pymor.discretizers.builtin.cg

This module provides some operators for continuous finite element discretizations.

Module Contents

class pymor.discretizers.builtin.cg.AdvectionOperatorP1(grid, boundary_info, advection_function=None, advection_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver=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) 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 – The Solver for the operator.

  • name – Name of the operator.

sparse = True[source]
class pymor.discretizers.builtin.cg.AdvectionOperatorQ1(grid, boundary_info, advection_function=None, advection_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver=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) 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 – The Solver for the operator.

  • name – Name of the operator.

sparse = True[source]
class pymor.discretizers.builtin.cg.BoundaryDirichletFunctional(grid, dirichlet_data, boundary_info, solver=None, name=None)[source]

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.

  • solver – The Solver for the operator.

  • name – The name of the functional.

source[source]
sparse = False[source]
class pymor.discretizers.builtin.cg.BoundaryL2ProductFunctional(grid, function, boundary_type=None, dirichlet_clear_dofs=False, boundary_info=None, solver=None, name=None)[source]

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.

  • solver – The Solver for the operator.

  • name – The name of the functional.

source[source]
sparse = False[source]
class pymor.discretizers.builtin.cg.DiffusionOperatorP1(grid, boundary_info, diffusion_function=None, diffusion_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver=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) 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 – The Solver for the operator.

  • name – Name of the operator.

sparse = True[source]
class pymor.discretizers.builtin.cg.DiffusionOperatorQ1(grid, boundary_info, diffusion_function=None, diffusion_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver=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) 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 – The Solver for the operator.

  • name – Name of the operator.

sparse = True[source]
class pymor.discretizers.builtin.cg.InterpolationOperator(grid, function, solver=None, name=None)[source]

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.

  • solver – The Solver for the operator.

linear = True[source]
source[source]
class pymor.discretizers.builtin.cg.L2ProductFunctionalP1(grid, function, dirichlet_clear_dofs=False, boundary_info=None, solver=None, name=None)[source]

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.

  • solver – The Solver for the operator.

  • name – The name of the functional.

source[source]
sparse = False[source]
class pymor.discretizers.builtin.cg.L2ProductFunctionalQ1(grid, function, dirichlet_clear_dofs=False, boundary_info=None, solver=None, name=None)[source]

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.

  • solver – The Solver for the operator.

  • name – The name of the functional.

source[source]
sparse = False[source]
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=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 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 – The Solver for the operator.

  • name – The name of the product.

sparse = True[source]
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=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 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 – The Solver for the operator.

  • name – The name of the product.

sparse = True[source]
class pymor.discretizers.builtin.cg.RobinBoundaryOperator(grid, boundary_info, robin_data=None, solver=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 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 – The Solver for the operator.

  • name – Name of the operator.

sparse = True[source]
pymor.discretizers.builtin.cg.CGVectorSpace(grid)[source]
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, solver=None)[source]

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.

  • solver – The Solver to be used.

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, solver=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 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).

  • solver – The Solver to be used.

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.LagrangeShapeFunctions[source]
pymor.discretizers.builtin.cg.LagrangeShapeFunctionsGrads[source]