pygeon.discretizations.fem.hdiv module

Module for the discretizations of the H(div) space.

class pygeon.discretizations.fem.hdiv.RT0(keyword='unitary_data')

Bases: Discretization

Discretization class for Raviart-Thomas of lowest order. Each degree of freedom is the integral over a mesh face.

The implementation of this class is inspired by the RT0 class in PorePy.

poly_order = 1

Polynomial degree of the basis functions

tensor_order = 1

Vector-valued discretization

ndof(sd)

Returns the number of faces.

Parameters:

sd (pg.Grid) – Grid, or a subclass.

Returns:

The number of degrees of freedom.

Return type:

int

assemble_mass_matrix(sd, data=None)

Assembles the mass matrix

Parameters:

• sd (pg.Grid) – Grid object or a subclass.

• data (dict | None) – Optional dictionary with physical parameters for scaling, in particular the pg.SECOND_ORDER_TENSOR that is the inverse of the diffusion tensor (permeability for porous media).

Returns:

The mass matrix.

Return type:

sps.csc_array

static local_inner_product(sd)

Compute the local inner product matrix for a given grid.

Parameters:

sd (pg.Grid) – The grid object containing the discretization information.

Returns:

Local inner product matrix.

Return type:

np.ndarray

static eval_basis(coord, sign, dim)

Evaluate the basis functions.

Parameters:

• coord (np.ndarray) – The coordinates of the opposite node for each face.

• sign (np.ndarray) – The sign associated to each of the face of the degree of freedom.

• dim (int) – The dimension of the grid.

Returns:

The value of the basis functions.

Return type:

np.ndarray

eval_at_cell_centers(sd)

Evaluate the finite element solution at the cell centers of the given grid.

Parameters:

sd (pg.Grid) – The grid on which to evaluate the solution.

Returns:

The finite element solution evaluated at the cell centers.

Return type:

sps.csc_array

assemble_lumped_matrix(sd, data=None)

Assembles the lumped mass matrix L such that B^T L^{-1} B is a TPFA method.

Parameters:

• sd (pg.Grid) – Grid object or a subclass.

• data (dict | None) – Optional dictionary with physical parameters for scaling. In particular the pg.SECOND_ORDER_TENSOR that is the inverse of the diffusion tensor (permeability for porous media).

Returns:

The lumped mass matrix.

Return type:

sps.csc_array

assemble_diff_matrix(sd)

Assembles the matrix corresponding to the differential operator, the divergence in this case.

Parameters:

sd (pg.Grid) – Grid object or a subclass.

Returns:

The differential matrix.

Return type:

sps.csc_array

interpolate(sd, func)

Interpolates a function onto the finite element space

Parameters:

• sd (pg.Grid) – Grid, or a subclass.

• func (Callable[[np.ndarray], np.ndarray]) – A function that returns the function values at coordinates.

Returns:

The values of the degrees of freedom.

Return type:

np.ndarray

assemble_nat_bc(sd, func, b_faces)

Assembles the natural boundary condition term (n dot q, func)_Gamma

Parameters:

• sd (pg.Grid) – The grid object representing the computational domain.

• func (Callable[[np.ndarray], np.ndarray]) – The function that defines the natural boundary condition.

• b_faces (np.ndarray) – The array of boundary faces.

Returns:

The assembled natural boundary condition term.

Return type:

np.ndarray

get_range_discr_class(dim)

Returns the range discretization class for the given dimension.

Parameters:

dim (int) – The dimension of the range space.

Returns:

The range discretization class.

Return type:

pg.Discretization

proj_to_PwPolynomials(sd)

Constructs the projection matrix from the current finite element space to the VecPwLinears space.

Parameters:

sd (pg.Grid) – The grid object.

Returns:

A sparse array in CSC format representing the projection from the current space to VecPwLinears.

Return type:

sps.csc_array

class pygeon.discretizations.fem.hdiv.BDM1(keyword='unitary_data')

Bases: Discretization

BDM1 is a class that represents the BDM1 (Brezzi-Douglas-Marini) finite element method. It provides methods for assembling matrices, projecting to and from the RT0 space, evaluating the solution at cell centers, interpolating a given function onto the grid, assembling the natural boundary condition term, and more.

poly_order = 1

Polynomial degree of the basis functions

tensor_order = 1

Vector-valued discretization

ndof(sd)

Return the number of degrees of freedom associated to the method. In this case the number of faces times the dimension.

Parameters:

sd (pp.Grid) – Grid object or a subclass.

Returns:

The number of degrees of freedom.

Return type:

int

Raises:

ValueError – If the input grid is not an instance of pp.Grid.

local_dofs_of_cell(sd, faces_loc)

Compute the local degrees of freedom (DOFs) indices for a cell.

Parameters:

• sd (pp.Grid) – Grid object or a subclass.

• faces_loc (np.ndarray) – Array of local face indices for the cell.

Returns:

Array of local DOF indices associated with the cell.

Return type:

np.ndarray

assemble_mass_matrix(sd, data=None)

Assembles the mass matrix for the given grid.

Parameters:

• sd (pg.Grid) – The grid for which the mass matrix is assembled.

• data (dict | None) – Additional data for the assembly process.

Returns:

The assembled mass matrix.

Return type:

sps.csc_array

eval_basis_at_node(sd, opposites, faces_loc, return_node_ind=False)

Compute the local basis function for the BDM1 finite element space.

Parameters:

• sd (pg.Grid) – The grid object.

• opposites (np.ndarray) – The local degrees of freedom.

• cell_nodes_loc (np.ndarray) – The local nodes of the cell.

• faces_loc (np.ndarray) – The local faces.

• return_node_ind (bool) – Whether to return the local indexing of the nodes, used in assemble_lumped_matrix.

Returns:

The local mass matrix.

Return type:

np.ndarray

static local_inner_product(dim)

Compute the local inner product matrix for the given dimension.

Parameters:

dim (int) – The dimension of the matrix.

Returns:

The computed local inner product matrix.

Return type:

np.ndarray

proj_to_RT0(sd)

Project the function space to the lowest order Raviart-Thomas (RT0) space.

Parameters:

sd (pg.Grid) – The grid object representing the computational domain.

Returns:

The projection matrix to the RT0 space.

Return type:

sps.csc_array

proj_from_RT0(sd)

Project the RT0 finite element space onto the faces of the given grid.

Parameters:

sd (pg.Grid) – The grid on which the projection is performed.

Returns:

The projection matrix.

Return type:

sps.csc_array

assemble_diff_matrix(sd)

Assembles the matrix corresponding to the differential operator.

Parameters:

sd (pg.Grid) – Grid object or a subclass.

Returns:

The differential matrix.

Return type:

sps.csc_array

eval_at_cell_centers(sd)

Evaluate the finite element solution at the cell centers of the given grid.

Parameters:

sd (pg.Grid) – The grid on which to evaluate the solution.

Returns:

The finite element solution evaluated at the cell centers.

Return type:

sps.csc_array

interpolate(sd, func)

Interpolates a given function onto the grid.

Parameters:

• sd (pg.Grid) – The grid on which to interpolate the function.

• func (Callable[[np.ndarray], np.ndarray]) – The function to be interpolated.

Returns:

The interpolated values on the grid.

Return type:

np.ndarray

assemble_nat_bc(sd, func, b_faces)

Assembles the natural boundary condition term (n dot q, func)_Gamma

Parameters:

• sd (pg.Grid) – The grid object representing the computational domain.

• func (Callable[[np.ndarray], np.ndarray]) – The function that defines the natural boundary condition.

• b_faces (np.ndarray) – The array of boundary faces.

Returns:

The assembled natural boundary condition term.

Return type:

np.ndarray

get_range_discr_class(dim)

Returns the range discretization class for the given dimension.

Parameters:

dim (int) – The dimension of the range space.

Returns:

The range discretization class.

Return type:

pg.Discretization

assemble_lumped_matrix(sd, data=None)

Assembles the lumped matrix for the given grid.

Parameters:

• sd (pg.Grid) – The grid object.

• data (dict | None) – Optional data dictionary.

Returns:

The assembled lumped matrix.

Return type:

sps.csc_array

proj_to_PwPolynomials(sd)

Constructs the projection matrix from the current finite element space to the VecPwLinears space.

Parameters:

sd (pg.Grid) – The grid object.

Returns:

A sparse array in CSC format representing the projection from the current space to VecPwLinears.

Return type:

sps.csc_array

class pygeon.discretizations.fem.hdiv.RT1(keyword='unitary_data')

Bases: Discretization

RT1 Discretization class for H(div) finite element method.

This class implements the Raviart-Thomas elements of order 1 (RT1) for discretizing vector fields in H(div) space. It provides methods for assembling mass matrices, differential matrices, evaluating basis functions, and interpolating functions onto the finite element space.

poly_order = 2

Polynomial degree of the basis functions

tensor_order = 1

Vector-valued discretization

ndof(sd)

Returns the number of degrees of freedom.

Parameters:

sd (pg.Grid) – Grid, or a subclass.

Returns:

The number of degrees of freedom.

Return type:

int

local_dofs_of_cell(sd, faces_loc, c)

Compute the local degrees of freedom (DOFs) indices for a cell.

Parameters:

• sd (pp.Grid) – Grid object or a subclass.

• faces_loc (np.ndarray) – Array of local face indices for the cell.

• c (int) – Cell index.

Returns:

Array of local DOF indices associated with the cell.

Return type:

np.ndarray

assemble_mass_matrix(sd, data=None)

Assembles the mass matrix

Parameters:

• sd (pg.Grid) – Grid object or a subclass.

• data (dict | None) – Optional dictionary with physical parameters for scaling, in particular the pg.SECOND_ORDER_TENSOR that is the inverse of the diffusion tensor (permeability for porous media).

Returns:

The mass matrix.

Return type:

sps.csc_array

local_inner_product(dim)

Assembles the local inner products based on the Lagrange2 element

Parameters:

dim (int) – Dimension of the grid.

Returns:

The local mass matrix.

Return type:

np.ndarray

reorder_faces(cell_faces, opposite_nodes, cell)

Reorders the local nodes, faces, and corresponding cell-face orientations

Parameters:

• cell_faces (sps.csc_array) – Cell_face connectivity of the grid.

• opposite_nodes (sps.csc_array) – Opposite nodes for each face.

• cell (int) – Cell index.

Returns:

The reordered local node indices np.ndarray: The reordered local face indices np.ndarray: The reordered cell-face orientation signs

Return type:

np.ndarray

eval_basis_functions(sd, nodes_loc, signs_loc, volume)

Evaluates the basis functions at the nodes and edges of a cell.

Parameters:

• sd (pg.Grid) – The grid.

• nodes_loc (np.ndarray) – Nodes of the cell.

• signs_loc (np.ndarray) – Cell-face orientation signs.

• volume (float) – Cell volume.

Returns:

An array Psi in which [i, 3j : 3(j + 1)] contains the values of basis function phi_i at evaluation point j

Return type:

np.ndarray

eval_basis_functions_at_center(sd, nodes_loc, volume)

Evaluates the basis functions at the center of a cell.

Parameters:

• sd (pg.Grid) – The grid.

• nodes_loc (np.ndarray) – Nodes of the cell.

• volume (float) – Cell volume.

Returns:

A (3 x dim) array with the values of the cell-based basis functions at the cell center.

Return type:

np.ndarray

eval_at_cell_centers(sd)

Evaluate the finite element solution at the cell centers of the given grid.

Parameters:

sd (pg.Grid) – The grid on which to evaluate the solution.

Returns:

The finite element solution evaluated at the cell centers.

Return type:

sps.csc_array

assemble_diff_matrix(sd)

Assembles the matrix corresponding to the differential operator, the divergence in this case.

Parameters:

sd (pg.Grid) – Grid object or a subclass.

Returns:

The differential matrix.

Return type:

sps.csc_array

compute_local_div_matrix(dim)

Assembles the local divergence matrix using local node and face ordering

Parameters:

dim (int) – Dimension of the grid.

Returns:

The local divergence matrix

Return type:

np.ndarray

interpolate(sd, func)

Interpolates a function onto the finite element space

Parameters:

• sd (pg.Grid) – Grid, or a subclass.

• func (Callable[[np.ndarray], np.ndarray]) – A function that returns the function values at coordinates.

Returns:

The values of the degrees of freedom.

Return type:

np.ndarray

assemble_nat_bc(sd, func, b_faces)

Assembles the natural boundary condition term (n dot q, func)_Gamma

Parameters:

• sd (pg.Grid) – The grid object representing the computational domain.

• func (Callable[[np.ndarray], np.ndarray]) – The function that defines the natural boundary condition.

• b_faces (np.ndarray) – The array of boundary faces.

Returns:

The assembled natural boundary condition term.

Return type:

np.ndarray

get_range_discr_class(dim)

Returns the range discretization class for the given dimension.

Parameters:

dim (int) – The dimension of the range space.

Returns:

The range discretization class.

Return type:

pg.Discretization

assemble_lumped_matrix(sd, data=None)

Assembles the lumped matrix for the given grid, using the integration rule from Egger & Radu (2020)

Parameters:

• sd (pg.Grid) – The grid object.

• data (dict | None) – Optional data dictionary.

Returns:

The assembled lumped matrix.

Return type:

sps.csc_array

proj_to_PwPolynomials(sd)

Constructs the projection matrix from the current finite element space to the VecPwQuadratics space.

Parameters:

sd (pg.Grid) – The grid object.

Returns:

A sparse array in CSC format representing the projection from the current space to VecPwQuadratics.

Return type:

sps.csc_array