pygeon.discretizations.fem.hcurl module

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

class pygeon.discretizations.fem.hcurl.Nedelec0(keyword='unitary_data')[source]

Bases: Discretization

Discretization class for the Nedelec of the first kind of lowest order. Each degree of freedom is the integral over a mesh edge in 3D.

poly_order = 1: Polynomial degree of the basis functions

tensor_order = 1: Vector-valued discretization

ndof(sd)[source]

Returns the number of degrees of freedom associated to the method. In this case, it returns the number of ridges in the given grid.

Parameters:: sd (pg.Grid) – The grid for which the number of degrees of freedom is calculated.
Returns:: The number of degrees of freedom.
Return type:: int

assemble_mass_matrix(sd, data=None)[source]

Computes the mass matrix for a lowest-order Nedelec discretization

Parameters:

sd (pg.Grid) – Grid, or a subclass, with geometry fields computed.
data (dict | None) – Dictionary to store the data. See self.matrix_rhs for required contents.

Returns:

Matrix obtained from the discretization.

Return type:

sps.csc_array

eval_basis_at_node(sd, ridges_loc)[source]

Compute the local basis function for the Nedelec0 finite element space.

Parameters:

sd (pg.Grid) – The grid object.
ridges_loc (np.ndarray) – The local ridges.

Returns:

The local mass matrix.

Return type:

np.ndarray

assemble_diff_matrix(sd)[source]

Assembles the differential matrix for the given grid.

Parameters:: sd (pg.Grid) – The grid for which the differential matrix is assembled.
Returns:: The assembled differential matrix.
Return type:: sps.csc_array

eval_at_cell_centers(sd)[source]

Evaluate the function at the cell centers of the given grid.

Parameters:: sd (pg.Grid) – The grid object representing the discretization.
Returns:: The evaluated function values at the cell centers.
Return type:: sps.csc_array

proj_to_PwPolynomials(sd)[source]

Returns the inclusion matrix that projects the finite element space onto the lowest order piecewise polynomial space without loss of information.

Parameters:: sd (pg.Grid) – The grid.
Returns:: The inclusion matrix.
Return type:: sps.csc_array

assemble_nat_bc(sd, func, b_faces)[source]

Assembles the natural boundary condition matrix for the given grid and function.

Parameters:

sd (pg.Grid) – The grid on which to assemble the matrix.
func (Callable) – The function defining the natural boundary condition.
b_faces (np.ndarray) – The array of boundary faces.

Returns:

The assembled natural boundary condition matrix.

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 for the given grid.
Return type:: pg.Discretization

interpolate(sd, func)[source]

Interpolates a given function onto the grid using the hcurl discretization.

Parameters:

sd (pg.Grid) – The grid on which to interpolate the function.
func (Callable) – The function to be interpolated.

Returns:

The interpolated values on the grid.

Return type:

np.ndarray

class pygeon.discretizations.fem.hcurl.Nedelec1(keyword='unitary_data')[source]

Bases: Discretization

Discretization class for the Nedelec of the second kind of lowest order. Each degree of freedom is a first moment over a mesh edge in 3D.

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, it returns twice the number of ridges in the given grid.

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

assemble_mass_matrix(sd, data=None)[source]

Assembles the mass matrix for the given grid and data.

Parameters:

sd (pg.Grid) – The grid for which the mass matrix is to be assembled.
data (dict | None) – Additional data required for the assembly process.

Returns:

The assembled mass matrix.

Return type:

sps.csc_array

assemble_lumped_matrix(sd, data=None)[source]

Assembles the lumped matrix for the given grid and data.

Parameters:

sd (pg.Grid) – The grid object.
data (dict | None) – Additional data. Defaults to None.

Returns:

The assembled lumped matrix.

Return type:

sps.csc_array

proj_to_Ne0(sd)[source]

Project the solution to the Nedelec of the first kind.

Parameters:: sd (pg.Grid) – The grid object representing the discretization.
Returns:: The projection matrix to the Nedelec of the first kind.
Return type:: sps.csc_array

assemble_diff_matrix(sd)[source]

Assembles the differential matrix for the H(curl) finite element space.

Parameters:: sd (pg.Grid) – The grid on which the finite element space is defined.
Returns:: The assembled differential matrix.
Return type:: sps.csc_array

interpolate(sd, func)[source]

Interpolates the given function func over the specified grid sd.

Parameters:

sd (pg.Grid) – The grid over which to interpolate the function.
func (Callable) – The function to be interpolated.

Returns:

The interpolated values.

Return type:

np.ndarray

eval_at_cell_centers(sd)[source]

Evaluate the basis functions at the cell centers and construct the global matrices.

Parameters:: sd (pg.Grid) – The grid object representing the mesh.
Returns:: The global matrices constructed from the basis functions evaluated at the cell centers.
Return type:: sps.csc_array

assemble_nat_bc(sd, func, b_faces)[source]

Assembles the natural boundary condition for the given grid, function, and: boundary faces.

Parameters:

sd (pg.Grid) – The grid on which to assemble the natural boundary condition.
func (Callable) – The function defining the natural boundary condition.
b_faces (np.ndarray) – The array of boundary faces.

Returns:

The assembled natural boundary condition.

Return type:

np.ndarray

proj_to_PwPolynomials(sd)[source]

Returns the inclusion matrix that projects the finite element space onto the lowest order piecewise polynomial space without loss of information.

Parameters:: sd (pg.Grid) – The grid.
Returns:: The inclusion matrix.
Return type:: sps.csc_array

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