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Reference documentation for deal.II version 9.1.1
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#include <deal.II/fe/fe_rt_bubbles.h>
Public Member Functions | |
FE_RT_Bubbles (const unsigned int k) | |
virtual std::string | get_name () const override |
virtual std::unique_ptr< FiniteElement< dim, dim > > | clone () 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 |
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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 override |
virtual Tensor< 1, dim > | shape_grad (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 2, dim > | shape_grad_grad (const unsigned int i, const Point< dim > &p) const override |
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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 |
const FiniteElement< dim, spacedim > & | operator[] (const unsigned int fe_index) const |
virtual bool | operator== (const FiniteElement< dim, spacedim > &fe) 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 final |
virtual FiniteElementDomination::Domination | compare_for_domination (const FiniteElement< dim, spacedim > &fe_other, const unsigned int codim=0) 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 |
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 |
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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) |
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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 rt_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) |
Additional Inherited Members | |
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enum | Conformity { unknown = 0x00, L2 = 0x01, Hcurl = 0x02, Hdiv = 0x04, H1 = Hcurl | Hdiv, H2 = 0x0e } |
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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 () |
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static ::ExceptionBase & | ExcInUse (int arg1, std::string arg2, std::string arg3) |
static ::ExceptionBase & | ExcNoSubscriber (std::string arg1, std::string arg2) |
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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 |
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static const unsigned int | space_dimension |
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static const unsigned int | dimension = dim |
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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 |
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static std::vector< unsigned int > | compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components) |
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MappingType | mapping_type |
PolynomialsRT_Bubbles< 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 |
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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 |
This class implements a curl-enhanced Raviart-Thomas elements, conforming with Hdiv space. The node functionals are defined as point values in Gauss-Lobatto points. These elements generate vector fields with normal components continuous between mesh cells. The purpose of this finite element is in localizing the interactions between degrees of freedom around the nodes when an appropriate quadrature rule is used, leading to a block-diagonal mass matrix (even with full-tensor coefficient).
The elements are defined through enrichment of classical Raviart-Thomas elements with extra curls, so that the Hdiv conformity is preserved, and the total number of degrees of freedom of FE_RT_Bubbles of order k is equal to the number of DoFs in dim copies of FE_Q of order k.
The matching pressure space for FE_RT_Bubbles of order k is FE_DGQ of order k-1. With the exact integration, this pair yields \((k+1)\)-st order of convergence in \(L_2\)-norm for a vector variable and \(k\)-th order in \(L_2\)-norm for a scalar one (same as \(BDM_k \times P_{k-1}\)).
For this enhanced Raviart-Thomas element, the node values are not cell and face moments with respect to certain polynomials, but the values in Gauss-Lobatto quadrature points. The nodal values on edges (faces in 3D) are evaluated first, according to the natural ordering of the edges (faces) of a cell. The interior degrees of freedom are evaluated last.
For an RT-Bubbles element of degree k, we choose (k+1)dim-1 Gauss-Lobatto points on each face. These points are ordered lexicographically with respect to the orientation of the face. In the interior of the cells, the values are computed using an anisotropic Gauss-Lobatto formula for integration. The mass matrix assembled with the use of this same quadrature rule, is block diagonal with blocks corresponding to quadrature points. See "Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra" for more details.
The elements of degree \(k=3\) in 2D and \(k=2\) in 3D are shown in the figures below (filled arrows indicate DoFs for which continuity across the edges (faces in 3D) is required).
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Left - \(2D,\,k=3\), right - \(3D,\,k=2\). |
Definition at line 90 of file fe_rt_bubbles.h.
FE_RT_Bubbles< dim >::FE_RT_Bubbles | ( | const unsigned int | k | ) |
Constructor for the RT_Bubbles element of degree k
.
Definition at line 39 of file fe_rt_bubbles.cc.
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overridevirtual |
Returns a string that uniquely identifies a finite element. This class returns FE_RT_Bubbles<dim>(degree)
, with dim
and degree
replaced by appropriate values.
Implements FiniteElement< dim, dim >.
Definition at line 105 of file fe_rt_bubbles.cc.
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overridevirtual |
A sort of virtual copy constructor, this function returns a copy of the finite element object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.
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 119 of file fe_rt_bubbles.cc.
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overridevirtual |
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 263 of file fe_rt_bubbles.cc.
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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 208 of file fe_rt_bubbles.cc.
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staticprivate |
Compute the vector used for the restriction_is_additive
field passed to the base class's constructor.
Definition at line 240 of file fe_rt_bubbles.cc.
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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 133 of file fe_rt_bubbles.cc.