A finite element. More...

#include <finite-element.h>

Public Member Functions
	FiniteElement (element::family family, cell::type cell_type, polyset::type poly_type, int degree, const std::vector< std::size_t > &value_shape, mdspan_t< const F, 2 > wcoeffs, const std::array< std::vector< mdspan_t< const F, 2 >>, 4 > &x, const std::array< std::vector< mdspan_t< const F, 4 >>, 4 > &M, int interpolation_nderivs, maps::type map_type, sobolev::space sobolev_space, bool discontinuous, int embedded_subdegree, int embedded_superdegree, element::lagrange_variant lvariant, element::dpc_variant dvariant, std::vector< int > dof_ordering={})
	Construct a finite element. More...

	FiniteElement (const FiniteElement &element)=default
	Copy constructor.

	FiniteElement (FiniteElement &&element)=default
	Move constructor.

	~FiniteElement ()=default
	Destructor.

FiniteElement &	operator= (const FiniteElement &element)=default
	Assignment operator.

FiniteElement &	operator= (FiniteElement &&element)=default
	Move assignment operator.

bool	operator== (const FiniteElement &e) const
	Check if two elements are the same. More...

std::size_t	hash () const
	Get a unique hash of this element.

std::array< std::size_t, 4 >	tabulate_shape (std::size_t nd, std::size_t num_points) const
	Array shape for tabulate basis values and derivatives at set of points. More...

std::pair< std::vector< F >, std::array< std::size_t, 4 > >	tabulate (int nd, impl::mdspan_t< const F, 2 > x) const
	Compute basis values and derivatives at set of points. More...

std::pair< std::vector< F >, std::array< std::size_t, 4 > >	tabulate (int nd, std::span< const F > x, std::array< std::size_t, 2 > shape) const
	Compute basis values and derivatives at set of points. More...

void	tabulate (int nd, impl::mdspan_t< const F, 2 > x, mdspan_t< F, 4 > basis) const
	Compute basis values and derivatives at set of points. More...

void	tabulate (int nd, std::span< const F > x, std::array< std::size_t, 2 > xshape, std::span< F > basis) const
	Compute basis values and derivatives at set of points. More...

cell::type	cell_type () const
	Get the element cell type. More...

polyset::type	polyset_type () const
	Get the element polyset type. More...

int	degree () const
	Get the element polynomial degree. More...

int	embedded_superdegree () const
	Lowest degree `n` such that the highest degree polynomial in this element is contained in a Lagrange (or vector Lagrange) element of degree `n`. More...

int	embedded_subdegree () const
	Highest degree `n` such that a Lagrange (or vector Lagrange) element of degree n is a subspace of this element. More...

const std::vector< std::size_t > &	value_shape () const
	Element value tensor shape. More...

int	dim () const
	Dimension of the finite element space. More...

element::family	family () const
	The finite element family. More...

element::lagrange_variant	lagrange_variant () const
	Lagrange variant of the element. More...

element::dpc_variant	dpc_variant () const
	DPC variant of the element. More...

maps::type	map_type () const
	Map type for the element. More...

sobolev::space	sobolev_space () const
	Underlying Sobolev space for this element. More...

bool	discontinuous () const
	Indicates whether this element is the discontinuous variant. More...

bool	dof_transformations_are_permutations () const
	Indicates if the degree-of-freedom transformations are all permutations.

bool	dof_transformations_are_identity () const
	Indicates is the dof transformations are all the identity.

std::pair< std::vector< F >, std::array< std::size_t, 3 > >	push_forward (impl::mdspan_t< const F, 3 > U, impl::mdspan_t< const F, 3 > J, std::span< const F > detJ, impl::mdspan_t< const F, 3 > K) const
	Map function values from the reference to a physical cell. More...

std::pair< std::vector< F >, std::array< std::size_t, 3 > >	pull_back (impl::mdspan_t< const F, 3 > u, impl::mdspan_t< const F, 3 > J, std::span< const F > detJ, impl::mdspan_t< const F, 3 > K) const
	Map function values from a physical cell to the reference. More...

template<typename O , typename P , typename Q , typename R >
std::function< void(O &, const P &, const Q &, F, const R &)>	map_fn () const
	Return a function that performs the appropriate push-forward/pull-back for the element type. More...

const std::vector< std::vector< std::vector< int > > > &	entity_dofs () const
	Get the dofs on each topological entity: (vertices, edges, faces, cell) in that order. More...

const std::vector< std::vector< std::vector< int > > > &	entity_closure_dofs () const
	Get the dofs on the closure of each topological entity: (vertices, edges, faces, cell) in that order. More...

std::pair< std::vector< F >, std::array< std::size_t, 3 > >	base_transformations () const
	Get the base transformations. More...

std::map< cell::type, std::pair< std::vector< F >, std::array< std::size_t, 3 > > >	entity_transformations () const
	Return the entity dof transformation matrices. More...

void	permute (std::span< std::int32_t > d, std::uint32_t cell_info) const
	Permute indices associated with degree-of-freedoms on the reference element ordering to the globally consistent physical element degree-of-freedom ordering. More...

void	permute_inv (std::span< std::int32_t > d, std::uint32_t cell_info) const
	Perform the inverse of the operation applied by permute(). More...

template<typename T >
void	T_apply (std::span< T > u, int n, std::uint32_t cell_info) const
	Transform basis functions from the reference element ordering and orientation to the globally consistent physical element ordering and orientation. More...

template<typename T >
void	Tt_apply (std::span< T > u, int n, std::uint32_t cell_info) const
	Apply the transpose of the operator applied by T_apply(). More...

template<typename T >
void	Tt_inv_apply (std::span< T > u, int n, std::uint32_t cell_info) const
	Apply the inverse transpose of the operator applied by T_apply(). More...

template<typename T >
void	Tinv_apply (std::span< T > u, int n, std::uint32_t cell_info) const
	Apply the inverse of the operator applied by T_apply(). More...

template<typename T >
void	Tt_apply_right (std::span< T > u, int n, std::uint32_t cell_info) const
	Right(post)-apply the transpose of the operator applied by T_apply(). More...

template<typename T >
void	T_apply_right (std::span< T > u, int n, std::uint32_t cell_info) const
	Right(post)-apply the operator applied by T_apply(). More...

template<typename T >
void	Tinv_apply_right (std::span< T > u, int n, std::uint32_t cell_info) const
	Right(post)-apply the inverse of the operator applied by T_apply(). More...

template<typename T >
void	Tt_inv_apply_right (std::span< T > u, int n, std::uint32_t cell_info) const
	Right(post)-apply the transpose inverse of the operator applied by T_apply(). More...

const std::pair< std::vector< F >, std::array< std::size_t, 2 > > &	points () const
	Return the interpolation points. More...

const std::pair< std::vector< F >, std::array< std::size_t, 2 > > &	interpolation_matrix () const
	Return a matrix of weights interpolation. More...

const std::pair< std::vector< F >, std::array< std::size_t, 2 > > &	dual_matrix () const
	Get the dual matrix. More...

const std::pair< std::vector< F >, std::array< std::size_t, 2 > > &	wcoeffs () const
	Get the coefficients that define the polynomial set in terms of the orthonormal polynomials. More...

const std::array< std::vector< std::pair< std::vector< F >, std::array< std::size_t, 2 > > >, 4 > &	x () const
	Get the interpolation points for each subentity. More...

const std::array< std::vector< std::pair< std::vector< F >, std::array< std::size_t, 4 > > >, 4 > &	M () const
	Get the interpolation matrices for each subentity. More...

const std::pair< std::vector< F >, std::array< std::size_t, 2 > > &	coefficient_matrix () const
	Get the matrix of coefficients. More...

bool	has_tensor_product_factorisation () const
	Indicates whether or not this element can be represented as a product of elements defined on lower-dimensional reference cells. More...

std::vector< std::vector< FiniteElement< F > > >	get_tensor_product_representation () const
	Get the tensor product representation of this element. More...

bool	interpolation_is_identity () const
	Indicates whether or not the interpolation matrix for this element is an identity matrix. More...

int	interpolation_nderivs () const
	The number of derivatives needed when interpolating.

const std::vector< int > &	dof_ordering () const
	Get dof layout.

Detailed Description

template<std::floating_point F>
class basix::FiniteElement< F >

A finite element.

The basis of a finite element is stored as a set of coefficients, which are applied to the underlying expansion set for that cell type, when tabulating.

Constructor & Destructor Documentation

◆ FiniteElement()

template<std::floating_point F>

basix::FiniteElement< F >::FiniteElement	(	element::family	family,
		cell::type	cell_type,
		polyset::type	poly_type,
		int	degree,
		const std::vector< std::size_t > &	value_shape,
		mdspan_t< const F, 2 >	wcoeffs,
		const std::array< std::vector< mdspan_t< const F, 2 >>, 4 > &	x,
		const std::array< std::vector< mdspan_t< const F, 4 >>, 4 > &	M,
		int	interpolation_nderivs,
		maps::type	map_type,
		sobolev::space	sobolev_space,
		bool	discontinuous,
		int	embedded_subdegree,
		int	embedded_superdegree,
		element::lagrange_variant	lvariant,
		element::dpc_variant	dvariant,
		std::vector< int >	dof_ordering = `{}`
	)

Construct a finite element.

Initialising a finite element calculates the basis functions of the finite element, in terms of the polynomial basis.

The below explanation uses Einstein notation.

The basis functions \({\phi_i}\) of a finite element are represented as a linear combination of polynomials \(\{p_j\}\) in an underlying polynomial basis that span the space of all d-dimensional polynomials up to order \(k \ (P_k^d)\):

\[ \phi_i = c_{ij} p_j \]

In some cases, the basis functions \(\{\phi_i\}\) do not span the full space \(P_k\), in which case we denote space spanned by the basis functions by \(\{q_k\}\), which can be represented by:

\[ q_i = b_{ij} p_j. \]

This leads to

\[ \phi_i = c^{\prime}_{ij} q_j = c^{\prime}_{ij} b_{jk} p_k, \]

and in matrix form:

\[ \phi = C^{\prime} B p \]

If the basis functions span the full space, then \( B \) is simply the identity.

The basis functions \(\phi_i\) are defined by a dual set of functionals \(\{f_i\}\). The basis functions are the functions in span{ \(q_k\)} such that

\[ f_i(\phi_j) = \delta_{ij} \]

and inserting the expression for \(\phi_{j}\):

\[ f_i(c^{\prime}_{jk}b_{kl}p_{l}) = c^{\prime}_{jk} b_{kl} f_i \left( p_{l} \right) \]

Defining a matrix D given by applying the functionals to each polynomial \(p_j\):

\[ [D] = d_{ij},\mbox{ where } d_{ij} = f_i(p_j), \]

we have:

\[ C^{\prime} B D^{T} = I \]

and

\[ C^{\prime} = (B D^{T})^{-1}. \]

Recalling that \(C = C^{\prime} B\), where \(C\) is the matrix form of \(c_{ij}\),

\[ C = (B D^{T})^{-1} B \]

This function takes the matrices \(B\) (wcoeffs) and \(D\) (M) as inputs and will internally compute \(C\).

The matrix \(BD^{T}\) can be obtained from an element by using dual_matrix(). The matrix \(C\) can be obtained from an element by using coefficient_matrix().

Example: Order 1 Lagrange elements on a triangle

On a triangle, the scalar expansion basis is:

\[ p_0 = \sqrt{2}/2 \qquad p_1 = \sqrt{3}(2x + y - 1) \qquad p_2 = 3y - 1 \]

These span the space \(P_1\).

Lagrange order 1 elements span the space P_1, so in this example, B (span_coeffs) is the identity matrix:

\[ B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

The functionals defining the Lagrange order 1 space are point evaluations at the three vertices of the triangle. The matrix D (dual) given by applying these to p_0 to p_2 is:

\[ \mbox{dual} = \begin{bmatrix} \sqrt{2}/2 & -\sqrt{3} & -1 \\ \sqrt{2}/2 & \sqrt{3} & -1 \\ \sqrt{2}/2 & 0 & 2 \end{bmatrix} \]

For this example, this function outputs the matrix:

\[ C = \begin{bmatrix} \sqrt{2}/3 & -\sqrt{3}/6 & -1/6 \\ \sqrt{2}/3 & \sqrt{3}/6 & -1/6 \\ \sqrt{2}/3 & 0 & 1/3 \end{bmatrix} \]

The basis functions of the finite element can be obtained by applying the matrix C to the vector \([p_0, p_1, p_2]\), giving:

\[ \begin{bmatrix} 1 - x - y \\ x \\ y \end{bmatrix} \]

Example: Order 1 Raviart-Thomas on a triangle

On a triangle, the 2D vector expansion basis is:

\[ \begin{matrix} p_0 & = & (\sqrt{2}/2, 0) \\ p_1 & = & (\sqrt{3}(2x + y - 1), 0) \\ p_2 & = & (3y - 1, 0) \\ p_3 & = & (0, \sqrt{2}/2) \\ p_4 & = & (0, \sqrt{3}(2x + y - 1)) \\ p_5 & = & (0, 3y - 1) \end{matrix} \]

These span the space \( P_1^2 \).

Raviart-Thomas order 1 elements span a space smaller than \( P_1^2 \), so B (span_coeffs) is not the identity. It is given by:

\[ B = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1/12 & \sqrt{6}/48 & -\sqrt{2}/48 & 1/12 & 0 & \sqrt{2}/24 \end{bmatrix} \]

Applying the matrix B to the vector \([p_0, p_1, ..., p_5]\) gives the basis of the polynomial space for Raviart-Thomas:

\[ \begin{bmatrix} \sqrt{2}/2 & 0 \\ 0 & \sqrt{2}/2 \\ \sqrt{2}x/8 & \sqrt{2}y/8 \end{bmatrix} \]

The functionals defining the Raviart-Thomas order 1 space are integral of the normal components along each edge. The matrix D (dual) given by applying these to \(p_0\) to \(p_5\) is:

\[ D = \begin{bmatrix} -\sqrt{2}/2 & -\sqrt{3}/2 & -1/2 & -\sqrt{2}/2 & -\sqrt{3}/2 & -1/2 \\ -\sqrt{2}/2 & \sqrt{3}/2 & -1/2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt{2}/2 & 0 & -1 \end{bmatrix} \]

In this example, this function outputs the matrix:

\[ C = \begin{bmatrix} -\sqrt{2}/2 & -\sqrt{3}/2 & -1/2 & -\sqrt{2}/2 & -\sqrt{3}/2 & -1/2 \\ -\sqrt{2}/2 & \sqrt{3}/2 & -1/2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt{2}/2 & 0 & -1 \end{bmatrix} \]

The basis functions of the finite element can be obtained by applying the matrix C to the vector \([p_0, p_1, ..., p_5]\), giving:

\[ \begin{bmatrix} -x & -y \\ x - 1 & y \\ -x & 1 - y \end{bmatrix} \]

Parameters

[in]	family	The element family
[in]	cell_type	The cell type
[in]	poly_type	The polyset type
[in]	degree	The degree of the element
[in]	interpolation_nderivs	The number of derivatives that need to be used during interpolation
[in]	value_shape	The value shape of the element
[in]	wcoeffs	Matrices for the kth value index containing the expansion coefficients defining a polynomial basis spanning the polynomial space for this element. Shape is (dim(finite element polyset), dim(Legendre polynomials))
[in]	x	Interpolation points. Indices are (tdim, entity index, point index, dim)
[in]	M	The interpolation matrices. Indices are (tdim, entity index, dof, vs, point_index, derivative)
[in]	map_type	The type of map to be used to map values from the reference to a cell
[in]	sobolev_space	The underlying Sobolev space for the element
[in]	discontinuous	Indicates whether or not this is the discontinuous version of the element
[in]	embedded_subdegree	The highest degree n such that a Lagrange (or vector Lagrange) element of degree n is a subspace of this element
[in]	embedded_superdegree	The highest degree n such that at least one polynomial of degree n is included in this element's polymonial set
[in]	lvariant	The Lagrange variant of the element
[in]	dvariant	The DPC variant of the element
[in]	dof_ordering	DOF reordering: a mapping from the reference order to a new permuted order

Member Function Documentation

◆ operator==()

template<std::floating_point F>

bool FiniteElement::operator== ( const FiniteElement< F > & e ) const

Check if two elements are the same.

Note: This operator compares the element properties, e.g. family, degree, etc, and not computed numerical data

Returns: True if elements are the same

◆ tabulate_shape()

template<std::floating_point F>

std::array<std::size_t, 4> basix::FiniteElement< F >::tabulate_shape	(	std::size_t	nd,
		std::size_t	num_points
	)		const

inline

Array shape for tabulate basis values and derivatives at set of points.

Parameters

[in]	nd	The order of derivatives, up to and including, to compute. Use 0 for the basis functions only.
[in]	num_points	Number of points that basis will be computed at.

Returns: The shape of the array to will filled when passed to tabulate().

◆ tabulate() [1/4]

template<std::floating_point F>

std::pair< std::vector< F >, std::array< std::size_t, 4 > > FiniteElement::tabulate	(	int	nd,
		impl::mdspan_t< const F, 2 >	x
	)		const

Compute basis values and derivatives at set of points.

Note: The version of tabulate() with the basis data as an out argument should be preferred for repeated call where performance is critical.

Parameters

[in]	nd	The order of derivatives, up to and including, to compute. Use 0 for the basis functions only.
[in]	x	The points at which to compute the basis functions. The shape of x is (number of points, geometric dimension).

Returns

The basis functions (and derivatives). The shape is (derivative, point, basis fn index, value index).

The first index is the derivative, with higher derivatives are stored in triangular (2D) or tetrahedral (3D) ordering, ie for the (x,y) derivatives in 2D: (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), (3,0)... The function basix::indexing::idx can be used to find the appropriate derivative.
The second index is the point index
The third index is the basis function index
The fourth index is the basis function component. Its has size one for scalar basis functions.

◆ tabulate() [2/4]

template<std::floating_point F>

std::pair< std::vector< F >, std::array< std::size_t, 4 > > FiniteElement::tabulate	(	int	nd,
		std::span< const F >	x,
		std::array< std::size_t, 2 >	shape
	)		const

Compute basis values and derivatives at set of points.

Note: The version of tabulate() with the basis data as an out argument should be preferred for repeated call where performance is critical

Parameters

[in]	nd	The order of derivatives, up to and including, to compute. Use 0 for the basis functions only.
[in]	x	The points at which to compute the basis functions (row-major storage).
[in]	shape	The shape `(number of points, geometric dimension)` of the `x` array.

Returns

The basis functions (and derivatives). The shape is (derivative, point, basis fn index, value index).

The first index is the derivative, with higher derivatives are stored in triangular (2D) or tetrahedral (3D) ordering, ie for the (x,y) derivatives in 2D: (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), (3,0)... The function indexing::idx can be used to find the appropriate derivative.
The second index is the point index
The third index is the basis function index
The fourth index is the basis function component. Its has size one for scalar basis functions.

◆ tabulate() [3/4]

template<std::floating_point F>

void FiniteElement::tabulate	(	int	nd,
		impl::mdspan_t< const F, 2 >	x,
		mdspan_t< F, 4 >	basis
	)		const

Compute basis values and derivatives at set of points.

Note: This function is designed to be called at runtime, so its performance is critical.

Parameters

[in]	nd	The order of derivatives, up to and including, to compute. Use 0 for the basis functions only.
[in]	x	The points at which to compute the basis functions. The shape of x is (number of points, geometric dimension).
[out]	basis	Memory location to fill. It must be allocated with shape `(num_derivatives, num_points, num basis functions, value_size)`. The function tabulate_shape() can be used to get the required shape. The first index is the derivative, with higher derivatives are stored in triangular (2D) or tetrahedral (3D) ordering, ie for the (x,y) derivatives in 2D: (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), (3,0)... The function indexing::idx can be used to find the appropriate derivative. The second index is the point index The third index is the basis function index The fourth index is the basis function component. Its has size one for scalar basis functions.

Todo:: Remove all internal dynamic memory allocation, pass scratch space as required

◆ tabulate() [4/4]

template<std::floating_point F>

void FiniteElement::tabulate	(	int	nd,
		std::span< const F >	x,
		std::array< std::size_t, 2 >	xshape,
		std::span< F >	basis
	)		const

Compute basis values and derivatives at set of points.

Note: This function is designed to be called at runtime, so its performance is critical.

Parameters

[in]	nd	The order of derivatives, up to and including, to compute. Use 0 for the basis functions only.
[in]	x	The points at which to compute the basis functions (row-major storage). The shape of `x` is `(number of points, geometric dimension)`.
[in]	xshape	The shape `(number of points, geometric dimension)` of `x`.
[out]	basis	Memory location to fill. It must be allocated with shape `(num_derivatives, num_points, num basis functions, value_size)`. The function tabulate_shape() can be used to get the required shape. The first index is the derivative, with higher derivatives are stored in triangular (2D) or tetrahedral (3D) ordering, ie for the (x,y) derivatives in 2D: (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), (3,0)... The function indexing::idx can be used to find the appropriate derivative. The second index is the point index The third index is the basis function index The fourth index is the basis function component. Its has size one for scalar basis functions.

◆ cell_type()

template<std::floating_point F>

cell::type basix::FiniteElement< F >::cell_type ( ) const

inline

Get the element cell type.

Returns: The cell type

◆ polyset_type()

template<std::floating_point F>

polyset::type basix::FiniteElement< F >::polyset_type ( ) const

inline

Get the element polyset type.

Returns: The polyset

◆ degree()

template<std::floating_point F>

int basix::FiniteElement< F >::degree ( ) const

inline

Get the element polynomial degree.

Returns: Polynomial degree

◆ embedded_superdegree()

template<std::floating_point F>

int basix::FiniteElement< F >::embedded_superdegree ( ) const

inline

Lowest degree n such that the highest degree polynomial in this element is contained in a Lagrange (or vector Lagrange) element of degree n.

Returns: Polynomial degree

◆ embedded_subdegree()

template<std::floating_point F>

int basix::FiniteElement< F >::embedded_subdegree ( ) const

inline

Highest degree n such that a Lagrange (or vector Lagrange) element of degree n is a subspace of this element.

Returns: Polynomial degree

◆ value_shape()

template<std::floating_point F>

const std::vector<std::size_t>& basix::FiniteElement< F >::value_shape ( ) const

inline

Element value tensor shape.

For example, returns {} for scalars, {3} for vectors in 3D, {2, 2} for a rank-2 tensor in 2D.

Returns: Value shape

◆ dim()

template<std::floating_point F>

int basix::FiniteElement< F >::dim ( ) const

inline

Dimension of the finite element space.

The dimension is the number of degrees-of-freedom for the element.

Returns: Number of degrees of freedom

◆ family()

template<std::floating_point F>

element::family basix::FiniteElement< F >::family ( ) const

inline

The finite element family.

Returns: The family

◆ lagrange_variant()

template<std::floating_point F>

element::lagrange_variant basix::FiniteElement< F >::lagrange_variant ( ) const

inline

Lagrange variant of the element.

Returns: The Lagrange variant.

◆ dpc_variant()

template<std::floating_point F>

element::dpc_variant basix::FiniteElement< F >::dpc_variant ( ) const

inline

DPC variant of the element.

Returns: The DPC variant.

◆ map_type()

template<std::floating_point F>

maps::type basix::FiniteElement< F >::map_type ( ) const

inline

Map type for the element.

Returns: The map type.

◆ sobolev_space()

template<std::floating_point F>

sobolev::space basix::FiniteElement< F >::sobolev_space ( ) const

inline

Underlying Sobolev space for this element.

Returns: The Sobolev space.

◆ discontinuous()

template<std::floating_point F>

bool basix::FiniteElement< F >::discontinuous ( ) const

inline

Indicates whether this element is the discontinuous variant.

Returns: True if this element is a discontinuous version of the element.

◆ push_forward()

template<std::floating_point F>

std::pair< std::vector< F >, std::array< std::size_t, 3 > > FiniteElement::push_forward	(	impl::mdspan_t< const F, 3 >	U,
		impl::mdspan_t< const F, 3 >	J,
		std::span< const F >	detJ,
		impl::mdspan_t< const F, 3 >	K
	)		const

Map function values from the reference to a physical cell.

This function can perform the mapping for multiple points, grouped by points that share a common Jacobian.

Parameters

[in]	U	The function values on the reference. The indices are `[Jacobian index, point index, components]`.
[in]	J	The Jacobian of the mapping. The indices are `[Jacobian index, J_i, J_j]`.
[in]	detJ	The determinant of the Jacobian of the mapping. It has length `J.shape(0)`
[in]	K	The inverse of the Jacobian of the mapping. The indices are `[Jacobian index, K_i, K_j]`.

Returns: The function values on the cell. The indices are [Jacobian index, point index, components].

◆ pull_back()

template<std::floating_point F>

std::pair< std::vector< F >, std::array< std::size_t, 3 > > FiniteElement::pull_back	(	impl::mdspan_t< const F, 3 >	u,
		impl::mdspan_t< const F, 3 >	J,
		std::span< const F >	detJ,
		impl::mdspan_t< const F, 3 >	K
	)		const

Map function values from a physical cell to the reference.

Parameters

[in]	u	The function values on the cell
[in]	J	The Jacobian of the mapping
[in]	detJ	The determinant of the Jacobian of the mapping
[in]	K	The inverse of the Jacobian of the mapping

Returns: The function values on the reference. The indices are [Jacobian index, point index, components].

◆ map_fn()

template<std::floating_point F>

template<typename O , typename P , typename Q , typename R >

std::function<void(O&, const P&, const Q&, F, const R&)> basix::FiniteElement< F >::map_fn ( ) const

inline

Return a function that performs the appropriate push-forward/pull-back for the element type.

Template Parameters

O	The type that hold the (computed) mapped data (ndim==2)
P	The type that hold the data to be mapped (ndim==2)
Q	The type that holds the Jacobian (or inverse) matrix (ndim==2)
R	The type that holds the inverse of the `Q` data (ndim==2)

Returns

A function that for a push-forward takes arguments

u [out] The data on the physical cell after the push-forward flattened with row-major layout, shape=(num_points, value_size)
U [in] The data on the reference cell physical field to push forward, flattened with row-major layout, shape=(num_points, ref_value_size)
J [in] The Jacobian matrix of the map ,shape=(gdim, tdim)
detJ [in] det(J)
K [in] The inverse of the Jacobian matrix, shape=(tdim, gdim)

For a pull-back the arguments should be:

U [out] The data on the reference cell after the pull-back, flattened with row-major layout, shape=(num_points, ref value_size)
u [in] The data on the physical cell that should be pulled back , flattened with row-major layout, shape=(num_points, value_size)
K [in] The inverse of the Jacobian matrix of the map ,shape=(tdim, gdim)
detJ_inv [in] 1/det(J)
J [in] The Jacobian matrix, shape=(gdim, tdim)

◆ entity_dofs()

template<std::floating_point F>

const std::vector<std::vector<std::vector<int> > >& basix::FiniteElement< F >::entity_dofs ( ) const

inline

Get the dofs on each topological entity: (vertices, edges, faces, cell) in that order.

For example, Lagrange degree 2 on a triangle has vertices: [[0], [1], [2]], edges: [[3], [4], [5]], cell: [[]]

Returns: Dofs associated with an entity of a given topological dimension. The shape is (tdim + 1, num_entities, num_dofs).

◆ entity_closure_dofs()

template<std::floating_point F>

const std::vector<std::vector<std::vector<int> > >& basix::FiniteElement< F >::entity_closure_dofs ( ) const

inline

Get the dofs on the closure of each topological entity: (vertices, edges, faces, cell) in that order.

For example, Lagrange degree 2 on a triangle has vertices: [[0], [1], [2]], edges: [[1, 2, 3], [0, 2, 4], [0, 1, 5]], cell: [[0, 1, 2, 3, 4, 5]]

Returns: Dofs associated with the closure of an entity of a given topological dimension. The shape is (tdim + 1, num_entities, num_dofs).

◆ base_transformations()

template<std::floating_point F>

std::pair< std::vector< F >, std::array< std::size_t, 3 > > FiniteElement::base_transformations

Get the base transformations.

The base transformations represent the effect of rotating or reflecting a subentity of the cell on the numbering and orientation of the DOFs. This returns a list of matrices with one matrix for each subentity permutation in the following order: Reversing edge 0, reversing edge 1, ... Rotate face 0, reflect face 0, rotate face 1, reflect face 1, ...

Example: Order 3 Lagrange on a triangle

This space has 10 dofs arranged like:

2
|\
6 4
|  \
5 9 3
|    \
0-7-8-1

For this element, the base transformations are: [Matrix swapping 3 and 4, Matrix swapping 5 and 6, Matrix swapping 7 and 8] The first row shows the effect of reversing the diagonal edge. The second row shows the effect of reversing the vertical edge. The third row shows the effect of reversing the horizontal edge.

Example: Order 1 Raviart-Thomas on a triangle

This space has 3 dofs arranged like:

  |\
  | \
  |  \
<-1   0
  |  / \
  | L ^ \
  |   |  \
   ---2---

These DOFs are integrals of normal components over the edges: DOFs 0 and 2 are oriented inward, DOF 1 is oriented outwards. For this element, the base transformation matrices are:

0: [[-1, 0, 0],
    [ 0, 1, 0],
    [ 0, 0, 1]]
1: [[1,  0, 0],
    [0, -1, 0],
    [0,  0, 1]]
2: [[1, 0,  0],
    [0, 1,  0],
    [0, 0, -1]]

The first matrix reverses DOF 0 (as this is on the first edge). The second matrix reverses DOF 1 (as this is on the second edge). The third matrix reverses DOF 2 (as this is on the third edge).

Example: DOFs on the face of Order 2 Nedelec first kind on a tetrahedron

On a face of this tetrahedron, this space has two face tangent DOFs:

|\        |\
| \       | \
|  \      | ^\
|   \     | | \
| 0->\    | 1  \
|     \   |     \
 ------    ------

For these DOFs, the subblocks of the base transformation matrices are:

rotation:   [[-1, 1],
             [ 1, 0]]
reflection: [[0, 1],
             [1, 0]]

Returns: The base transformations for this element. The shape is (ntranformations, ndofs, ndofs)

◆ entity_transformations()

template<std::floating_point F>

std::map<cell::type, std::pair<std::vector<F>, std::array<std::size_t, 3> > > basix::FiniteElement< F >::entity_transformations ( ) const

inline

Return the entity dof transformation matrices.

Returns: The entity transformations for the sub-entities of this element. The shape for each cell is (ntransformations, ndofs, ndofs)

◆ permute()

template<std::floating_point F>

void basix::FiniteElement< F >::permute	(	std::span< std::int32_t >	d,
		std::uint32_t	cell_info
	)		const

inline

Permute indices associated with degree-of-freedoms on the reference element ordering to the globally consistent physical element degree-of-freedom ordering.

Given an array \(\tilde{d}\) that holds an integer associated with each degree-of-freedom and following the reference element degree-of-freedom ordering, this function computes

\[ d = P \tilde{d}, \]

where \(P\) is a permutation matrix and \(d\) holds the integers in \(\tilde{d}\) but permuted to follow the globally consistent physical element degree-of-freedom ordering. The permutation is computed in-place.

Note: This function is designed to be called at runtime, so its performance is critical.

Parameters

[in,out]	d	Indices associated with each reference element degree-of-freedom (in). Indices associated with each physical element degree-of-freedom (out).
	cell_info	Permutation info for the cell

◆ permute_inv()

template<std::floating_point F>

void basix::FiniteElement< F >::permute_inv	(	std::span< std::int32_t >	d,
		std::uint32_t	cell_info
	)		const

inline

Perform the inverse of the operation applied by permute().

Given an array \(d\) that holds an integer associated with each degree-of-freedom and following the globally consistent physical element degree-of-freedom ordering, this function computes

\[ \tilde{d} = P^{T} d, \]

where \(P^{T}\) is a permutation matrix and \(\tilde{d}\) holds the integers in \(d\) but permuted to follow the reference element degree-of-freedom ordering. The permutation is computed in-place.

Parameters

[in,out]	d	Indices associated with each physical element degree-of-freedom [in]. Indices associated with each reference element degree-of-freedom [out].
	cell_info	Permutation info for the cell

◆ T_apply()

template<std::floating_point F>

template<typename T >

void basix::FiniteElement< F >::T_apply	(	std::span< T >	u,
		int	n,
		std::uint32_t	cell_info
	)		const

Transform basis functions from the reference element ordering and orientation to the globally consistent physical element ordering and orientation.

Consider that the value of a finite element function \(f_{h}\) at a point is given by

\[ f_{h} = \phi^{T} c, \]

where \(f_{h}\) has shape \(r \times 1\), \(\phi\) has shape \(d \times r\) and holds the finite element basis functions, and \(c\) has shape \(d \times 1\) and holds the degrees-of-freedom. The basis functions and degree-of-freedom are with respect to the physical element orientation. If the degrees-of-freedom on the physical element orientation are given by

\[ \phi = T \tilde{\phi}, \]

where \(T\) is a \(d \times d\) matrix, it follows from \(f_{h} = \phi^{T} c = \tilde{\phi}^{T} T^{T} c\) that

\[ \tilde{c} = T^{T} c. \]

This function applies \(T\) to data. The transformation is performed in-place. The operator \(T\) is orthogonal for many elements, but not all.

Parameters

[in,out]	u	Data to transform. The shape is `(m, n)`, where `m` is the number of dgerees-of-freedom and the storage is row-major.
[in]	n	Number of columns in `data`.
	cell_info	Permutation info for the cell

◆ Tt_apply()

template<std::floating_point F>

template<typename T >

void basix::FiniteElement< F >::Tt_apply	(	std::span< T >	u,
		int	n,
		std::uint32_t	cell_info
	)		const

Apply the transpose of the operator applied by T_apply().

The transformation

\[ u \leftarrow T^{T} u \]

is performed in-place.

Parameters

[in,out]	u	Data to transform. The shape is `(m, n)`, where `m` is the number of dgerees-of-freedom an d the storage is row-major.
[in]	n	Number of columns in `data`.
[in]	cell_info	Permutation info for the cell,

◆ Tt_inv_apply()

template<std::floating_point F>

template<typename T >

void basix::FiniteElement< F >::Tt_inv_apply	(	std::span< T >	u,
		int	n,
		std::uint32_t	cell_info
	)		const

Apply the inverse transpose of the operator applied by T_apply().

The transformation

\[ u \leftarrow T^{-T} u \]

is performed in-place.

Parameters

[in,out]	u	Data to transform. The shape is `(m, n)`, where `m` is the number of dgerees-of-freedom and the storage is row-major.
[in]	n	Number of columns in `data`.
[in]	cell_info	Permutation info for the cell.

◆ Tinv_apply()

template<std::floating_point F>

template<typename T >

void basix::FiniteElement< F >::Tinv_apply	(	std::span< T >	u,
		int	n,
		std::uint32_t	cell_info
	)		const

Apply the inverse of the operator applied by T_apply().

The transformation

\[ u \leftarrow T^{-1} u \]

is performed in-place.

Parameters

[in,out]	u	Data to transform. The shape is `(m, n)`, where `m` is the number of dgerees-of-freedom and the storage is row-major.
[in]	n	Number of columns in `data`.
[in]	cell_info	Permutation info for the cell.

◆ Tt_apply_right()

template<std::floating_point F>

template<typename T >

void basix::FiniteElement< F >::Tt_apply_right	(	std::span< T >	u,
		int	n,
		std::uint32_t	cell_info
	)		const

Right(post)-apply the transpose of the operator applied by T_apply().

Computes

\[ u^{T} \leftarrow u^{T} T^{T} \]

in-place.

Parameters

[in,out]	u	Data to transform. The shape is `(m, n)`, where `m` is the number of dgerees-of-freedom and the storage is row-major.
[in]	n	Number of columns in `data`.
[in]	cell_info	Permutation info for the cell.

◆ T_apply_right()

template<std::floating_point F>

template<typename T >

void basix::FiniteElement< F >::T_apply_right	(	std::span< T >	u,
		int	n,
		std::uint32_t	cell_info
	)		const

Right(post)-apply the operator applied by T_apply().

Computes

\[ u^{T} \leftarrow u^{T} T \]

in-place.

Parameters

[in,out]	u	Data to transform. The shape is `(m, n)`, where `m` is the number of dgerees-of-freedom and the storage is row-major.
[in]	n	Number of columns in `data`.
[in]	cell_info	Permutation info for the cell.

◆ Tinv_apply_right()

template<std::floating_point F>

template<typename T >

void basix::FiniteElement< F >::Tinv_apply_right	(	std::span< T >	u,
		int	n,
		std::uint32_t	cell_info
	)		const

Right(post)-apply the inverse of the operator applied by T_apply().

Computes

\[ u^{T} \leftarrow u^{T} T^{-1} \]

in-place.

Parameters

[in,out]	u	Data to transform. The shape is `(m, n)`, where `m` is the number of dgerees-of-freedom and the storage is row-major.
[in]	n	Number of columns in `data`.
	cell_info	Permutation info for the cell.

◆ Tt_inv_apply_right()

template<std::floating_point F>

template<typename T >

void basix::FiniteElement< F >::Tt_inv_apply_right	(	std::span< T >	u,
		int	n,
		std::uint32_t	cell_info
	)		const

Right(post)-apply the transpose inverse of the operator applied by T_apply().

Computes

\[ u^{T} \leftarrow u^{T} T^{-T} \]

in-place.

Parameters

[in,out]	u	Data to transform. The shape is `(m, n)`, where `m` is the number of dgerees-of-freedom and the storage is row-major.
[in]	n	Number of columns in `data`.
	cell_info	Permutation info for the cell.

◆ points()

template<std::floating_point F>

const std::pair<std::vector<F>, std::array<std::size_t, 2> >& basix::FiniteElement< F >::points ( ) const

inline

Return the interpolation points.

The interpolation points are the coordinates on the reference element where a function need to be evaluated in order to interpolate it in the finite element space.

Returns: Array of coordinate with shape (num_points, tdim)

◆ interpolation_matrix()

template<std::floating_point F>

const std::pair<std::vector<F>, std::array<std::size_t, 2> >& basix::FiniteElement< F >::interpolation_matrix ( ) const

inline

Return a matrix of weights interpolation.

To interpolate a function in this finite element, the functions should be evaluated at each point given by points(). These function values should then be multiplied by the weight matrix to give the coefficients of the interpolated function.

The shape of the returned matrix will be (dim, num_points * value_size), where dim is the number of DOFs in the finite element, num_points is the number of points returned by points(), and value_size is the value size of the finite element.

For example, to interpolate into a Lagrange space, the following should be done:

i_m = element.interpolation_matrix()
pts = element.points()
values = vector(pts.shape(0))
FOR i, p IN ENUMERATE(pts):
    values[i] = f.evaluate_at(p)
coefficients = i_m * values

To interpolate into a Raviart-Thomas space, the following should be done:

i_m = element.interpolation_matrix()
pts = element.points()
vs = prod(element.value_shape())
values = VECTOR(pts.shape(0) * vs)
FOR i, p IN ENUMERATE(pts):
    values[i::pts.shape(0)] = f.evaluate_at(p)
coefficients = i_m * values

To interpolate into a Lagrange space with a block size, the following should be done:

i_m = element.interpolation_matrix()
pts = element.points()
coefficients = VECTOR(element.dim() * block_size)
FOR b IN RANGE(block_size):
    values = vector(pts.shape(0))
    FOR i, p IN ENUMERATE(pts):
        values[i] = f.evaluate_at(p)[b]
    coefficients[::block_size] = i_m * values

Returns: The interpolation matrix. Shape is (ndofs, number of interpolation points).

◆ dual_matrix()

template<std::floating_point F>

const std::pair<std::vector<F>, std::array<std::size_t, 2> >& basix::FiniteElement< F >::dual_matrix ( ) const

inline

Get the dual matrix.

This is the matrix \(BD^{T}\), as described in the documentation of the FiniteElement() constructor.

Returns: The dual matrix. Shape is (ndofs, ndofs) = (dim(), dim()).

◆ wcoeffs()

template<std::floating_point F>

const std::pair<std::vector<F>, std::array<std::size_t, 2> >& basix::FiniteElement< F >::wcoeffs ( ) const

inline

Get the coefficients that define the polynomial set in terms of the orthonormal polynomials.

The polynomials spanned by each finite element in Basix are represented as a linear combination of the orthonormal polynomials of a given degree on the cell. Each row of this matrix defines a polynomial in the set spanned by the finite element.

For example, the orthonormal polynomials of degree <= 1 on a triangle are (where a, b, c, d are some constants):

(sqrt(2), 0)
(a*x - b, 0)
(c*y - d, 0)
(0, sqrt(2))
(0, a*x - b)
(0, c*y - d)

For a degree 1 Raviart-Thomas element, the first two rows of wcoeffs would be the following, as (1, 0) and (0, 1) are spanned by the element

[1, 0, 0, 0, 0, 0]
[0, 0, 0, 1, 0, 0]

The third row of wcoeffs in this example would give coefficients that represent (x, y) in terms of the orthonormal polynomials:

[-b/(a*sqrt(2)), 1/a, 0, -d/(c*sqrt(2)), 0, 1/c]

These coefficients are only stored for custom elements. This function will throw an exception if called on a non-custom element.

Returns: Coefficient matrix. Shape is (dim(finite element polyset), dim(Lagrange polynomials)).

◆ x()

template<std::floating_point F>

const std::array< std::vector<std::pair<std::vector<F>, std::array<std::size_t, 2> > >, 4>& basix::FiniteElement< F >::x ( ) const

inline

Get the interpolation points for each subentity.

The indices of this data are (tdim, entity index, point index, dim).

◆ M()

template<std::floating_point F>

const std::array< std::vector<std::pair<std::vector<F>, std::array<std::size_t, 4> > >, 4>& basix::FiniteElement< F >::M ( ) const

inline

Get the interpolation matrices for each subentity.

The shape of this data is (tdim, entity index, dof, value size, point_index, derivative).

These matrices define how to evaluate the DOF functionals associated with each sub-entity of the cell. Given a function f, the functionals associated with the e-th entity of dimension d can be computed as follows:

matrix = element.M()[d][e]
pts = element.x()[d][e]
nderivs = element
values = f.eval_derivs(nderivs, pts)
result = ZEROS(matrix.shape(0))
FOR i IN RANGE(matrix.shape(0)):
    FOR j IN RANGE(matrix.shape(1)):
        FOR k IN RANGE(matrix.shape(2)):
            FOR l IN RANGE(matrix.shape(3)):
                result[i] += matrix[i, j, k, l] * values[l][k][j]

For example, for a degree 1 Raviart-Thomas (RT) element on a triangle, the DOF functionals are integrals over the edges of the dot product of the function with the normal to the edge. In this case, x() would contain quadrature points for each edge, and M() would be a 1 by 2 by npoints by 1 array for each edge. For each point, the [0, :, point, 0] slice of this would be the quadrature weight multiplied by the normal. For all entities that are not edges, the entries in x() and M() for a degree 1 RT element would have size 0.

These matrices are only stored for custom elements. This function will throw an exception if called on a non-custom element

Returns: The interpolation matrices. The indices of this data are (tdim, entity index, dof, vs, point_index, derivative).

◆ coefficient_matrix()

template<std::floating_point F>

const std::pair<std::vector<F>, std::array<std::size_t, 2> >& basix::FiniteElement< F >::coefficient_matrix ( ) const

inline

Get the matrix of coefficients.

Returns: The coefficient matrix. Shape is (ndofs, ndofs).

◆ has_tensor_product_factorisation()

template<std::floating_point F>

bool basix::FiniteElement< F >::has_tensor_product_factorisation ( ) const

inline

Indicates whether or not this element can be represented as a product of elements defined on lower-dimensional reference cells.

If the product exists, this element's basis functions can be computed as a tensor product of the basis elements of the elements in the product.

If such a factorisation exists, get_tensor_product_representation() can be used to get these elements.

◆ get_tensor_product_representation()

template<std::floating_point F>

std::vector<std::vector<FiniteElement<F> > > basix::FiniteElement< F >::get_tensor_product_representation ( ) const

inline

Get the tensor product representation of this element.

Exceptions

std::runtime_error Thrown if no such factorisation exists.

The tensor product representation will be a vector of vectors of finite elements. Each tuple contains a vector of finite elements, and a vector of integers. The vector of finite elements gives the elements on an interval that appear in the tensor product representation. The vector of integers gives the permutation between the numbering of the tensor product DOFs and the number of the DOFs of this Basix element.

Returns: The tensor product representation

◆ interpolation_is_identity()

template<std::floating_point F>

bool basix::FiniteElement< F >::interpolation_is_identity ( ) const

inline

Indicates whether or not the interpolation matrix for this element is an identity matrix.

Returns: True if the interpolation matrix is the identity and false otherwise.

The documentation for this class was generated from the following files:

/home/runner/work/basix/basix/cpp/basix/finite-element.h
/home/runner/work/basix/basix/cpp/basix/finite-element.cpp

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Public Member Functions

Detailed Description

template<std::floating_point F> class basix::FiniteElement< F >

Constructor & Destructor Documentation

◆ FiniteElement()

Example: Order 1 Lagrange elements on a triangle

Example: Order 1 Raviart-Thomas on a triangle

Member Function Documentation

◆ operator==()

◆ tabulate_shape()

◆ tabulate() [1/4]

◆ tabulate() [2/4]

◆ tabulate() [3/4]

◆ tabulate() [4/4]

◆ cell_type()

◆ polyset_type()

◆ degree()

◆ embedded_superdegree()

◆ embedded_subdegree()

◆ value_shape()

◆ dim()

◆ family()

◆ lagrange_variant()

◆ dpc_variant()

◆ map_type()

◆ sobolev_space()

◆ discontinuous()

◆ push_forward()

◆ pull_back()

◆ map_fn()

◆ entity_dofs()

◆ entity_closure_dofs()

◆ base_transformations()

Example: Order 3 Lagrange on a triangle

Example: Order 1 Raviart-Thomas on a triangle

Example: DOFs on the face of Order 2 Nedelec first kind on a tetrahedron

◆ entity_transformations()

◆ permute()

◆ permute_inv()

◆ T_apply()

◆ Tt_apply()

◆ Tt_inv_apply()

◆ Tinv_apply()

◆ Tt_apply_right()

◆ T_apply_right()

◆ Tinv_apply_right()

◆ Tt_inv_apply_right()

◆ points()

◆ interpolation_matrix()

◆ dual_matrix()

◆ wcoeffs()

◆ x()

◆ M()

◆ coefficient_matrix()

◆ has_tensor_product_factorisation()

◆ get_tensor_product_representation()

◆ interpolation_is_identity()

template<std::floating_point F>
class basix::FiniteElement< F >