonal entries are set to one.- coefficient_function
Coefficient
Function
for product withshape_range == ()
. IfNone
, constant one is assumed.- solver_options
The
solver_options
for the operator.- name
The name of the product.
- class pymor.discretizers.builtin.cg.DiffusionOperatorP1(grid, boundary_info, diffusion_function=None, diffusion_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Diffusion
Operator
for linear finite elements.The operator is of the form
(Lu)(x) = c ∇ ⋅ [ d(x) ∇ u(x) ]
The function
d
can be scalar- or matrix-valued.Parameters
- grid
The
Grid
for which to assemble the operator.- boundary_info
BoundaryInfo
for the treatment of Dirichlet boundary conditions.- diffusion_function
The
Function
d(x)
withshape_range == ()
orshape_range = (grid.dim, grid.dim)
. IfNone
, constant one is assumed.- diffusion_constant
The constant
c
. IfNone
,c
is set to one.- dirichlet_clear_columns
If
True
, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero.- dirichlet_clear_diag
If
True
, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one.- solver_options
The
solver_options
for the operator.- name
Name of the operator.
- class pymor.discretizers.builtin.cg.DiffusionOperatorQ1(grid, boundary_info, diffusion_function=None, diffusion_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Diffusion
Operator
for bilinear finite elements.The operator is of the form
(Lu)(x) = c ∇ ⋅ [ d(x) ∇ u(x) ]
The function
d
can be scalar- or matrix-valued.Parameters
- grid
The
Grid
for which to assemble the operator.- boundary_info
BoundaryInfo
for the treatment of Dirichlet boundary conditions.- diffusion_function
The
Function
d(x)
withshape_range == ()
orshape_range = (grid.dim, grid.dim)
. IfNone
, constant one is assumed.- diffusion_constant
The constant
c
. IfNone
,c
is set to one.- dirichlet_clear_columns
If
True
, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero.- dirichlet_clear_diag
If
True
, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one.- solver_options
The
solver_options
for the operator.- name
Name of the operator.
- class pymor.discretizers.builtin.cg.AdvectionOperatorP1(grid, boundary_info, advection_function=None, advection_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Linear advection
Operator
for linear finite elements.The operator is of the form
(Lu)(x) = c ∇ ⋅ [ v(x) u(x) ]
The function
v
has to be vector-valued.Parameters
- grid
The
Grid
for which to assemble the operator.- boundary_info
BoundaryInfo
for the treatment of Dirichlet boundary conditions.- advection_function
The
Function
v(x)
withshape_range = (grid.dim, )
. IfNone
, constant one is assumed.- advection_constant
The constant
c
. IfNone
,c
is set to one.- dirichlet_clear_columns
If
True
, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero.- dirichlet_clear_diag
If
True
, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one.- solver_options
The
solver_options
for the operator.- name
Name of the operator.
- class pymor.discretizers.builtin.cg.AdvectionOperatorQ1(grid, boundary_info, advection_function=None, advection_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Linear advection
Operator
for bilinear finite elements.The operator is of the form
(Lu)(x) = c ∇ ⋅ [ v(x) u(x) ]
The function
v
has to be vector-valued.Parameters
- grid
The
Grid
for which to assemble the operator.- boundary_info
BoundaryInfo
for the treatment of Dirichlet boundary conditions.- advection_function
The
Function
v(x)
withshape_range = (grid.dim, )
. IfNone
, constant one is assumed.- advection_constant
The constant
c
. IfNone
,c
is set to one.- dirichlet_clear_columns
If
True
, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero.- dirichlet_clear_diag
If
True
, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one.- solver_options
The
solver_options
for the operator.- name
Name of the operator.
- class pymor.discretizers.builtin.cg.RobinBoundaryOperator(grid, boundary_info, robin_data=None, solver_options=None, name=None)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Robin boundary
Operator
for linear finite elements.The operator represents the contribution of Robin boundary conditions to the stiffness matrix, where the boundary condition is supposed to be given in the form
-[ d(x) ∇u(x) ] ⋅ n(x) = c(x) (u(x) - g(x))
d
andn
are the diffusion function (seeDiffusionOperatorP1
) and the unit outer normal inx
, whilec
is the (scalar) Robin parameter function andg
is the (also scalar) Robin boundary value function.Parameters
- grid
The
Grid
over which to assemble the operator.- boundary_info
BoundaryInfo
for the treatment of Dirichlet boundary conditions.- robin_data
Tuple providing two
Functions
that represent the Robin parameter and boundary value function. IfNone
, the resulting operator is zero.- solver_options
The
solver_options
for the operator.- name
Name of the operator.
- class pymor.discretizers.builtin.cg.InterpolationOperator(grid, function)[source]¶
Bases:
pymor.operators.numpy.NumpyMatrixBasedOperator
Vector-like Lagrange interpolation
Operator
for continuous finite element spaces.
- pymor.discretizers.builtin.cg.discretize_stationary_cg(analytical_problem, diameter=None, domain_discretizer=None, grid_type=None, grid=None, boundary_info=None, preassemble=True, mu_energy_product=None)[source]¶
Discretizes a
StationaryProblem
using finite elements.Parameters
- analytical_problem
The
StationaryProblem
to discretize.- diameter
If not
None
,diameter
is passed as an argument to thedomain_discretizer
.- domain_discretizer
Discretizer to be used for discretizing the analytical domain. This has to be a function
domain_discretizer(domain_description, diameter, ...)
. IfNone
,discretize_domain_default
is used.- grid_type
If not
None
, this parameter is forwarded todomain_discretizer
to specify the type of the generatedGrid
.- grid
Instead of using a domain discretizer, the
Grid
can also be passed directly using this parameter.- boundary_info
A
BoundaryInfo
specifying the boundary types of the grid boundary entities. Must be provided ifgrid
is specified.- preassemble
If
True
, preassemble all operators in the resultingModel
.- mu_energy_product
If not
None
,parameter values
for which to assemble the symmetric part of theOperator
of the resultingModel
fom
(ignoring the advection part). Thus, assuming no advection and a symmetric diffusion tensor,fom.products['energy']
is equal tofom.operator.assemble(mu)
, except for the fact that the former has cleared Dirichlet rows and columns, while the latter only has cleared Dirichlet rows).
Returns
- m
The
Model
that has been generated.- data
Dictionary with the following entries:
- grid
The generated
Grid
.- boundary_info
The generated
BoundaryInfo
.- unassembled_m
In case
preassemble
isTrue
, the generatedModel
before preassembling operators.
- pymor.discretizers.builtin.cg.discretize_instationary_cg(analytical_problem, diameter=None, domain_discretizer=None, grid_type=None, grid=None, boundary_info=None, num_values=None, time_stepper=None, nt=None, preassemble=True)[source]¶
Discretizes an
InstationaryProblem
with aStationaryProblem
as stationary part using finite elements.Parameters
- analytical_problem
The
InstationaryProblem
to discretize.- diameter
If not
None
,diameter
is passed as an argument to thedomain_discretizer
.- domain_discretizer
Discretizer to be used for discretizing the analytical domain. This has to be a function
domain_discretizer(domain_description, diameter, ...)
. IfNone
,discretize_domain_default
is used.- grid_type
If not
None
, this parameter is forwarded todomain_discretizer
to specify the type of the generatedGrid
.- grid
Instead of using a domain discretizer, the
Grid
can also be passed directly using this parameter.- boundary_info
A
BoundaryInfo
specifying the boundary types of the grid boundary entities. Must be provided ifgrid
is specified.- num_values
The number of returned vectors of the solution trajectory. If
None
, each intermediate vector that is calculated is returned.- time_stepper
The
time-stepper
to be used bysolve
.- nt
If
time_stepper
is not specified, the number of time steps for implicit Euler time stepping.- preassemble
If
True
, preassemble all operators in the resultingModel
.
Returns
- m
The
Model
that has been generated.- data
Dictionary with the following entries:
- grid
The generated
Grid
.- boundary_info
The generated
BoundaryInfo
.- unassembled_m
In case
preassemble
isTrue
, the generatedModel
before preassembling operators.