pymor.discretizers.builtin.cg

This module provides some operators for continuous finite element discretizations.

Module Contents

pymor.discretizers.builtin.cg.LagrangeShapeFunctions

pymor.discretizers.builtin.cg.LagrangeShapeFunctionsGrads

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)

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) with shape_range = (grid.dim, ). If None, constant one is assumed.

advection_constant

The constant c. If None, 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.

sparse = True

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)

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) with shape_range = (grid.dim, ). If None, constant one is assumed.

advection_constant

The constant c. If None, 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.

sparse = True

class pymor.discretizers.builtin.cg.BoundaryDirichletFunctional(grid, dirichlet_data, boundary_info, name=None)

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Linear 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.

source

sparse = False

class pymor.discretizers.builtin.cg.BoundaryL2ProductFunctional(grid, function, boundary_type=None, dirichlet_clear_dofs=False, boundary_info=None, name=None)

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Linear 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 or dirichlet_clear_dofs is True, the BoundaryInfo determining which boundary entity belongs to which physical boundary.

name

The name of the functional.

source

sparse = False

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)

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) with shape_range == () or shape_range = (grid.dim, grid.dim). If None, constant one is assumed.

diffusion_constant

The constant c. If None, 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.

sparse = True

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)

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) with shape_range == () or shape_range = (grid.dim, grid.dim). If None, constant one is assumed.

diffusion_constant

The constant c. If None, 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.

sparse = True

class pymor.discretizers.builtin.cg.InterpolationOperator(grid, function)

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Vector-like Lagrange interpolation Operator for continuous finite element spaces.

Parameters

grid

The Grid on which to interpolate.

function

The Function to interpolate.

linear = True

source

class pymor.discretizers.builtin.cg.L2ProductFunctionalP1(grid, function, dirichlet_clear_dofs=False, boundary_info=None, name=None)

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Linear 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 case dirichlet_clear_dofs is set to True.

name

The name of the functional.

source

sparse = False

class pymor.discretizers.builtin.cg.L2ProductFunctionalQ1(grid, function, dirichlet_clear_dofs=False, boundary_info=None, name=None)

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Bilinear 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 case dirichlet_clear_dofs is set to True.

name

The name of the functional.

source

sparse = False

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)

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 either dirichlet_clear_rows or dirichlet_clear_columns is True, the diagonal entries are set to one.

coefficient_function

Coefficient Function for product with shape_range == (). If None, constant one is assumed.

solver_options

The solver_options for the operator.

name

The name of the product.

sparse = True

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)

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 either dirichlet_clear_rows or dirichlet_clear_columns is True, the diagonal entries are set to one.

coefficient_function

Coefficient Function for product with shape_range == (). If None, constant one is assumed.

solver_options

The solver_options for the operator.

name

The name of the product.

sparse = True

class pymor.discretizers.builtin.cg.RobinBoundaryOperator(grid, boundary_info, robin_data=None, solver_options=None, name=None)

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 and n are the diffusion function (see DiffusionOperatorP1) and the unit outer normal in x, while c is the (scalar) Robin parameter function and g 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 (c(x), g(x)) providing two Functions that represent the Robin parameter and boundary value function. If None, the resulting operator is zero.

solver_options

The solver_options for the operator.

name

Name of the operator.

sparse = True

pymor.discretizers.builtin.cg.CGVectorSpace(grid, id='STATE')

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)

Finite Element discretization of an InstationaryProblem.

Discretizes an InstationaryProblem with a StationaryProblem as the stationary part using finite elements.

Parameters

analytical_problem

The InstationaryProblem to discretize.

diameter

If not None, diameter is passed as an argument to the domain_discretizer.

domain_discretizer

Discretizer to be used for discretizing the analytical domain. This has to be a function domain_discretizer(domain_description, diameter, ...). If None, discretize_domain_default is used.

grid_type

If not None, this parameter is forwarded to domain_discretizer to specify the type of the generated Grid.

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 if grid 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 by solve of InstationaryModel.

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 resulting Model.

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 is True, the generated Model before preassembling operators.

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)

Discretizes a StationaryProblem using finite elements.

Parameters

analytical_problem

The StationaryProblem to discretize.

diameter

If not None, diameter is passed as an argument to the domain_discretizer.

domain_discretizer

Discretizer to be used for discretizing the analytical domain. This has to be a function domain_discretizer(domain_description, diameter, ...). If None, discretize_domain_default is used.

grid_type

If not None, this parameter is forwarded to domain_discretizer to specify the type of the generated Grid.

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 if grid is specified.

preassemble

If True, preassemble all operators in the resulting Model.

mu_energy_product

If not None, parameter values for which to assemble the symmetric part of the Operator of the resulting Model fom (ignoring the advection part). Thus, assuming no advection and a symmetric diffusion tensor, fom.products['energy'] is equal to fom.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 is True, the generated Model before preassembling operators.