Reference documentation for deal.II version 9.2.0
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#include <deal.II/fe/fe_nedelec.h>
Public Member Functions | |
FE_Nedelec (const unsigned int order) | |
virtual std::string | get_name () const override |
virtual bool | has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const override |
virtual bool | hp_constraints_are_implemented () const override |
virtual FiniteElementDomination::Domination | compare_for_domination (const FiniteElement< dim > &fe_other, const unsigned int codim=0) const override final |
virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_vertex_dof_identities (const FiniteElement< dim > &fe_other) const override |
virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_line_dof_identities (const FiniteElement< dim > &fe_other) const override |
virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_quad_dof_identities (const FiniteElement< dim > &fe_other) const override |
virtual void | get_face_interpolation_matrix (const FiniteElement< dim > &source, FullMatrix< double > &matrix) const override |
virtual void | get_subface_interpolation_matrix (const FiniteElement< dim > &source, const unsigned int subface, FullMatrix< double > &matrix) const override |
virtual const FullMatrix< double > & | get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override |
virtual const FullMatrix< double > & | get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override |
virtual void | convert_generalized_support_point_values_to_dof_values (const std::vector< Vector< double >> &support_point_values, std::vector< double > &nodal_values) const override |
virtual std::pair< Table< 2, bool >, std::vector< unsigned int > > | get_constant_modes () const override |
virtual std::size_t | memory_consumption () const override |
virtual std::unique_ptr< FiniteElement< dim, dim > > | clone () const override |
void | get_subface_interpolation_matrix (const FiniteElement< 1, 1 > &, const unsigned int, FullMatrix< double > &) const |
Public Member Functions inherited from FE_PolyTensor< dim > | |
FE_PolyTensor (const TensorPolynomialsBase< dim > &polynomials, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components) | |
FE_PolyTensor (const FE_PolyTensor &fe) | |
virtual UpdateFlags | requires_update_flags (const UpdateFlags update_flags) const override |
virtual double | shape_value (const unsigned int i, const Point< dim > &p) const override |
virtual double | shape_value_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 1, dim > | shape_grad (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 1, dim > | shape_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 2, dim > | shape_grad_grad (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 2, dim > | shape_grad_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
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) | |
FiniteElement (FiniteElement< dim, spacedim > &&)=default | |
FiniteElement (const FiniteElement< dim, spacedim > &)=default | |
virtual | ~FiniteElement () override=default |
std::pair< std::unique_ptr< FiniteElement< dim, spacedim > >, unsigned int > | operator^ (const unsigned int multiplicity) const |
virtual std::unique_ptr< FiniteElement< dim, spacedim > > | clone () const=0 |
virtual std::string | get_name () const=0 |
const FiniteElement< dim, spacedim > & | operator[] (const unsigned int fe_index) 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 |
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 final |
virtual FiniteElementDomination::Domination | compare_for_domination (const FiniteElement< dim, spacedim > &fe_other, const unsigned int codim=0) const |
virtual bool | operator== (const FiniteElement< dim, spacedim > &fe) const |
bool | operator!= (const FiniteElement< dim, spacedim > &) 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 |
const FiniteElement< dim, spacedim > & | get_sub_fe (const ComponentMask &mask) const |
virtual const FiniteElement< dim, spacedim > & | get_sub_fe (const unsigned int first_component, const unsigned int n_selected_components) 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 |
GeometryPrimitive | get_associated_geometry_primitive (const unsigned int cell_dof_index) const |
virtual std::size_t | memory_consumption () const |
Public Member Functions inherited from Subscriptor | |
Subscriptor () | |
Subscriptor (const Subscriptor &) | |
Subscriptor (Subscriptor &&) noexcept | |
virtual | ~Subscriptor () |
Subscriptor & | operator= (const Subscriptor &) |
Subscriptor & | operator= (Subscriptor &&) noexcept |
void | subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const |
void | unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const |
unsigned int | n_subscriptions () const |
template<typename StreamType > | |
void | list_subscribers (StreamType &stream) 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 order) |
void | initialize_restriction () |
void | initialize_support_points (const unsigned int) |
void | initialize_support_points (const unsigned int order) |
void | initialize_support_points (const unsigned int order) |
void | initialize_restriction () |
Static Private Member Functions | |
static std::vector< unsigned int > | get_dpo_vector (const unsigned int degree, bool dg=false) |
Private Attributes | |
Table< 2, double > | boundary_weights |
Threads::Mutex | mutex |
Friends | |
template<int dim1> | |
class | FE_Nedelec |
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, std::string arg2, std::string arg3) |
static ::ExceptionBase & | ExcNoSubscriber (std::string arg1, std::string 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 FE_PolyTensor< dim > | |
bool | single_mapping_kind () const |
MappingKind | get_mapping_kind (const unsigned int i) const |
virtual std::unique_ptr< typename FiniteElement< dim, dim >::InternalDataBase > | get_data (const UpdateFlags update_flags, const Mapping< dim, dim > &, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &) const override |
virtual void | fill_fe_values (const typename Triangulation< dim, dim >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< dim > &quadrature, const Mapping< dim, dim > &mapping, const typename Mapping< dim, dim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, dim > &mapping_data, const typename FiniteElement< dim, dim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &output_data) const override |
virtual void | fill_fe_face_values (const typename Triangulation< dim, dim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim - 1 > &quadrature, const Mapping< dim, dim > &mapping, const typename Mapping< dim, dim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, dim > &mapping_data, const typename FiniteElement< dim, dim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &output_data) const override |
virtual void | fill_fe_subface_values (const typename Triangulation< dim, dim >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< dim - 1 > &quadrature, const Mapping< dim, dim > &mapping, const typename Mapping< dim, dim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, dim > &mapping_data, const typename FiniteElement< dim, dim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &output_data) const override |
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 std::unique_ptr< InternalDataBase > | get_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0 |
virtual std::unique_ptr< InternalDataBase > | get_face_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim - 1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const |
virtual std::unique_ptr< InternalDataBase > | get_subface_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim - 1 > &quadrature, ::internal::FEValuesImplementation::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::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::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::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::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::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::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< dim > | |
std::vector< MappingKind > | mapping_kind |
const std::unique_ptr< const TensorPolynomialsBase< dim > > | poly_space |
FullMatrix< double > | inverse_node_matrix |
std::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 |
Implementation of Nédélec elements. The Nédélec space is designed to solve problems in which the solution only lives in the space \(H^\text{curl}=\{ {\mathbf u} \in L_2: \text{curl}\, {\mathbf u} \in L_2\}\), rather than in the more commonly used space \(H^1=\{ u \in L_2: \nabla u \in L_2\}\). In other words, the solution must be a vector field whose curl is square integrable, but for which the gradient may not be square integrable. The typical application for this space (and these elements) is to the Maxwell equations and corresponding simplifications, such as the reduced version of the Maxwell equation that only involves the electric field \(\mathbf E\) which has to satisfy the equation \(\text{curl}\, \text{curl}\, {\mathbf E} = 0\) in the time independent case when no currents are present, or the equation \(\text{curl}\,\text{curl}\,{\mathbf A} = 4\pi{\mathbf j}\) that the magnetic vector potential \(\mathbf A\) has to satisfy in the time independent case.
The defining characteristic of functions in \(H^\text{curl}\) is that they are in general discontinuous – but that if you draw a line in 2d (or a surface in 3d), then the tangential component(s) of the vector field must be continuous across the line (or surface) even though the normal component may not be. As a consequence, the Nédélec element is constructed in such a way that (i) it is vector-valued, (ii) the shape functions are discontinuous, but (iii) the tangential component(s) of the vector field represented by each shape function are continuous across the faces of cells.
Other properties of the Nédélec element are that (i) it is not a primitive element; (ii) the shape functions are defined so that certain integrals over the faces are either zero or one, rather than the common case of certain point values being either zero or one.
We follow the commonly used – though confusing – definition of the "degree" of Nédélec elements. Specifically, the "degree" of the element denotes the polynomial degree of the largest complete polynomial subspace contained in the finite element space, even if the space may contain shape functions of higher polynomial degree. The lowest order element is consequently FE_Nedelec(0), i.e., the Raviart-Thomas element "of degree zero", even though the functions of this space are in general polynomials of degree one in each variable. This choice of "degree" implies that the approximation order of the function itself is degree+1, as with usual polynomial spaces. The numbering so chosen implies the sequence
\[ Q_{k+1} \stackrel{\text{grad}}{\rightarrow} \text{Nedelec}_k \stackrel{\text{curl}}{\rightarrow} \text{RaviartThomas}_k \stackrel{\text{div}}{\rightarrow} DGQ_{k} \]
Note that this follows the convention of Brezzi and Raviart, though not the one used in the original paper by Nédélec.
This class is not implemented for the codimension one case (spacedim != dim
).
The interpolation operators associated with the Nédélec element are constructed such that interpolation and computing the curl are commuting operations on rectangular mesh cells. We require this from interpolating arbitrary functions as well as the restriction matrices.
The node values for an element of degree k on the reference cell are:
dim
-1 dimensional FE_Nedelec polynomials of degree k-1. The node values above rely on integrals, which will be computed by quadrature rules themselves. The generalized support points are a set of points such that this quadrature can be performed with sufficient accuracy. The points needed are those of QGaussk+1 on each edge and QGaussk+2 on each face and in the interior of the cell (or none for N1).
Definition at line 147 of file fe_nedelec.h.
FE_Nedelec< dim >::FE_Nedelec | ( | const unsigned int | order | ) |
Constructor for the Nedelec element of given order
. The maximal polynomial degree of the shape functions is order+1
(in each variable; the total polynomial degree may be higher). If order = 0
, the element is linear and has degrees of freedom only on the edges. If order >=1
the element has degrees of freedom on the edges, faces and volume. For example the 3D version of FE_Nedelec has 12 degrees of freedom for order = 0
and 54 for degree = 1
. It is important to have enough quadrature points in order to perform the quadrature with sufficient accuracy. For example QGauss<dim>(order + 2) can be used for the quadrature formula, where order
is the order of FE_Nedelec.
Definition at line 71 of file fe_nedelec.cc.
|
overridevirtual |
Return a string that uniquely identifies a finite element. This class returns FE_Nedelec<dim>(degree)
, with dim
and degree
replaced by appropriate values.
Definition at line 207 of file fe_nedelec.cc.
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overridevirtual |
This function returns true
, if the shape function shape_index
has non-zero function values somewhere on the face face_index
.
Reimplemented from FiniteElement< dim, dim >.
Definition at line 2035 of file fe_nedelec.cc.
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overridevirtual |
Return whether this element implements its hanging node constraints in the new way, which has to be used to make elements "hp compatible".
For the FE_Nedelec
class the result is always true (independent of the degree of the element), as it implements the complete set of functions necessary for hp capability.
Definition at line 2310 of file fe_nedelec.cc.
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finaloverridevirtual |
Return whether this element dominates another one given as argument fe_other
, whether it is the other way around, whether neither dominates, or if either could dominate. The codim
parameter describes the codimension of the investigated subspace and specifies that it is subject to this comparison. For example, if codim==0
then this function compares which element dominates at the cell level. If codim==1
, then the elements are compared at faces, i.e., the comparison happens between the function spaces of the two finite elements as restricted to a face. Larger values of codim
work correspondingly.
For a definition of domination, see FiniteElementDomination::Domination and in particular the hp paper.
Definition at line 2274 of file fe_nedelec.cc.
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overridevirtual |
If, on a vertex, several finite elements are active, the hp code first assigns the degrees of freedom of each of these FEs different global indices. It then calls this function to find out which of them should get identical values, and consequently can receive the same global DoF index. This function therefore returns a list of identities between DoFs of the present finite element object with the DoFs of fe_other
, which is a reference to a finite element object representing one of the other finite elements active on this particular vertex. The function computes which of the degrees of freedom of the two finite element objects are equivalent, both numbered between zero and the corresponding value of dofs_per_vertex of the two finite elements. The first index of each pair denotes one of the vertex dofs of the present element, whereas the second is the corresponding index of the other finite element.
Definition at line 2317 of file fe_nedelec.cc.
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overridevirtual |
Same as hp_vertex_dof_indices(), except that the function treats degrees of freedom on lines.
Definition at line 2326 of file fe_nedelec.cc.
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overridevirtual |
Same as hp_vertex_dof_indices(), except that the function treats degrees of freedom on lines.
Definition at line 2368 of file fe_nedelec.cc.
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overridevirtual |
Return the matrix interpolating from a face of one element to the face of the neighboring element. The size of the matrix is then source.dofs_per_face
times this->dofs_per_face
.
Derived elements will have to implement this function. They may only provide interpolation matrices for certain source finite elements, for example those from the same family. If they don't implement interpolation from a given element, then they must throw an exception of type FiniteElement<dim>::ExcInterpolationNotImplemented
.
Definition at line 2422 of file fe_nedelec.cc.
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overridevirtual |
Return the matrix interpolating from a face of one element to the subface of the neighboring element. The size of the matrix is then source.dofs_per_face
times this->dofs_per_face
.
Derived elements will have to implement this function. They may only provide interpolation matrices for certain source finite elements, for example those from the same family. If they don't implement interpolation from a given element, then they must throw an exception of type ExcInterpolationNotImplemented
.
Definition at line 2524 of file fe_nedelec.cc.
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Projection from a fine grid space onto a coarse grid space. If this projection operator is associated with a matrix P
, then the restriction of this matrix P_i
to a single child cell is returned here.
The matrix P
is the concatenation or the sum of the cell matrices P_i
, depending on the restriction_is_additive_flags. This distinguishes interpolation (concatenation) and projection with respect to scalar products (summation).
Row and column indices are related to coarse grid and fine grid spaces, respectively, consistent with the definition of the associated operator.
Reimplemented from FiniteElement< dim, dim >.
Definition at line 3010 of file fe_nedelec.cc.
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Embedding matrix between grids.
The identity operator from a coarse grid space into a fine grid space is associated with a matrix P
. The restriction of this matrix P_i
to a single child cell is returned here.
The matrix P
is the concatenation, not the sum of the cell matrices P_i
. That is, if the same non-zero entry j,k
exists in two different child matrices P_i
, the value should be the same in both matrices and it is copied into the matrix P
only once.
Row and column indices are related to fine grid and coarse grid spaces, respectively, consistent with the definition of the associated operator.
These matrices are used by routines assembling the prolongation matrix for multi-level methods. Upon assembling the transfer matrix between cells using this matrix array, zero elements in the prolongation matrix are discarded and will not fill up the transfer matrix.
Reimplemented from FiniteElement< dim, dim >.
Definition at line 2955 of file fe_nedelec.cc.
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Given the values of a function \(f(\mathbf x)\) at the (generalized) support points of the reference cell, 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 (see also Node values or node functionals). 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
.
[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. |
Reimplemented from FiniteElement< dim, dim >.
Definition at line 3073 of file fe_nedelec.cc.
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Return a list of constant modes of the element.
Definition at line 4019 of file fe_nedelec.cc.
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Definition at line 4035 of file fe_nedelec.cc.
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Definition at line 225 of file fe_nedelec.cc.
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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
.
If the optional argument dg
is true, the vector returned will have all degrees of freedom assigned to the cell, none on the faces and edges.
Definition at line 1999 of file fe_nedelec.cc.
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private |
Initialize the generalized_support_points
field of the FiniteElement class and fill the tables with interpolation weights (boundary_weights and interior_weights). Called from the constructor.
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private |
Initialize the interpolation from functions on refined mesh cells onto the father cell. According to the philosophy of the Nédélec element, this restriction operator preserves the curl of a function weakly.
Definition at line 515 of file fe_nedelec.cc.
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private |
Definition at line 244 of file fe_nedelec.cc.
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private |
Definition at line 253 of file fe_nedelec.cc.
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private |
Definition at line 335 of file fe_nedelec.cc.
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Definition at line 502 of file fe_nedelec.cc.
void FE_Nedelec< 1 >::get_subface_interpolation_matrix | ( | const FiniteElement< 1, 1 > & | , |
const unsigned int | , | ||
FullMatrix< double > & | |||
) | const |
Definition at line 2497 of file fe_nedelec.cc.
Definition at line 373 of file fe_nedelec.h.
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private |
These are the factors multiplied to a function in the generalized_face_support_points when computing the integration.
See the glossary entry on generalized support points for more information.
Definition at line 364 of file fe_nedelec.h.
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mutableprivate |
Mutex for protecting initialization of restriction and embedding matrix.
Definition at line 369 of file fe_nedelec.h.