porepy.numerics.interface_laws.elliptic_discretization module

The module contains the base class for all elliptic discretization methods, to be used in both mono and mixed-dimensional problems. One instance for the scalar and one for the vector equation.

The classes may be interpreted as a “contract” which elliptic discretization methods must honor. This is particularly useful to ensure uniformity when coupling discretizations between dimensions by interface laws.

class EllipticDiscretization(keyword)[source]

Bases: Discretization

This is the parent class of all discretizations for second order elliptic problems. The class cannot be used itself, but should rather be seen as a declaration of which methods are assumed implemented for all specific discretization schemes.

Subclasses are intended used both on single grids, and as components in a mixed-dimensional, or more generally multiple grid problem. In the latter case, this class will provide the discretization on individual grids; a full discretization will also need discretizations of the edge problems (coupling between grids) and a way to assemble the grids.

The class methods should together take care of both the discretization (defined as construction of operators used when writing down the continuous form of the equations, e.g. divergence, transmissibility matrix etc), and assembly of (whole or part of) the system matrix. For problems on multiple grids, an assembly will only be a partial (considering a single grid)- the full system matrix for multiple grids is handled somewhere else.

Known realizations of the elliptic discretization are:

Tpfa: Finite volume method using a wwo-point flux approximation. Mpfa: Finite volume method using a multi-point flux approximation. RT0: Mixed finite element method, using the lowest order Raviart-Thomas

elements.

MVEM: Mixed formulation of the lowest order virtual element method.

Parameters

keyword (str) –

keyword

This is used to identify operators and parameters that the discretization will work on.

Type

str

abstract assemble_int_bound_flux(sd, sd_data, intf, intf_data, 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

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

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

  • intf_data (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.

  • 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.

  • intf (MortarGrid) –

  • rhs (ndarray) –

Return type

None

abstract assemble_int_bound_pressure_cell(sd, sd_data, intf, intf_data, 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.

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

  • intf_data (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.

  • intf (MortarGrid) –

Return type

None

abstract assemble_int_bound_pressure_trace(sd, sd_data, intf, intf_data, 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

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

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

  • intf_data (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.

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

  • intf (MortarGrid) –

Return type

None

abstract assemble_int_bound_pressure_trace_between_interfaces(sd, sd_data, proj_primary, proj_secondary, cc, matrix, rhs)[source]

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

Parameters

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

  • sd_data – Data dictionary for the node in the mixed-dimensional grid.

  • proj_primary (spmatrix) –

  • proj_secondary (spmatrix) –

  • cc (ndarray) –

  • matrix (ndarray) –

  • rhs (ndarray) –

Return type

None

abstract assemble_int_bound_pressure_trace_rhs(sd, sd_data, intf, intf_data, 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

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

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

  • intf_data (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.

abstract assemble_int_bound_source(sd, sd_data, intf, intf_data, 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

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

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

  • data_intf (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.

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

  • sd_data (dict) –

  • intf (MortarGrid) –

  • intf_data (dict) –

  • rhs (ndarray) –

Return type

None

abstract enforce_neumann_int_bound(sd, intf, intf_data, matrix, self_ind)[source]

Enforce Neumann boundary conditions on a given system matrix.

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

The discretization matrix should be modified in place.

Parameters

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

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

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

  • intf (MortarGrid) –

  • intf_data (dict) –

  • self_ind (int) –

Return type

None

extract_flux(sd, solution_array, data)[source]

Abstract method. Extract the pressure part of a solution.

The implementation will depend what are the primary variables of the specific implementation.

TODO: We should incrude the boundary condition as well?

Parameters

  • sd (grid) – To which the solution array belongs.

  • solution_array (np.array) – Solution for this grid obtained from either a mono-dimensional or a mixed-dimensional problem. Will correspond to the pressure solution.

  • data (dictionary) – Data dictionary associated with the grid.

Returns

Flux vector.

Return type

np.array (g.num_faces)

extract_pressure(sd, solution_array, data)[source]

Abstract method. Extract the pressure part of a solution.

The implementation will depend what the primary variables of the specific implementation are.

Parameters

  • sd (pp.Grid) – To which the solution array belongs.

  • solution_array (np.array) – Solution for this grid obtained from either a mono-dimensional or a mixed-dimensional problem.

  • data (dictionary) – Data dictionary associated with the grid. Not used, but included for consistency reasons.

Returns

Pressure solution vector. Will be identical

to solution_array.

Return type

np.array (g.num_cells)

abstract ndof(sd)[source]

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

Parameters

sd (grid) – Computational grid.

Returns

the number of degrees of freedom.

Return type

int