
Public Member Functions | |
| FiniteElement (const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< std::vector< bool > > &nonzero_components) | |
| virtual | ~FiniteElement () |
| virtual std::string | get_name () const =0 |
| const FiniteElement< dim > & | operator[] (const unsigned int fe_index) const |
| virtual std::vector< std::pair < unsigned int, unsigned int > > | hp_vertex_dof_identities (const FiniteElement< dim > &fe_other) const |
| virtual std::vector< std::pair < unsigned int, unsigned int > > | hp_line_dof_identities (const FiniteElement< dim > &fe_other) const |
| virtual std::vector< std::pair < unsigned int, unsigned int > > | hp_quad_dof_identities (const FiniteElement< dim > &fe_other) const |
| virtual FiniteElementDomination::Domination | compare_for_face_domination (const FiniteElement< dim > &fe_other) const |
| bool | operator== (const FiniteElement< dim > &) const |
| virtual unsigned int | memory_consumption () const |
Shape function access | |
| virtual double | shape_value (const unsigned int i, const Point< dim > &p) const |
| virtual double | shape_value_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const |
| virtual Tensor< 1, dim > | shape_grad (const unsigned int i, const Point< dim > &p) const |
| virtual Tensor< 1, dim > | shape_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const |
| virtual Tensor< 2, dim > | shape_grad_grad (const unsigned int i, const Point< dim > &p) const |
| virtual Tensor< 2, dim > | shape_grad_grad_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 |
Transfer and constraint matrices | |
| const FullMatrix< double > & | get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const |
| 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 > &source, FullMatrix< double > &matrix) const |
Functions to support hp | |
| virtual void | get_face_interpolation_matrix (const FiniteElement< dim > &source, FullMatrix< double > &matrix) const |
| virtual void | get_subface_interpolation_matrix (const FiniteElement< dim > &source, const unsigned int subface, FullMatrix< double > &matrix) const |
Index computations | |
| 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 |
| unsigned int | adjust_line_dof_index_for_line_orientation (const unsigned int index, const bool line_orientation) const |
| const std::vector< bool > & | get_nonzero_components (const unsigned int i) const |
| unsigned int | n_nonzero_components (const unsigned int i) const |
| bool | is_primitive (const unsigned int i) const |
| bool | is_primitive () const |
| virtual unsigned int | n_base_elements () const =0 |
| virtual const FiniteElement < dim > & | base_element (const unsigned int index) const =0 |
| virtual unsigned int | element_multiplicity (const unsigned int index) const =0 |
| 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 |
| unsigned int | 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, unsigned int > | system_to_block_index (const unsigned int component) const |
| unsigned int | component_to_block_index (const unsigned int component) const |
Support points and interpolation | |
| 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 |
| virtual void | interpolate (std::vector< double > &local_dofs, const std::vector< double > &values) const |
| virtual void | interpolate (std::vector< double > &local_dofs, const std::vector< Vector< double > > &values, unsigned int offset=0) const |
| virtual void | interpolate (std::vector< double > &local_dofs, const VectorSlice< const std::vector< std::vector< double > > > &values) const |
Protected Member Functions | |
| void | reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false) |
| TableIndices< 2 > | interface_constraints_size () const |
| void | compute_2nd (const Mapping< dim > &mapping, const typename Triangulation< dim >::cell_iterator &cell, const unsigned int offset, typename Mapping< dim >::InternalDataBase &mapping_internal, InternalDataBase &fe_internal, FEValuesData< dim > &data) const |
| virtual UpdateFlags | update_once (const UpdateFlags flags) const =0 |
| virtual UpdateFlags | update_each (const UpdateFlags flags) const =0 |
| virtual FiniteElement< dim > * | clone () const =0 |
Static Protected Member Functions | |
| static std::vector< unsigned int > | compute_n_nonzero_components (const std::vector< std::vector< bool > > &nonzero_components) |
Protected Attributes | |
| const bool | cached_primitivity |
| 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 |
Private Member Functions | |
| virtual Mapping< dim > ::InternalDataBase * | get_data (const UpdateFlags flags, const Mapping< dim > &mapping, const Quadrature< dim > &quadrature) const =0 |
| virtual Mapping< dim > ::InternalDataBase * | get_face_data (const UpdateFlags flags, const Mapping< dim > &mapping, const Quadrature< dim-1 > &quadrature) const |
| virtual Mapping< dim > ::InternalDataBase * | get_subface_data (const UpdateFlags flags, const Mapping< dim > &mapping, const Quadrature< dim-1 > &quadrature) const |
| virtual void | fill_fe_values (const Mapping< dim > &mapping, const typename Triangulation< dim >::cell_iterator &cell, const Quadrature< dim > &quadrature, typename Mapping< dim >::InternalDataBase &mapping_internal, typename Mapping< dim >::InternalDataBase &fe_internal, FEValuesData< dim > &data) const =0 |
| virtual void | fill_fe_face_values (const Mapping< dim > &mapping, const typename Triangulation< dim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim-1 > &quadrature, typename Mapping< dim >::InternalDataBase &mapping_internal, typename Mapping< dim >::InternalDataBase &fe_internal, FEValuesData< dim > &data) const =0 |
| virtual void | fill_fe_subface_values (const Mapping< dim > &mapping, const typename Triangulation< dim >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< dim-1 > &quadrature, typename Mapping< dim >::InternalDataBase &mapping_internal, typename Mapping< dim >::InternalDataBase &fe_internal, FEValuesData< dim > &data) const =0 |
Private Attributes | |
| 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 |
| std::vector< unsigned int > | first_block_of_base_table |
| 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< std::vector < bool > > | nonzero_components |
| const std::vector< unsigned int > | n_nonzero_components_table |
Static Private Attributes | |
| static const double | fd_step_length |
Friends | |
| class | InternalDataBase |
| class | FEValuesBase< dim > |
| class | FEValues< dim > |
| class | FEFaceValues< dim > |
| class | FESubfaceValues< dim > |
| class | FESystem |
| class | FE_PolyTensor |
| class | hp::FECollection< dim > |
Classes | |
| class | ExcBoundaryFaceUsed |
| class | ExcComponentIndexInvalid |
| class | ExcConstraintsVoid |
| class | ExcEmbeddingVoid |
| class | ExcFEHasNoSupportPoints |
| class | ExcFENotPrimitive |
| class | ExcInterpolationNotImplemented |
| class | ExcJacobiDeterminantHasWrongSign |
| class | ExcProjectionVoid |
| class | ExcShapeFunctionNotPrimitive |
| class | ExcUnitShapeValuesDoNotExist |
| class | ExcWrongInterfaceMatrixSize |
| class | InternalDataBase |
The interface of this class is very restrictive. The reason is that finite element values should be accessed only by use of FEValues objects. These, together with FiniteElement are responsible to provide an optimized implementation.
This class declares the shape functions and their derivatives on the unit cell
. The means to transform them onto a given cell in physical space is provided by the FEValues class with a Mapping object.
The different matrices are initialized with the correct size, such that in the derived (concrete) finite element classes, their entries only have to be filled in; no resizing is needed. If the matrices are not defined by a concrete finite element, they should be resized to zero. This way functions using them can find out, that they are missing. On the other hand, it is possible to use finite element classes without implementation of the full functionality, if only part of it is needed. The functionality under consideration here is hanging nodes constraints and grid transfer, respectively.
For vector valued elements shape functions may have nonzero entries in one or several components of the vector valued function. If the element is primitive, there is indeed a single component with a nonzero entry for each shape function. This component can be determined by system_to_component_index(), the number of components is FiniteElementData::n_components().
Furthermore, you may want to split your linear system into blocks for the use in BlockVector, BlockSparseMatrix, BlockMatrixArray and so on. If you use non-primitive elements, you cannot determine the block number by system_to_component_index(). Instead, you can use system_to_block_index(), which will automatically take care of the additional components occupied by vector valued elements. The number of generated blocks can be determined by FiniteElementData::n_blocks().
If you decide to operate by base element and multiplicity, the function first_block_of_base() will be helpful.
Since a FiniteElement does not have information on the actual grid cell, it can only provide support points on the unit cell. Support points on the actual grid cell must be computed by mapping these points. The class used for this kind of operation is FEValues. In most cases, code of the following type will serve to provide the mapped support points.
Quadrature<dim> dummy_quadrature (fe.get_unit_support_points()); FEValues<dim> fe_values (mapping, fe, dummy_quadrature, update_quadrature_points); fe_values.reinit (cell); Point<dim>& mapped_point = fe_values.quadrature_point (i);
Alternatively, the points can be transformed one-by-one:
const vector<Point<dim> >& unit_points = fe.get_unit_support_points(); Point<dim> mapped_point = mapping.transform_unit_to_real_cell (cell, unit_points[i]);
The following sections list the information to be provided by derived classes, depending on the space dimension. They are followed by a list of functions helping to generate these values.
Finite elements in one dimension need only set the restriction and prolongation matrices. The constructor of this class in one dimension presets the interface_constraints matrix to have dimension zero. Changing this behaviour in derived classes is generally not a reasonable idea and you risk getting into trouble.
In addition to the fields already present in 1D, a constraint matrix is needed, if the finite element has node values located on edges or vertices. These constraints are represented by an
-matrix interface_constraints, where m is the number of degrees of freedom on the refined side without the corner vertices (those dofs on the middle vertex plus those on the two lines), and n is that of the unrefined side (those dofs on the two vertices plus those on the line). The matrix is thus a rectangular one. The
size of the interface_constraints matrix can also be accessed through the interface_constraints_size() function.
The mapping of the dofs onto the indices of the matrix on the unrefined side is as follows: let
be the number of dofs on a vertex,
that on a line, then
refers to the dofs on vertex zero of the unrefined line,
to those on vertex one,
to those on the line.
Similarly,
refers to the dofs on the middle vertex of the refined side (vertex one of child line zero, vertex zero of child line one),
refers to the dofs on child line zero,
refers to the dofs on child line one. Please note that we do not need to reserve space for the dofs on the end vertices of the refined lines, since these must be mapped one-to-one to the appropriate dofs of the vertices of the unrefined line.
It should be noted that it is not possible to distribute a constrained degree of freedom to other degrees of freedom which are themselves constrained. Only one level of indirection is allowed. It is not known at the time of this writing whether this is a constraint itself.
For the interface constraints, almost the same holds as for the 2D case. The numbering for the indices
on the mother face is obvious and keeps to the usual numbering of degrees of freedom on quadrilaterals.
The numbering of the degrees of freedom on the interior of the refined faces for the index
is as follows: let
and
be as above, and
be the number of degrees of freedom per quadrilateral (and therefore per face), then
denote the dofs on the vertex at the center,
for the dofs on the vertices at the center of the bounding lines of the quadrilateral,
are for the degrees of freedom on the four lines connecting the center vertex to the outer boundary of the mother face,
for the degrees of freedom on the small lines surrounding the quad, and
for the dofs on the four child faces. Note the direction of the lines at the boundary of the quads, as shown below.
The order of the twelve lines and the four child faces can be extracted from the following sketch, where the overall order of the different dof groups is depicted:
* *--15--4--16--* * | | | * 10 19 6 20 12 * | | | * 1--7---0--8---2 * | | | * 9 17 5 18 11 * | | | * *--13--3--14--* *
The three-dimensional case has a few pitfalls available for derived classes that want to implement constraint matrices. Consider the following case:
* *-------* * / /| * / / | * / / | * *-------* | * | | *-------* * | | / /| * | 1 | / / | * | |/ / | * *-------*-------* | * | | | * * | | | / * | 2 | 3 | / * | | |/ * *-------*-------* *
For the constraint matrix this means the following: if a degree of freedom on one edge of a face is constrained by some other nodes on the same edge with some weights, then the weights have to be exactly the same as those for constrained nodes on the three other edges with respect to the corresponding nodes on these edges. If this isn't the case, you will get into trouble with the ConstraintMatrix class that is the primary consumer of the constraint information: while that class is able to handle constraints that are entered more than once (as is necessary for the case above), it insists that the weights are exactly the same.
Construction of a finite element and computation of the matrices described above may be a tedious task, in particular if it has to be performed for several space dimensions. Therefore, some functions in FETools have been provided to help with these tasks.
First, aready the basis of the shape function space may be difficult to implement for arbitrary order and dimension. On the other hand, if the node values are given, then the duality relation between node functionals and basis functions defines the basis. As a result, the shape function space may be defined with arbitrary "raw" basis functions, such that the actual finite element basis is computed from linear combinations of them. The coefficients of these combinations are determined by the duality of node values.
Using this matrix allows the construction of the basis of shape functions in two steps.
The function computing the matrix M for you is FETools::compute_node_matrix(). It relies on the existence of generalized_support_points and implementation of interpolate() with VectorSlice argument.
The piece of code in the constructor of a finite element responsible for this looks like
FullMatrix<double> M(this->dofs_per_cell, this->dofs_per_cell); FETools::compute_node_matrix(M, *this); this->inverse_node_matrix.reinit(this->dofs_per_cell, this->dofs_per_cell); this->inverse_node_matrix.invert(M);
Once the shape functions are set up, the grid transfer matrices for Multigrid accessed by get_prolongation_matrix() can be computed automatically, using FETools::compute_embedding_matrices().
This can be achieved by
for (unsigned int i=0; i<GeometryInfo<dim>::children_per_cell; ++i) this->prolongation[i].reinit (this->dofs_per_cell, this->dofs_per_cell); FETools::compute_embedding_matrices (*this, this->prolongation);
missing...
Constraint matrices can be computed semi-automatically using FETools::compute_face_embedding_matrices(). This function computes the representation of the coarse mesh functions by fine mesh functions for each child of a face separately. These matrices must be convoluted into a single rectangular constraint matrix, eliminating degrees of freedom on common vertices and edges as well as on the coarse grid vertices. See the discussion above for details.
| FiniteElement< dim >::FiniteElement | ( | const FiniteElementData< dim > & | fe_data, | |
| const std::vector< bool > & | restriction_is_additive_flags, | |||
| const std::vector< std::vector< bool > > & | nonzero_components | |||
| ) |
Constructor
| virtual FiniteElement< dim >::~FiniteElement | ( | ) | [virtual] |
Virtual destructor. Makes sure that pointers to this class are deleted properly.
| virtual std::string FiniteElement< dim >::get_name | ( | ) | const [pure virtual] |
Return a string that uniquely identifies a finite element. The general convention is that this is the class name, followed by the space dimension in angle brackets, and the polynomial degree and whatever else is necessary in parentheses. For example, FE_Q<2>(3) is the value returned for a cubic element in 2d.
Systems of elements have their own naming convention, see the FESystem class.
Implemented in FE_ABF< dim >, FE_DGP< dim >, FE_DGPMonomial< dim >, FE_DGPNonparametric< dim >, FE_DGQ< dim >, FE_DGQArbitraryNodes< dim >, FE_Nedelec< dim >, FE_Q< dim >, FE_Q_Hierarchical< dim >, FE_RaviartThomas< dim >, FE_RaviartThomasNodal< dim >, and FESystem< dim >.
| const FiniteElement< dim > & FiniteElement< dim >::operator[] | ( | const unsigned int | fe_index | ) | const [inline] |
This operator returns a reference to the present object if the argument given equals to zero. While this does not seem particularly useful, it is helpful in writing code that works with both DoFHandler and the hp version hp::DoFHandler, since one can then write code like this:
* dofs_per_cell * = dof_handler->get_fe()[cell->active_fe_index()].dofs_per_cell; *
This code doesn't work in both situations without the present operator because DoFHandler::get_fe() returns a finite element, whereas hp::DoFHandler::get_fe() returns a collection of finite elements that doesn't offer a dofs_per_cell member variable: one first has to select which finite element to work on, which is done using the operator[]. Fortunately, cell->active_fe_index() also works for non-hp classes and simply returns zero in that case. The present operator[] accepts this zero argument, by returning the finite element with index zero within its collection (that, of course, consists only of the present finite element anyway).
References Assert.
| virtual double FiniteElement< dim >::shape_value | ( | const unsigned int | i, | |
| const Point< dim > & | p | |||
| ) | const [virtual] |
Return the value of the ith shape function at the point p. p is a point on the reference element. If the finite element is vector-valued, then return the value of the only non-zero component of the vector value of this shape function. If the shape function has more than one non-zero component (which we refer to with the term non-primitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_value_component() function.
An ExcUnitShapeValuesDoNotExist is thrown if the shape values of the FiniteElement under consideration depends on the shape of the cell in real space.
Reimplemented in FE_DGPNonparametric< dim >, FE_Poly< POLY, dim >, FE_PolyTensor< POLY, dim >, FESystem< dim >, FE_Poly< PolynomialSpace< dim >, dim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
| virtual double FiniteElement< dim >::shape_value_component | ( | const unsigned int | i, | |
| const Point< dim > & | p, | |||
| const unsigned int | component | |||
| ) | const [virtual] |
Just like for shape_value(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the value of the component-th vector component of the ith shape function at point p.
Reimplemented in FE_DGPNonparametric< dim >, FE_Nedelec< dim >, FE_Poly< POLY, dim >, FE_PolyTensor< POLY, dim >, FESystem< dim >, FE_Poly< PolynomialSpace< dim >, dim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
| virtual Tensor<1,dim> FiniteElement< dim >::shape_grad | ( | const unsigned int | i, | |
| const Point< dim > & | p | |||
| ) | const [virtual] |
Return the gradient of the ith shape function at the point p. p is a point on the reference element, and likewise the gradient is the gradient on the unit cell with respect to unit cell coordinates. If the finite element is vector-valued, then return the value of the only non-zero component of the vector value of this shape function. If the shape function has more than one non-zero component (which we refer to with the term non-primitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_grad_component() function.
An ExcUnitShapeValuesDoNotExist is thrown if the shape values of the FiniteElement under consideration depends on the shape of the cell in real space.
Reimplemented in FE_DGPNonparametric< dim >, FE_Poly< POLY, dim >, FE_PolyTensor< POLY, dim >, FESystem< dim >, FE_Poly< PolynomialSpace< dim >, dim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
| virtual Tensor<1,dim> FiniteElement< dim >::shape_grad_component | ( | const unsigned int | i, | |
| const Point< dim > & | p, | |||
| const unsigned int | component | |||
| ) | const [virtual] |
Just like for shape_grad(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the gradient of the component-th vector component of the ith shape function at point p.
Reimplemented in FE_DGPNonparametric< dim >, FE_Nedelec< dim >, FE_Poly< POLY, dim >, FE_PolyTensor< POLY, dim >, FESystem< dim >, FE_Poly< PolynomialSpace< dim >, dim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
| virtual Tensor<2,dim> FiniteElement< dim >::shape_grad_grad | ( | const unsigned int | i, | |
| const Point< dim > & | p | |||
| ) | const [virtual] |
Return the tensor of second derivatives of the ith shape function at point p on the unit cell. The derivatives are derivatives on the unit cell with respect to unit cell coordinates. If the finite element is vector-valued, then return the value of the only non-zero component of the vector value of this shape function. If the shape function has more than one non-zero component (which we refer to with the term non-primitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_grad_grad_component() function.
An ExcUnitShapeValuesDoNotExist is thrown if the shape values of the FiniteElement under consideration depends on the shape of the cell in real space.
Reimplemented in FE_DGPNonparametric< dim >, FE_Poly< POLY, dim >, FE_PolyTensor< POLY, dim >, FESystem< dim >, FE_Poly< PolynomialSpace< dim >, dim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
| virtual Tensor<2,dim> FiniteElement< dim >::shape_grad_grad_component | ( | const unsigned int | i, | |
| const Point< dim > & | p, | |||
| const unsigned int | component | |||
| ) | const [virtual] |
Just like for shape_grad_grad(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the gradient of the component-th vector component of the ith shape function at point p.
Reimplemented in FE_DGPNonparametric< dim >, FE_Nedelec< dim >, FE_Poly< POLY, dim >, FE_PolyTensor< POLY, dim >, FESystem< dim >, FE_Poly< PolynomialSpace< dim >, dim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
| virtual bool FiniteElement< dim >::has_support_on_face | ( | const unsigned int | shape_index, | |
| const unsigned int | face_index | |||
| ) | const [virtual] |
Check for non-zero values on a face in order to optimize out matrix elements.
This function returns true, if the shape function shape_index has non-zero values on the face face_index.
A default implementation is provided in this basis class which always returns true. This is the safe way to go.
Reimplemented in FE_ABF< dim >, FE_DGP< dim >, FE_DGPMonomial< dim >, FE_DGPNonparametric< dim >, FE_DGQ< dim >, FE_Nedelec< dim >, FE_Q< dim >, FE_Q_Hierarchical< dim >, FE_RaviartThomas< dim >, FESystem< dim >, and FE_Q_Hierarchical< dim >.
| const FullMatrix<double>& FiniteElement< dim >::get_restriction_matrix | ( | const unsigned int | child, | |
| const RefinementCase< dim > & | refinement_case = RefinementCase< dim >::isotropic_refinement | |||
| ) | const |
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.
If projection matrices are not implemented in the derived finite element class, this function aborts with ExcProjectionVoid. You can check whether this is the case by calling the restriction_is_implemented() or the isotropic_restriction_is_implemented() function.
| const FullMatrix<double>& FiniteElement< dim >::get_prolongation_matrix | ( | const unsigned int | child, | |
| const RefinementCase< dim > & | refinement_case = RefinementCase< dim >::isotropic_refinement | |||
| ) | const |
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 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.
If projection matrices are not implemented in the derived finite element class, this function aborts with ExcEmbeddingVoid. You can check whether this is the case by calling the prolongation_is_implemented() or the isotropic_prolongation_is_implemented() function.
| bool FiniteElement< dim >::prolongation_is_implemented | ( | ) | const |
Return whether this element implements its prolongation matrices. The return value also indicates whether a call to the get_prolongation_matrix() function will generate an error or not.
Note, that this function returns true only if the prolongation matrices of the isotropic and all anisotropic refinement cases are implemented. If you are interested in the prolongation matrices for isotropic refinement only, use the isotropic_prolongation_is_implemented function instead.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_prolongation_matrix() will succeed; however, one then still needs to cope with the lack of information this just expresses.
| bool FiniteElement< dim >::isotropic_prolongation_is_implemented | ( | ) | const |
Return whether this element implements its prolongation matrices for isotropic children. The return value also indicates whether a call to the get_prolongation_matrix function will generate an error or not.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_prolongation_matrix() will succeed; however, one then still needs to cope with the lack of information this just expresses.
| bool FiniteElement< dim >::restriction_is_implemented | ( | ) | const |
Return whether this element implements its restriction matrices. The return value also indicates whether a call to the get_restriction_matrix() function will generate an error or not.
Note, that this function returns true only if the restriction matrices of the isotropic and all anisotropic refinement cases are implemented. If you are interested in the restriction matrices for isotropic refinement only, use the isotropic_restriction_is_implemented function instead.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_restriction_matrix() will succeed; however, one then still needs to cope with the lack of information this just expresses.
| bool FiniteElement< dim >::isotropic_restriction_is_implemented | ( | ) | const |
Return whether this element implements its restriction matrices for isotropic children. The return value also indicates whether a call to the get_restriction_matrix function will generate an error or not.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_restriction_matrix() will succeed; however, one then still needs to cope with the lack of information this just expresses.
| bool FiniteElement< dim >::restriction_is_additive | ( | const unsigned int | index | ) | const [inline] |
Access the restriction_is_additive_flags field. See there for more information on its contents.
The index must be between zero and the number of shape functions of this element.
References Assert, FiniteElementData< dim >::dofs_per_cell, and FiniteElement< dim >::restriction_is_additive_flags.
| const FullMatrix<double>& FiniteElement< dim >::constraints | ( | const internal::SubfaceCase< dim > & | subface_case = internal::SubfaceCase< dim >::case_isotropic |
) | const |
Return a readonly reference to the matrix which describes the constraints at the interface between a refined and an unrefined cell.
The matrix is obviously empty in only one space dimension, since there are no constraints then.
Note that some finite elements do not (yet) implement hanging node constraints. If this is the case, then this function will generate an exception, since no useful return value can be generated. If you should have a way to live with this, then you might want to use the constraints_are_implemented() function to check up front whethehr this function will succeed or generate the exception.
| bool FiniteElement< dim >::constraints_are_implemented | ( | const internal::SubfaceCase< dim > & | subface_case = internal::SubfaceCase< dim >::case_isotropic |
) | const |
Return whether this element implements its hanging node constraints. The return value also indicates whether a call to the constraints() function will generate an error or not.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case hanging node constraints are not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs hanging node constraints and they are not implemented. This function could be used to check whether a call to constraints() will succeed; however, one then still needs to cope with the lack of information this just expresses.
| virtual bool FiniteElement< dim >::hp_constraints_are_implemented | ( | ) | const [virtual] |
Return whether this element implements its hanging node constraints in the new way, which has to be used to make elements "hp compatible". That means, the element properly implements the get_face_interpolation_matrix and get_subface_interpolation_matrix methods. Therefore the return value also indicates whether a call to the get_face_interpolation_matrix() method and the get_subface_interpolation_matrix() method will generate an error or not.
Currently the main purpose of this function is to allow the make_hanging_node_constraints method to decide whether the new procedures, which are supposed to work in the hp framework can be used, or if the old well verified but not hp capable functions should be used. Once the transition to the new scheme for computing the interface constraints is complete, this function will be superfluous and will probably go away.
Derived classes should implement this function accordingly. The default assumption is that a finite element does not provide hp capable face interpolation, and the default implementation therefore returns false.
Reimplemented in FE_DGP< dim >, FE_DGPMonomial< dim >, FE_DGPNonparametric< dim >, FE_DGQ< dim >, FE_Q< dim >, FE_RaviartThomasNodal< dim >, and FESystem< dim >.
| virtual void FiniteElement< dim >::get_interpolation_matrix | ( | const FiniteElement< dim > & | source, | |
| FullMatrix< double > & | matrix | |||
| ) | const [virtual] |
Return the matrix interpolating from the given finite element to the present one. The size of the matrix is then dofs_per_cell times source.dofs_per_cell.
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.
Reimplemented in FE_DGPMonomial< dim >, FE_DGQ< dim >, FE_Q< dim >, and FESystem< dim >.
| virtual void FiniteElement< dim >::get_face_interpolation_matrix | ( | const FiniteElement< dim > & | source, | |
| FullMatrix< double > & | matrix | |||
| ) | const [virtual] |
Return the matrix interpolating from a face of 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 ExcInterpolationNotImplemented.
Reimplemented in FE_DGP< dim >, FE_DGPMonomial< dim >, FE_DGPNonparametric< dim >, FE_DGQ< dim >, FE_Q< dim >, FE_RaviartThomasNodal< dim >, and