porepy.numerics.vem.dual_elliptic module

Module contains common functionalities for discretization based on the mixed variational formulation.

class DualElliptic(keyword, name)[source]

Bases: EllipticDiscretization

Parent class for methods based on the mixed variational form of the elliptic equation. The class should not be used by itself, but provides a sheared implementation of central methods.

Known subclasses that can be used for actual discretization are MVEM and RT0.

Parameters

  • keyword (str) –

  • name (str) –

assemble_int_bound_flux(sd, data, intf, data_edge, cc, matrix, rhs, self_ind, use_secondary_proj=False)[source]

Abstract method. Assemble the contribution from an internal boundary, manifested as a flux boundary condition.

The intended use is when the internal boundary is coupled to another node in a mixed-dimensional method. Specific usage depends on the interface condition between the nodes; this method will typically be used to impose flux continuity on a higher-dimensional domain.

Implementations of this method will use an interplay between the grid on the node and the mortar grid on the relevant edge.

Parameters

  • g (Grid) – Grid which the condition should be imposed on.

  • data (dictionary) – Data dictionary for the node in the mixed-dimensional grid.

  • data_edge (dictionary) – Data dictionary for the edge in the mixed-dimensional grid.

  • cc (block matrix, 3x3) – Block matrix for the coupling condition. The first and second rows and columns are identified with the primary and secondary side; the third belongs to the edge variable. The discretization of the relevant term is done in-place in cc.

  • matrix (block matrix 3x3) – Discretization matrix for the edge and the two adjacent nodes.

  • rhs (block_array 3x1) – Right hand side contribution for the edge and the two adjacent nodes.

  • self_ind (int) – Index in cc and matrix associated with this node. Should be either 1 or 2.

  • use_secondary_proj (boolean) – If True, the secondary side projection operator is used. Needed for periodic boundary conditions.

  • sd (Grid) –

  • intf (MortarGrid) –

Return type

None

assemble_int_bound_pressure_cell(sd, data, intf, data_edge, cc, matrix, rhs, self_ind)[source]

Abstract method. Assemble the contribution from an internal boundary, manifested as a condition on the cell pressure.

The intended use is when the internal boundary is coupled to another node in a mixed-dimensional method. Specific usage depends on the interface condition between the nodes; this method will typically be used to impose flux continuity on a lower-dimensional domain.

Implementations of this method will use an interplay between the grid on the node and the mortar grid on the relevant edge.

Parameters

  • sd (Grid) – Grid which the condition should be imposed on.

  • data (dictionary) – Data dictionary for the node in the mixed-dimensional grid.

  • data_edge (dictionary) – Data dictionary for the edge in the mixed-dimensional grid.

  • grid_swap (boolean) – If True, the grid g is identified with the @ secondary side of the mortar grid in data_adge.

  • cc (block matrix, 3x3) – Block matrix for the coupling condition. The first and second rows and columns are identified with the primary and secondary side; the third belongs to the edge variable. The discretization of the relevant term is done in-place in cc.

  • matrix (block matrix 3x3) – Discretization matrix for the edge and the two adjacent nodes.

  • rhs (block_array 3x1) – Right hand side contribution for the edge and the two adjacent nodes.

  • self_ind (int) – Index in cc and matrix associated with this node. Should be either 1 or 2.

  • intf (MortarGrid) –

Return type

None

assemble_int_bound_pressure_trace(sd, data, intf, data_edge, cc, matrix, rhs, self_ind, use_secondary_proj=False, assemble_matrix=True, assemble_rhs=True)[source]

Abstract method. Assemble the contribution from an internal boundary, manifested as a condition on the boundary pressure.

The intended use is when the internal boundary is coupled to another node in a mixed-dimensional method. Specific usage depends on the interface condition between the nodes; this method will typically be used to impose flux continuity on a higher-dimensional domain.

Implementations of this method will use an interplay between the grid on the node and the mortar grid on the relevant edge.

Parameters

  • g (Grid) – Grid which the condition should be imposed on.

  • data (dictionary) – Data dictionary for the node in the mixed-dimensional grid.

  • data_edge (dictionary) – Data dictionary for the edge in the mixed-dimensional grid.

  • cc (block matrix, 3x3) – Block matrix for the coupling condition. The first and second rows and columns are identified with the primary and secondary side; the third belongs to the edge variable. The discretization of the relevant term is done in-place in cc.

  • matrix (block matrix 3x3) – Discretization matrix for the edge and the two adjacent nodes.

  • rhs (block_array 3x1) – Right hand side contribution for the edge and the two adjacent nodes.

  • self_ind (int) – Index in cc and matrix associated with this node. Should be either 1 or 2.

  • use_secondary_proj (boolean) – If True, the secondary side projection operator is used. Needed for periodic boundary conditions.

  • sd (Grid) –

  • intf (MortarGrid) –

Return type

None

assemble_int_bound_pressure_trace_between_interfaces(g, data_grid, data_primary_edge, data_secondary_edge, cc, matrix, rhs)[source]

Assemble the contribution from an internal boundary, manifested as a condition on the boundary pressure.

No contribution for this method.

Parameters

  • g (Grid) – Grid which the condition should be imposed on.

  • data_grid (dictionary) – Data dictionary for the node in the mixed-dimensional grid.

  • data_primary_edge (dictionary) – Data dictionary for the primary edge in the mixed-dimensional grid.

  • data_secondary_edge (dictionary) – Data dictionary for the secondary edge in the mixed-dimensional grid.

  • cc (block matrix, 3x3) – Block matrix of size 3 x 3, whwere each block represents coupling between variables on this interface. Index 0, 1 and 2 represent the primary grid, the primary and secondary interface, respectively.

  • matrix (block matrix 3x3) – Discretization matrix for the edge and the two adjacent nodes.

  • rhs (block_array 3x1) – Block matrix of size 3 x 1, representing the right hand side of this coupling. Index 0, 1 and 2 represent the primary grid, the primary and secondary interface, respectively.

Return type

None

assemble_int_bound_pressure_trace_rhs(sd, data, data_edge, cc, rhs, self_ind, use_secondary_proj=False)[source]

Assemble the rhs contribution from an internal boundary, manifested as a condition on the boundary pressure.

For details, see self.assemble_int_bound_pressure_trace()

Parameters

  • g (Grid) – Grid which the condition should be imposed on.

  • data (dictionary) – Data dictionary for the node in the mixed-dimensional grid.

  • data_edge (dictionary) – Data dictionary for the edge in the mixed-dimensional grid.

  • cc (block matrix, 3x3) – Block matrix for the coupling condition. The first and second rows and columns are identified with the primary and secondary side; the third belongs to the edge variable. The discretization of the relevant term is done in-place in cc.

  • matrix (block matrix 3x3) – Discretization matrix for the edge and the two adjacent nodes.

  • rhs (block_array 3x1) – Right hand side contribution for the edge and the two adjacent nodes.

  • self_ind (int) – Index in cc and matrix associated with this node. Should be either 1 or 2.

  • use_secondary_proj (boolean) – If True, the secondary side projection operator is used. Needed for periodic boundary conditions.

assemble_int_bound_source(sd, data, intf, data_edge, cc, matrix, rhs, self_ind)[source]

Abstract method. Assemble the contribution from an internal boundary, manifested as a source term.

The intended use is when the internal boundary is coupled to another node in a mixed-dimensional method. Specific usage depends on the interface condition between the nodes; this method will typically be used to impose flux continuity on a lower-dimensional domain.

Implementations of this method will use an interplay between the grid on the node and the mortar grid on the relevant edge.

Parameters

  • g (Grid) – Grid which the condition should be imposed on.

  • data (dictionary) – Data dictionary for the node in the mixed-dimensional grid.

  • data_edge (dictionary) – Data dictionary for the edge in the mixed-dimensional grid.

  • cc (block matrix, 3x3) – Block matrix for the coupling condition. The first and second rows and columns are identified with the primary and secondary side; the third belongs to the edge variable. The discretization of the relevant term is done in-place in cc.

  • matrix (block matrix 3x3) – Discretization matrix for the edge and the two adjacent nodes.

  • rhs (block_array 3x1) – Right hand side contribution for the edge and the two adjacent nodes.

  • self_ind (int) – Index in cc and matrix associated with this node. Should be either 1 or 2.

  • sd (Grid) –

  • intf (MortarGrid) –

Return type

None

assemble_matrix(g, data)[source]

Assemble matrix from an existing discretization.

Parameters

  • sd (pp.Grid) – Computational grid, with geometry fields computed.

  • data (dictionary) – With data stored.

  • g (Grid) –

Returns

System matrix of this discretization. The

size of the matrix will depend on the specific discretization.

Return type

scipy.sparse.csr_matrix

assemble_matrix_rhs(sd, data)[source]

Return the matrix and right-hand side for a discretization of a second order elliptic equation.

Parameters

  • sd (pp.Grid) – Computational grid, with geometry fields computed.

  • data (dictionary) – With data stored.

Returns

System matrix of this discretization. The size of

the matrix will depend on the specific discretization.

np.ndarray: Right-hand side vector with representation of boundary

conditions. The size of the vector will depend on the discretization.

Return type

scipy.sparse.csr_matrix

assemble_neumann_robin(sd, data, M, bc_weight=False)[source]

Impose Neumann and Robin boundary discretization on an already assembled system matrix.

Parameters
Return type

tuple[scipy.sparse._csr.csr_matrix, int]

assemble_rhs(sd, data, bc_weight=1.0)[source]

Return the righ-hand side for a discretization of a second order elliptic equation.

Parameters

  • sd (Grid) – Computational grid, with geometry fields computed.

  • data (dictionary) – With data stored.

  • bc_weight (float) – to use the infinity norm of the matrix to impose the boundary conditions. Default 1.

Returns

Right hand side vector with representation of boundary

conditions. The size of the vector will depend on the discretization.

Return type

np.ndarray

enforce_neumann_int_bound(sd, intf, data_edge, matrix, self_ind)[source]

Enforce Neumann boundary conditions on a given system matrix.

Methods based on a mixed variational form need this function to implement essential boundary conditions.

The discretization matrix should be modified in place.

Parameters

  • g (Grid) – On which the equation is discretized

  • data (dictionary) – Of data related to the discretization.

  • matrix (scipy.sparse.matrix) – Discretization matrix to be modified.

  • self_ind (int) – Index in local block system of this grid and variable.

  • sd (Grid) –

  • intf (MortarGrid) –

  • data_edge (dict) –

Return type

None

extract_flux(sd, solution_array, data)[source]

Extract the velocity from a dual virtual element solution.

Parameters

sd : grid, or a subclass, with geometry fields computed. up : array (sd.num_faces+sd.num_cells)

Solution, stored as [velocity,pressure]

data: data dictionary associated with the grid.

Unused, but included for consistency reasons.

Return

uarray (sd.num_faces)

Velocity at each face.

Parameters
Return type

ndarray

extract_pressure(sd, solution_array, data)[source]

Extract the pressure from a dual virtual element solution.

Parameters

sd : grid, or a subclass, with geometry fields computed. solution_array : array (sd.num_faces + sd.num_cells)

Solution, stored as [velocity,pressure]

data: data dictionary associated with the grid.

Unused, but included for consistency reasons.

Return

parray (sd.num_cells)

Pressure at each cell.

Parameters
Return type

ndarray

ndof(sd)[source]

Return the number of degrees of freedom associated to the method.

Parameters

sd (pp.Grid) – A grid.

Returns

The number of degrees of freedom.

Return type

int

project_flux(sd, u, data)[source]

Project the velocity computed with a dual solver to obtain a piecewise constant vector field, one triplet for each cell.

We assume the following two sub-dictionaries to be present in the data dictionary:

matrix_dictionary, for storage of discretization matrices.

Stored in data[pp.DISCRETIZATION_MATRICES][self.keyword] with matrix named self.vector_proj_key and constructed in the discretize method.

Parameters

sd : grid, or a subclass, with geometry fields computed. u : array (sd.num_faces) Velocity at each face. data: data of the current grid.

Return

P0u : ndarray (3, sd.num_faces) Velocity at each cell.

Parameters
Return type

ndarray

project_flux(mdg, discr, flux, P0_flux, mortar_key='mortar_solution')[source]

Save in the grid bucket a piece-wise vector representation of the flux for each grid.

Parameters

mdg: the grid bucket discr: discretization class flux: identifier of the flux, already split, in the grid bucket P0_flux: identifier of the reconstructed flux which will be added to the grid bucket. mortar_key (optional): identifier of the mortar variable, already split, in the

grid bucket. The default value is “mortar_solution”.

Parameters
Return type

None