pygeon.discretizations.fem.hdiv module

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

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

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)[source]

Returns the number of faces.

Parameters:: sd (pg.Grid) – Grid, or a subclass.
Returns:: The number of degrees of freedom.
Return type:: int

assemble_adv_matrix(sd, data=None)[source]

Assembles and returns the advection matrix for mixed finite elements (RT0-P0), which is given by \((D^{-1}\boldsymbol{v} \cdot \boldsymbol{q}, p)\).

The trial functions \(\boldsymbol{q}\) are lowest-order Raviart-Thomas (RT0) and test functions \(p\) are piecewise constant (P0). \(\boldsymbol{v}\) is a given vector field and \(D^{-1}\) is a given second-order tensor, both assumed constant per cell. If not provided, \(\boldsymbol{v}\) defaults to \((0, 0, 0)\), and \(D^{-1}\) defaults to the identity tensor.

Parameters:

sd (pg.Grid) – Grid object or a subclass.
data (dict | None) – Optional data for scaling, in particular pg.SECOND_ORDER_TENSOR (inverse diffusion or permeability tensor) and pg.VECTOR_FIELD (advection velocity field).

Returns:

The advection matrix obtained from the discretization.

Return type:

sps.csc_array

assemble_lumped_matrix(sd, data=None)[source]

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)[source]

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)[source]

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)[source]

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)[source]

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)[source]

Constructs the projection matrix to the VecPwLinears space. This function is cached to speed up repetitive calls for the same grid.

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')[source]

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)[source]

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.

static local_inner_product(dim)[source]

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)[source]

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)[source]

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)[source]

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

interpolate(sd, func)[source]

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)[source]

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)[source]

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)[source]

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')[source]

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)[source]

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)[source]

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)[source]

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)[source]

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)[source]

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)[source]

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, 3*j : 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)[source]

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)[source]

Assembles the matrix for evaluating the discretization at the cell centers.

Parameters:: sd (pg.Grid) – Grid object or a subclass.
Returns:: The evaluation matrix.
Return type:: sps.csc_array

assemble_diff_matrix(sd)[source]

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)[source]

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)[source]

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)[source]

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)[source]

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)[source]

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)[source]

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