#include <deal.II/fe/fe_bdm.h>

Inheritance diagram for FE_BDM< dim >:

Public Member Functions
	FE_BDM (const unsigned int p)

virtual std::string	get_name () const

virtual FiniteElement< dim > *	clone () const

virtual void	convert_generalized_support_point_values_to_nodal_values (const std::vector< Vector< double > > &support_point_values, std::vector< double > &nodal_values) const

virtual void	interpolate (std::vector< double > &local_dofs, const std::vector< double > &values) const 1

virtual void	interpolate (std::vector< double > &local_dofs, const std::vector< Vector< double > > &values, const unsigned int offset=0) const 1

virtual void	interpolate (std::vector< double > &local_dofs, const VectorSlice< const std::vector< std::vector< double > > > &values) const 1

Public Member Functions inherited from FE_PolyTensor< PolynomialsBDM< dim >, dim >
	FE_PolyTensor (const unsigned int degree, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components)

virtual double	shape_value (const unsigned int i, const Point< dim > &p) const

virtual Tensor< 1, dim >	shape_grad (const unsigned int i, const Point< dim > &p) const

virtual Tensor< 2, dim >	shape_grad_grad (const unsigned int i, const Point< dim > &p) const

Public Member Functions inherited from FiniteElement< dim, dim >
	FiniteElement (const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components)

virtual	~FiniteElement ()

const FiniteElement< dim, spacedim > &	operator[] (const unsigned int fe_index) const

bool	operator== (const FiniteElement< dim, spacedim > &) const

virtual std::size_t	memory_consumption () const

virtual Tensor< 3, dim >	shape_3rd_derivative (const unsigned int i, const Point< dim > &p) const

virtual Tensor< 3, dim >	shape_3rd_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const

virtual Tensor< 4, dim >	shape_4th_derivative (const unsigned int i, const Point< dim > &p) const

virtual Tensor< 4, dim >	shape_4th_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const

virtual bool	has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const

virtual const FullMatrix< double > &	get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const

virtual const FullMatrix< double > &	get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const

bool	prolongation_is_implemented () const

bool	isotropic_prolongation_is_implemented () const

bool	restriction_is_implemented () const

bool	isotropic_restriction_is_implemented () const

bool	restriction_is_additive (const unsigned int index) const

const FullMatrix< double > &	constraints (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const

bool	constraints_are_implemented (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const

virtual bool	hp_constraints_are_implemented () const

virtual void	get_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const

virtual void	get_face_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const

virtual void	get_subface_interpolation_matrix (const FiniteElement< dim, spacedim > &source, const unsigned int subface, FullMatrix< double > &matrix) const

virtual std::vector< std::pair< unsigned int, unsigned int > >	hp_vertex_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const

virtual std::vector< std::pair< unsigned int, unsigned int > >	hp_line_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const

virtual std::vector< std::pair< unsigned int, unsigned int > >	hp_quad_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const

virtual FiniteElementDomination::Domination	compare_for_face_domination (const FiniteElement< dim, spacedim > &fe_other) const

std::pair< unsigned int, unsigned int >	system_to_component_index (const unsigned int index) const

unsigned int	component_to_system_index (const unsigned int component, const unsigned int index) const

std::pair< unsigned int, unsigned int >	face_system_to_component_index (const unsigned int index) const

unsigned int	adjust_quad_dof_index_for_face_orientation (const unsigned int index, const bool face_orientation, const bool face_flip, const bool face_rotation) const

virtual unsigned int	face_to_cell_index (const unsigned int face_dof_index, const unsigned int face, const bool face_orientation=true, const bool face_flip=false, const bool face_rotation=false) const

unsigned int	adjust_line_dof_index_for_line_orientation (const unsigned int index, const bool line_orientation) const

const ComponentMask &	get_nonzero_components (const unsigned int i) const

unsigned int	n_nonzero_components (const unsigned int i) const

bool	is_primitive () const

bool	is_primitive (const unsigned int i) const

unsigned int	n_base_elements () const

virtual const FiniteElement< dim, spacedim > &	base_element (const unsigned int index) const

unsigned int	element_multiplicity (const unsigned int index) const

std::pair< std::pair< unsigned int, unsigned int >, unsigned int >	system_to_base_index (const unsigned int index) const

std::pair< std::pair< unsigned int, unsigned int >, unsigned int >	face_system_to_base_index (const unsigned int index) const

types::global_dof_index	first_block_of_base (const unsigned int b) const

std::pair< unsigned int, unsigned int >	component_to_base_index (const unsigned int component) const

std::pair< unsigned int, unsigned int >	block_to_base_index (const unsigned int block) const

std::pair< unsigned int, types::global_dof_index >	system_to_block_index (const unsigned int component) const

unsigned int	component_to_block_index (const unsigned int component) const

ComponentMask	component_mask (const FEValuesExtractors::Scalar &scalar) const

ComponentMask	component_mask (const FEValuesExtractors::Vector &vector) const

ComponentMask	component_mask (const FEValuesExtractors::SymmetricTensor< 2 > &sym_tensor) const

ComponentMask	component_mask (const BlockMask &block_mask) const

BlockMask	block_mask (const FEValuesExtractors::Scalar &scalar) const

BlockMask	block_mask (const FEValuesExtractors::Vector &vector) const

BlockMask	block_mask (const FEValuesExtractors::SymmetricTensor< 2 > &sym_tensor) const

BlockMask	block_mask (const ComponentMask &component_mask) const

virtual std::pair< Table< 2, bool >, std::vector< unsigned int > >	get_constant_modes () const

const std::vector< Point< dim > > &	get_unit_support_points () const

bool	has_support_points () const

virtual Point< dim >	unit_support_point (const unsigned int index) const

const std::vector< Point< dim-1 > > &	get_unit_face_support_points () const

bool	has_face_support_points () const

virtual Point< dim-1 >	unit_face_support_point (const unsigned int index) const

const std::vector< Point< dim > > &	get_generalized_support_points () const

bool	has_generalized_support_points () const

const std::vector< Point< dim-1 > > &	get_generalized_face_support_points () const

bool	has_generalized_face_support_points () const

GeometryPrimitive	get_associated_geometry_primitive (const unsigned int cell_dof_index) const

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&)

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&)

void	subscribe (const char *identifier=0) const

void	unsubscribe (const char *identifier=0) const

unsigned int	n_subscriptions () const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Public Member Functions inherited from FiniteElementData< dim >
	FiniteElementData (const std::vector< unsigned int > &dofs_per_object, const unsigned int n_components, const unsigned int degree, const Conformity conformity=unknown, const BlockIndices &block_indices=BlockIndices())

unsigned int	n_dofs_per_vertex () const

unsigned int	n_dofs_per_line () const

unsigned int	n_dofs_per_quad () const

unsigned int	n_dofs_per_hex () const

unsigned int	n_dofs_per_face () const

unsigned int	n_dofs_per_cell () const

template<int structdim>
unsigned int	n_dofs_per_object () const

unsigned int	n_components () const

unsigned int	n_blocks () const

const BlockIndices &	block_indices () const

unsigned int	tensor_degree () const

bool	conforms (const Conformity) const

bool	operator== (const FiniteElementData &) const

Private Member Functions
void	initialize_support_points (const unsigned int bdm_degree)

Static Private Member Functions
static std::vector< unsigned int >	get_dpo_vector (const unsigned int degree)

static std::vector< bool >	get_ria_vector (const unsigned int degree)

Private Attributes
std::vector< std::vector< double > >	test_values_face

std::vector< std::vector< double > >	test_values_cell

Additional Inherited Members
Public Types inherited from FiniteElementData< dim >
enum	Conformity { unknown = 0x00, L2 = 0x01, Hcurl = 0x02, Hdiv = 0x04, H1 = Hcurl \| Hdiv, H2 = 0x0e }

Static Public Member Functions inherited from FiniteElement< dim, dim >
static ::ExceptionBase &	ExcShapeFunctionNotPrimitive (int arg1)

static ::ExceptionBase &	ExcFENotPrimitive ()

static ::ExceptionBase &	ExcUnitShapeValuesDoNotExist ()

static ::ExceptionBase &	ExcFEHasNoSupportPoints ()

static ::ExceptionBase &	ExcEmbeddingVoid ()

static ::ExceptionBase &	ExcProjectionVoid ()

static ::ExceptionBase &	ExcWrongInterfaceMatrixSize (int arg1, int arg2)

static ::ExceptionBase &	ExcInterpolationNotImplemented ()

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, char *arg2, std::string &arg3)

static ::ExceptionBase &	ExcNoSubscriber (char arg1, char arg2)

Public Attributes inherited from FiniteElementData< dim >
const unsigned int	dofs_per_vertex

const unsigned int	dofs_per_line

const unsigned int	dofs_per_quad

const unsigned int	dofs_per_hex

const unsigned int	first_line_index

const unsigned int	first_quad_index

const unsigned int	first_hex_index

const unsigned int	first_face_line_index

const unsigned int	first_face_quad_index

const unsigned int	dofs_per_face

const unsigned int	dofs_per_cell

const unsigned int	components

const unsigned int	degree

const Conformity	conforming_space

const BlockIndices	block_indices_data

Static Public Attributes inherited from FiniteElement< dim, dim >
static const unsigned int	space_dimension

Static Public Attributes inherited from FiniteElementData< dim >
static const unsigned int	dimension = dim

Protected Member Functions inherited from FiniteElement< dim, dim >
void	reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false)

TableIndices< 2 >	interface_constraints_size () const

virtual InternalDataBase *	get_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim > &quadrature, ::internal::FEValues::FiniteElementRelatedData< dim, spacedim > &output_data) const=0

virtual InternalDataBase *	get_face_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim-1 > &quadrature, ::internal::FEValues::FiniteElementRelatedData< dim, spacedim > &output_data) const

virtual InternalDataBase *	get_subface_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim-1 > &quadrature, ::internal::FEValues::FiniteElementRelatedData< dim, spacedim > &output_data) const

virtual void	fill_fe_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< dim > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValues::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValues::FiniteElementRelatedData< dim, spacedim > &output_data) const=0

virtual void	fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim-1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValues::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValues::FiniteElementRelatedData< dim, spacedim > &output_data) const=0

virtual void	fill_fe_subface_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< dim-1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValues::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValues::FiniteElementRelatedData< dim, spacedim > &output_data) const=0

Static Protected Member Functions inherited from FiniteElement< dim, dim >
static std::vector< unsigned int >	compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components)

Protected Attributes inherited from FE_PolyTensor< PolynomialsBDM< dim >, dim >
MappingType	mapping_type

PolynomialsBDM< dim >	poly_space

FullMatrix< double >	inverse_node_matrix

Threads::Mutex	cache_mutex

Point< dim >	cached_point

std::vector< Tensor< 1, dim > >	cached_values

std::vector< Tensor< 2, dim > >	cached_grads

std::vector< Tensor< 3, dim > >	cached_grad_grads

Protected Attributes inherited from FiniteElement< dim, dim >
std::vector< std::vector< FullMatrix< double > > >	restriction

std::vector< std::vector< FullMatrix< double > > >	prolongation

FullMatrix< double >	interface_constraints

std::vector< Point< dim > >	unit_support_points

std::vector< Point< dim-1 > >	unit_face_support_points

std::vector< Point< dim > >	generalized_support_points

std::vector< Point< dim-1 > >	generalized_face_support_points

Table< 2, int >	adjust_quad_dof_index_for_face_orientation_table

std::vector< int >	adjust_line_dof_index_for_line_orientation_table

std::vector< std::pair< unsigned int, unsigned int > >	system_to_component_table

std::vector< std::pair< unsigned int, unsigned int > >	face_system_to_component_table

std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > >	system_to_base_table

std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > >	face_system_to_base_table

BlockIndices	base_to_block_indices

std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > >	component_to_base_table

const std::vector< bool >	restriction_is_additive_flags

const std::vector< ComponentMask >	nonzero_components

const std::vector< unsigned int >	n_nonzero_components_table

const bool	cached_primitivity

Detailed Description

template<int dim>
class FE_BDM< dim >

The Brezzi-Douglas-Marini element.

Degrees of freedom

Todo:: The 3D version exhibits some numerical instabilities, in particular for higher order

Todo:: Restriction matrices are missing.

The matching pressure space for FE_BDM of order k is the element FE_DGP of order k-1.

The BDM element of order p has p+1 degrees of freedom on each face. These are implemented as the function values in the p+1 Gauss points on each face.

Additionally, for order greater or equal 2, we have additional p(p-1), the number of vector valued polynomials in P_p, interior degrees of freedom. These are the vector function values in the first p(p-1)/2 of the p² Gauss points in the cell.

Definition at line 56 of file fe_bdm.h.

Constructor & Destructor Documentation

◆ FE_BDM()

template<int dim>

FE_BDM< dim >::FE_BDM ( const unsigned int p )

Constructor for the BDM element of degree p.

Definition at line 37 of file fe_bdm.cc.

Member Function Documentation

◆ get_name()

template<int dim>

std::string FE_BDM< dim >::get_name ( ) const

virtual

Return a string that uniquely identifies a finite element. This class returns FE_BDM<dim>(degree), with dim and degree replaced by appropriate values.

Implements FiniteElement< dim, dim >.

Definition at line 96 of file fe_bdm.cc.

◆ clone()

template<int dim>

FiniteElement< dim > * FE_BDM< dim >::clone ( ) const

virtual

A sort of virtual copy constructor. Some places in the library, for example the constructors of FESystem as well as the hp::FECollection class, need to make copies of finite elements without knowing their exact type. They do so through this function.

Implements FiniteElement< dim, dim >.

Definition at line 118 of file fe_bdm.cc.

◆ convert_generalized_support_point_values_to_nodal_values()

template<int dim>

void FE_BDM< dim >::convert_generalized_support_point_values_to_nodal_values	(	const std::vector< Vector< double > > &	support_point_values,
		std::vector< double > &	nodal_values
	)		const

virtual

Given the values of a function \(f(\mathbf x)\) at the (generalized) support points, this function then computes what the nodal values of the element are, i.e., \(\Psi_i[f]\), where \(\Psi_i\) are the node functionals of the element. The values \(\Psi_i[f]\) are then the expansion coefficients for the shape functions of the finite element function that interpolates the given function \(f(x)\), i.e., \( f_h(\mathbf x) = \sum_i \Psi_i[f] \varphi_i(\mathbf x) \) is the finite element interpolant of \(f\) with the current element. The operation described here is used, for example, in the FETools::compute_node_matrix() function.

In more detail, let us assume that the generalized support points (see this glossary entry ) of the current element are \(\hat{\mathbf x}_i\) and that the node functionals associated with the current element are \(\Psi_i[\cdot]\). Then, the fact that the element is based on generalized support points, implies that if we apply \(\Psi_i\) to a (possibly vector-valued) finite element function \(\varphi\), the result must have the form \(\Psi_i[\varphi] = f_i(\varphi(\hat{\mathbf x}_i))\) – in other words, the value of the node functional \(\Psi_i\) applied to \(\varphi\) only depends on the values of \(\varphi\) at \(\hat{\mathbf x}_i\) and not on values anywhere else, or integrals of \(\varphi\), or any other kind of information.

The exact form of \(f_i\) depends on the element. For example, for scalar Lagrange elements, we have that in fact \(\Psi_i[\varphi] = \varphi(\hat{\mathbf x}_i)\). If you combine multiple scalar Lagrange elements via an FESystem object, then \(\Psi_i[\varphi] = \varphi(\hat{\mathbf x}_i)_{c(i)}\) where \(c(i)\) is the result of the FiniteElement::system_to_component_index() function's return value's first component. In these two cases, \(f_i\) is therefore simply the identity (in the scalar case) or a function that selects a particular vector component of its argument. On the other hand, for Raviart-Thomas elements, one would have that \(f_i(\mathbf y) = \mathbf y \cdot \mathbf n_i\) where \(\mathbf n_i\) is the normal vector of the face at which the shape function is defined.

Given all of this, what this function does is the following: If you input a list of values of a function \(\varphi\) at all generalized support points (where each value is in fact a vector of values with as many components as the element has), then this function returns a vector of values obtained by applying the node functionals to these values. In other words, if you pass in \(\{\varphi(\hat{\mathbf x}_i)\}_{i=0}^{N-1}\) then you will get out a vector \(\{\Psi[\varphi]\}_{i=0}^{N-1}\) where \(N\) equals dofs_per_cell.

Parameters

[in]	support_point_values	An array of size `dofs_per_cell` (which equals the number of points the get_generalized_support_points() function will return) where each element is a vector with as many entries as the element has vector components. This array should contain the values of a function at the generalized support points of the current element.
[out]	nodal_values	An array of size `dofs_per_cell` that contains the node functionals of the element applied to the given function.

Note: Given what the function is supposed to do, the function clearly can only work for elements that actually implement (generalized) support points. Elements that do not have generalized support points – e.g., elements whose nodal functionals evaluate integrals or moments of functions (such as FE_Q_Hierarchical) – can in general not make sense of the operation that is required for this function. They consequently may not implement it.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 128 of file fe_bdm.cc.

◆ interpolate() [1/3]

template<int dim>

void FE_BDM< dim >::interpolate	(	std::vector< double > &	local_dofs,
		const std::vector< double > &	values
	)		const

virtual

Interpolate a set of scalar values, computed in the generalized support points.

Note: This function is implemented in FiniteElement for the case that the element has support points. In this case, the resulting coefficients are just the values in the support points. All other elements must reimplement it.

Deprecated:: Use convert_generalized_support_point_values_to_nodal_values() instead.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 198 of file fe_bdm.cc.

◆ interpolate() [2/3]

template<int dim>

void FE_BDM< dim >::interpolate	(	std::vector< double > &	local_dofs,
		const std::vector< Vector< double > > &	values,
		const unsigned int	offset = `0`
	)		const

virtual

Interpolate a set of vector values, computed in the generalized support points.

Since a finite element often only interpolates part of a vector, offset is used to determine the first component of the vector to be interpolated. Maybe consider changing your data structures to use the next function.

Deprecated:: Use convert_generalized_support_point_values_to_nodal_values() instead.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 208 of file fe_bdm.cc.

◆ interpolate() [3/3]

template<int dim>

void FE_BDM< dim >::interpolate	(	std::vector< double > &	local_dofs,
		const VectorSlice< const std::vector< std::vector< double > > > &	values
	)		const

virtual

Interpolate a set of vector values, computed in the generalized support points.

Deprecated:: Use convert_generalized_support_point_values_to_nodal_values() instead.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 220 of file fe_bdm.cc.

◆ get_dpo_vector()

template<int dim>

std::vector< unsigned int > FE_BDM< dim >::get_dpo_vector ( const unsigned int degree )

staticprivate

Only for internal use. Its full name is get_dofs_per_object_vector function and it creates the dofs_per_object vector that is needed within the constructor to be passed to the constructor of FiniteElementData.

Definition at line 292 of file fe_bdm.cc.

◆ get_ria_vector()

template<int dim>

std::vector< bool > FE_BDM< dim >::get_ria_vector ( const unsigned int degree )

staticprivate

Compute the vector used for the restriction_is_additive field passed to the base class's constructor.

Definition at line 317 of file fe_bdm.cc.

◆ initialize_support_points()

template<int dim>

void FE_BDM< dim >::initialize_support_points ( const unsigned int bdm_degree )

private

Initialize the FiniteElement<dim>::generalized_support_points and FiniteElement<dim>::generalized_face_support_points fields. Called from the constructor. See the glossary entry on generalized support points for more information.

Definition at line 392 of file fe_bdm.cc.

Member Data Documentation

◆ test_values_face

template<int dim>

std::vector<std::vector<double> > FE_BDM< dim >::test_values_face

private

The values in the face support points of the polynomials needed as test functions. The outer vector is indexed by quadrature points, the inner by the test function. The test function space is PolynomialsP<dim-1>.

Definition at line 120 of file fe_bdm.h.

◆ test_values_cell

template<int dim>

std::vector<std::vector<double> > FE_BDM< dim >::test_values_cell

private

The values in the interior support points of the polynomials needed as test functions. The outer vector is indexed by quadrature points, the inner by the test function. The test function space is PolynomialsP<dim>.

Definition at line 126 of file fe_bdm.h.

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

deal.II/fe/fe_bdm.h
source/fe/fe_bdm.cc

Public Member Functions

Private Member Functions

Static Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

template<int dim> class FE_BDM< dim >

Degrees of freedom

Constructor & Destructor Documentation

◆ FE_BDM()

Member Function Documentation

◆ get_name()

◆ clone()

◆ convert_generalized_support_point_values_to_nodal_values()

◆ interpolate() [1/3]

◆ interpolate() [2/3]

◆ interpolate() [3/3]

◆ get_dpo_vector()

◆ get_ria_vector()

◆ initialize_support_points()

Member Data Documentation

◆ test_values_face

◆ test_values_cell

template<int dim>
class FE_BDM< dim >