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// $Id: fe.h 20602 2010-02-13 17:44:17Z bangerth $
// Version: $Name$
//
// Copyright (C) 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
// to the file deal.II/doc/license.html for the text and
// further information on this license.
//
//---------------------------------------------------------------------------
#ifndef __deal2__fe_h
#define __deal2__fe_h
#include <base/config.h>
#include <base/geometry_info.h>
#include <fe/fe_base.h>
DEAL_II_NAMESPACE_OPEN
template <int dim, int spacedim> class FEValuesData;
template <int dim, int spacedim> class FEValuesBase;
template <int dim, int spacedim> class FEValues;
template <int dim, int spacedim> class FEFaceValues;
template <int dim, int spacedim> class FESubfaceValues;
template <int dim, int spacedim> class FESystem;
template <class POLY, int dim, int spacedim> class FE_PolyTensor;
namespace hp
{
template <int dim, int spacedim> class FECollection;
}
/**
* Base class for finite elements in arbitrary dimensions. This class
* provides several fields which describe a specific finite element
* and which are filled by derived classes. It more or less only
* offers the fields and access functions which makes it possible to
* copy finite elements without knowledge of the actual type (linear,
* quadratic, etc). In particular, the functions to fill the data
* fields of FEValues and its derived classes are declared.
*
* 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 $[0,1]^d$. 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.
*
* The <tt>spacedim</tt> parameter has to be used if one wants to
* solve problems in the boundary element method formulation or in an
* equivalent one, as it is explained in the Triangulation class. If
* not specified, this parameter takes the default value <tt>=dim</tt>
* so that this class can be used to solve problems in the finite
* element method formulation.
*
* <h3>Components and blocks</h3>
*
* For vector valued elements shape functions may have nonzero entries
* in one or several @ref GlossComponent "components" of the vector
* valued function. If the element is @ref GlossPrimitive "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 @ref
* GlossBlock "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.
*
* <h3>Support points</h3>
*
* Since a FiniteElement does not have information on the actual grid
* cell, it can only provide @ref GlossSupport "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.
*
* @code
* 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);
* @endcode
*
* Alternatively, the points can be transformed one-by-one:
* @code
* 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]);
* @endcode
* This is a shortcut, and as all shortcuts should be used cautiously.
* If the mapping of all support points is needed, the first variant should
* be preferred for efficiency.
*
* <h3>Notes on the implementation of derived classes</h3>
*
* The following sections list the information to be provided by
* derived classes, depending on the dimension. They are
* followed by a list of functions helping to generate these values.
*
* <h4>Finite elements in one dimension</h4>
*
* 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.
*
* <h4>Finite elements in two dimensions</h4>
*
* 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 $m\times
* n$-matrix #interface_constraints, where <i>m</i> 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
* <i>n</i> 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 $m\times n$ 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 $d_v$ be the number of dofs on a
* vertex, $d_l$ that on a line, then $n=0...d_v-1$ refers to the dofs
* on vertex zero of the unrefined line, $n=d_v...2d_v-1$ to those on
* vertex one, $n=2d_v...2d_v+d_l-1$ to those on the line.
*
* Similarly, $m=0...d_v-1$ refers to the dofs on the middle vertex of
* the refined side (vertex one of child line zero, vertex zero of
* child line one), $m=d_v...d_v+d_l-1$ refers to the dofs on child
* line zero, $m=d_v+d_l...d_v+2d_l-1$ 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.
*
*
* <h4>Finite elements in three dimensions</h4>
*
* For the interface constraints, almost the same holds as for the 2D case.
* The numbering for the indices $n$ 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 $m$ is as follows: let $d_v$ and $d_l$ be as above,
* and $d_q$ be the number of degrees of freedom per quadrilateral (and
* therefore per face), then $m=0...d_v-1$ denote the dofs on the vertex at
* the center, $m=d_v...5d_v-1$ for the dofs on the vertices at the center
* of the bounding lines of the quadrilateral,
* $m=5d_v..5d_v+4*d_l-1$ are for the degrees of freedom on
* the four lines connecting the center vertex to the outer boundary of the
* mother face, $m=5d_v+4*d_l...5d_v+4*d_l+8*d_l-1$ for the degrees of freedom
* on the small lines surrounding the quad,
* and $m=5d_v+12*d_l...5d_v+12*d_l+4*d_q-1$ 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:
* @verbatim
* *--15--4--16--*
* | | |
* 10 19 6 20 12
* | | |
* 1--7---0--8---2
* | | |
* 9 17 5 18 11
* | | |
* *--13--3--14--*
* @endverbatim
* The numbering of vertices and lines, as well as the numbering of
* children within a line is consistent with the one described in
* Triangulation. Therefore, this numbering is seen from the
* outside and inside, respectively, depending on the face.
*
* The three-dimensional case has a few pitfalls available for derived classes
* that want to implement constraint matrices. Consider the following case:
* @verbatim
* *-------*
* / /|
* / / |
* / / |
* *-------* |
* | | *-------*
* | | / /|
* | 1 | / / |
* | |/ / |
* *-------*-------* |
* | | | *
* | | | /
* | 2 | 3 | /
* | | |/
* *-------*-------*
* @endverbatim
* Now assume that we want to refine cell 2. We will end up with two faces
* with hanging nodes, namely the faces between cells 1 and 2, as well as
* between cells 2 and 3. Constraints have to be applied to the degrees of
* freedom on both these faces. The problem is that there is now an edge
* (the top right one of cell 2) which is part of both faces. The hanging
* node(s) on this edge are therefore constrained twice, once from both
* faces. To be meaningful, these constraints of course have to be
* consistent: both faces have to constrain the hanging nodes on the edge to
* the same nodes on the coarse edge (and only on the edge, as there can
* then be no constraints to nodes on the rest of the face), and they have
* to do so with the same weights. This is sometimes tricky since the nodes
* on the edge may have different local numbers.
*
* 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.
*
* <h4>Helper functions</h4>
*
* 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 dimensions. Therefore, some
* functions in FETools have been provided to help with these tasks.
*
* <h5>Computing the correct basis from "raw" basis functions</h5>
*
* 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 @ref GlossNodes "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.
* <ol>
*
* <li>Define the space of shape functions using an arbitrary basis
* <i>w<sub>j</sub></i> and compute the matrix <i>M</i> of node
* functionals <i>N<sub>i</sub></i> applied to these basis functions,
* such that its entries are <i>m<sub>ij</sub> =
* N<sub>i</sub>(w<sub>j</sub>)</i>.
*
* <li>Compute the basis <i>v<sub>j</sub></i> of the finite element
* shape function space by applying <i>M<sup>-1</sup></i> to the basis
* <i>w<sub>j</sub></i>.
* </ol>
*
* The function computing the matrix <i>M</i> 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
* @code
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);
* @endcode
* Don't forget to make sure that #unit_support_points or
* #generalized_support_points are initialized before this!
*
* <h5>Computing the #prolongation matrices for multigrid</h5>
*
* 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
* @code
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);
* @endcode
*
* <h5>Computing the #restriction matrices for error estimators</h5>
*
* missing...
*
* <h5>Computing #interface_constraints</h5>
*
* 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.
*
* @ingroup febase fe
*
* @author Wolfgang Bangerth, Guido Kanschat, Ralf Hartmann, 1998, 2000, 2001, 2005
*/
template <int dim, int spacedim=dim>
class FiniteElement : public Subscriptor,
public FiniteElementData<dim>
{
public:
/**
* Base class for internal data.
* Adds data for second derivatives to
* Mapping::InternalDataBase()
*
* For information about the
* general purpose of this class,
* see the documentation of the
* base class.
*
* @author Guido Kanschat, 2001
*/
class InternalDataBase : public Mapping<dim,spacedim>::InternalDataBase
{
public:
/**
* Destructor. Needed to
* avoid memory leaks with
* difference quotients.
*/
virtual ~InternalDataBase ();
/**
* Initialize some pointers
* used in the computation of
* second derivatives by
* finite differencing of
* gradients.
*/
void initialize_2nd (const FiniteElement<dim,spacedim> *element,
const Mapping<dim,spacedim> &mapping,
const Quadrature<dim> &quadrature);
/**
* Storage for FEValues
* objects needed to
* approximate second
* derivatives.
*
* The ordering is <i>p+hx</i>,
* <i>p+hy</i>, <i>p+hz</i>,
* <i>p-hx</i>, <i>p-hy</i>,
* <i>p-hz</i>, where unused
* entries in lower dimensions
* are missing.
*/
std::vector<FEValues<dim,spacedim>*> differences;
};
public:
/**
* Constructor
*/
FiniteElement (const FiniteElementData<dim> &fe_data,
const std::vector<bool> &restriction_is_additive_flags,
const std::vector<std::vector<bool> > &nonzero_components);
/**
* Virtual destructor. Makes sure
* that pointers to this class
* are deleted properly.
*/
virtual ~FiniteElement ();
/**
* Return a string that uniquely
* identifies a finite
* element. The general
* convention is that this is the
* class name, followed by the
* dimension in angle
* brackets, and the polynomial
* degree and whatever else is
* necessary in parentheses. For
* example, <tt>FE_Q<2>(3)</tt> is the
* value returned for a cubic
* element in 2d.
*
* Systems of elements have their
* own naming convention, see the
* FESystem class.
*/
virtual std::string get_name () const = 0;
/**
* 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:
* @verbatim
* dofs_per_cell
* = dof_handler->get_fe()[cell->active_fe_index()].dofs_per_cell;
* @endverbatim
*
* 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
* <code>dofs_per_cell</code>
* member variable: one first has
* to select which finite element
* to work on, which is done
* using the
* operator[]. Fortunately,
* <code>cell-@>active_fe_index()</code>
* 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).
*/
const FiniteElement<dim,spacedim> & operator[] (const unsigned int fe_index) const;
/**
* @name Shape function access
* @{
*/
/**
* Return the value of the
* @p ith shape function at the
* point @p p. @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.
*/
virtual double shape_value (const unsigned int i,
const Point<dim> &p) const;
/**
* 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
* @p component-th vector
* component of the @p ith shape
* function at point @p p.
*/
virtual double shape_value_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the gradient of the
* @p ith shape function at the
* point @p p. @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.
*/
virtual Tensor<1,dim> shape_grad (const unsigned int i,
const Point<dim> &p) const;
/**
* 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
* @p component-th vector
* component of the @p ith shape
* function at point @p p.
*/
virtual Tensor<1,dim> shape_grad_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the tensor of second
* derivatives of the @p ith
* shape function at point @p 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.
*/
virtual Tensor<2,dim> shape_grad_grad (const unsigned int i,
const Point<dim> &p) const;
/**
* 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
* @p component-th vector
* component of the @p ith shape
* function at point @p p.
*/
virtual Tensor<2,dim> shape_grad_grad_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Check for non-zero values on a
* face in order to optimize out
* matrix elements.
*
* This function returns
* @p true, if the shape
* function @p shape_index has
* non-zero values on the face
* @p face_index.
*
* A default implementation is
* provided in this basis class
* which always returns @p
* true. This is the safe way to
* go.
*/
virtual bool has_support_on_face (const unsigned int shape_index,
const unsigned int face_index) const;
//@}
/**
* @name Transfer and constraint matrices
* @{
*/
/**
* Projection from a fine grid
* space onto a coarse grid
* space. If this projection
* operator is associated with a
* matrix @p P, then the
* restriction of this matrix
* @p P_i to a single child cell
* is returned here.
*
* The matrix @p P is the
* concatenation or the sum of
* the cell matrices @p 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> &
get_restriction_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 P. The
* restriction of this matrix @p P_i to
* a single child cell is
* returned here.
*
* The matrix @p P is the
* concatenation, not the sum of
* the cell matrices
* @p P_i. That is, if the same
* non-zero entry <tt>j,k</tt> exists
* in in two different child
* matrices @p P_i, the value
* should be the same in both
* matrices and it is copied into
* the matrix @p 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.
*/
const FullMatrix<double> &
get_prolongation_matrix (const unsigned int child,
const RefinementCase<dim> &refinement_case=RefinementCase<dim>::isotropic_refinement) 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 <code>true</code> 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
* <tt>get_prolongation_matrix()</tt> will
* succeed; however, one then still needs
* to cope with the lack of information
* this just expresses.
*/
bool 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 @p
* 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
* <tt>get_prolongation_matrix()</tt> will
* succeed; however, one then still needs
* to cope with the lack of information
* this just expresses.
*/
bool isotropic_prolongation_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 <code>true</code> 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
* <tt>get_restriction_matrix()</tt> will
* succeed; however, one then still needs
* to cope with the lack of information
* this just expresses.
*/
bool 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 @p
* 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
* <tt>get_restriction_matrix()</tt> will
* succeed; however, one then still needs
* to cope with the lack of information
* this just expresses.
*/
bool isotropic_restriction_is_implemented () const;
/**
* 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.
*/
bool restriction_is_additive (const unsigned int index) 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 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.
*/
const FullMatrix<double> & constraints (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 <tt>constraints()</tt>
* will succeed; however, one
* then still needs to cope with
* the lack of information this
* just expresses.
*/
bool constraints_are_implemented (const internal::SubfaceCase<dim> &subface_case=internal::SubfaceCase<dim>::case_isotropic) const;
/**
* 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 @p
* false.
*/
virtual bool hp_constraints_are_implemented () const;
/**
* Return the matrix
* interpolating from the given
* finite element to the present
* one. The size of the matrix is
* then #dofs_per_cell times
* <tt>source.#dofs_per_cell</tt>.
*
* 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.
*/
virtual void
get_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
FullMatrix<double> &matrix) const;
//@}
/**
* @name Functions to support hp
* @{
*/
/**
* 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 <tt>source.#dofs_per_face</tt> times
* <tt>this->#dofs_per_face</tt>.
*
* 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.
*/
virtual void
get_face_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
FullMatrix<double> &matrix) const;
/**
* Return the matrix
* interpolating from a face of
* of one element to the subface of
* the neighboring element.
* The size of the matrix is
* then <tt>source.#dofs_per_face</tt> times
* <tt>this->#dofs_per_face</tt>.
*
* 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.
*/
virtual void
get_subface_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
const unsigned int subface,
FullMatrix<double> &matrix) const;
//@}
/**
* 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 @p 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, and returns a list
* of pairs of global dof indices
* in @p identities. 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.
*/
virtual
std::vector<std::pair<unsigned int, unsigned int> >
hp_vertex_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
/**
* Same as
* hp_vertex_dof_indices(),
* except that the function
* treats degrees of freedom on
* lines.
*/
virtual
std::vector<std::pair<unsigned int, unsigned int> >
hp_line_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
/**
* Same as
* hp_vertex_dof_indices(),
* except that the function
* treats degrees of freedom on
* quads.
*/
virtual
std::vector<std::pair<unsigned int, unsigned int> >
hp_quad_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
/**
* Return whether this element dominates
* the one given as argument when they
* meet at a common face,
* whether it is the other way around,
* whether neither dominates, or if
* either could dominate.
*
* For a definition of domination, see
* FiniteElementBase::Domination and in
* particular the @ref hp_paper "hp paper".
*/
virtual
FiniteElementDomination::Domination
compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const;
//@}
/**
* Comparison operator. We also
* check for equality of the
* constraint matrix, which is
* quite an expensive operation.
* Do therefore use this function
* with care, if possible only
* for debugging purposes.
*
* Since this function is not
* that important, we avoid an
* implementational question
* about comparing arrays and do
* not compare the matrix arrays
* #restriction and
* #prolongation.
*/
bool operator == (const FiniteElement<dim,spacedim> &) const;
/**
* @name Index computations
* @{
*/
/**
* Compute vector component and
* index of this shape function
* within the shape functions
* corresponding to this
* component from the index of a
* shape function within this
* finite element.
*
* If the element is scalar, then
* the component is always zero,
* and the index within this
* component is equal to the
* overall index.
*
* If the shape function
* referenced has more than one
* non-zero component, then it
* cannot be associated with one
* vector component, and an
* exception of type
* ExcShapeFunctionNotPrimitive
* will be raised.
*
* Note that if the element is
* composed of other (base)
* elements, and a base element
* has more than one component
* but all its shape functions
* are primitive (i.e. are
* non-zero in only one
* component), then this mapping
* contains valid
* information. However, the
* index of a shape function of
* this element within one
* component (i.e. the second
* number of the respective entry
* of this array) does not
* indicate the index of the
* respective shape function
* within the base element (since
* that has more than one
* vector-component). For this
* information, refer to the
* #system_to_base_table field
* and the
* system_to_base_index()
* function.
*
* The use of this function is
* explained extensively in the
* step-8 and @ref
* step_20 "step-20" tutorial
* programs as well as in the
* @ref vector_valued module.
*/
std::pair<unsigned int, unsigned int>
system_to_component_index (const unsigned int index) const;
/**
* Compute the shape function for
* the given vector component and
* index.
*
* If the element is scalar, then
* the component must be zero,
* and the index within this
* component is equal to the
* overall index.
*
* This is the opposite operation
* from the system_to_component_index()
* function.
*/
unsigned int component_to_system_index(const unsigned int component,
const unsigned int index) const;
/**
* Same as
* system_to_component_index(),
* but do it for shape functions
* and their indices on a
* face. The range of allowed
* indices is therefore
* 0..#dofs_per_face.
*
* You will rarely need this
* function in application
* programs, since almost all
* application codes only need to
* deal with cell indices, not
* face indices. The function is
* mainly there for use inside
* the library.
*/
std::pair<unsigned int, unsigned int>
face_system_to_component_index (const unsigned int index) const;
/**
* For faces with non-standard
* face_orientation in 3D, the dofs on
* faces (quads) have to be permuted in
* order to be combined with the correct
* shape functions. Given a local dof @p
* index on a quad, return the local index,
* if the face has non-standard
* face_orientation, face_flip or
* face_rotation. In 2D and 1D there is no
* need for permutation and consequently
* an exception is thrown.
*/
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;
/**
* For lines with non-standard
* line_orientation in 3D, the dofs on
* lines have to be permuted in order to be
* combined with the correct shape
* functions. Given a local dof @p index on
* a line, return the local index, if the
* line has non-standard
* line_orientation. In 2D and 1D there is
* no need for permutation, so the given
* index is simply returned.
*/
unsigned int adjust_line_dof_index_for_line_orientation (const unsigned int index,
const bool line_orientation) const;
/**
* Return in which of the vector
* components of this finite
* element the @p ith shape
* function is non-zero. The
* length of the returned array
* is equal to the number of
* vector components of this
* element.
*
* For most finite element
* spaces, the result of this
* function will be a vector with
* exactly one element being
* @p true, since for most
* spaces the individual vector
* components are independent. In
* that case, the component with
* the single zero is also the
* first element of what
* system_to_component_index()
* returns.
*
* Only for those
* spaces that couple the
* components, for example to
* make a shape function
* divergence free, will there be
* more than one @p true entry.
*/
const std::vector<bool> &
get_nonzero_components (const unsigned int i) const;
/**
* Return in how many vector
* components the @p ith shape
* function is non-zero. This
* value equals the number of
* entries equal to @p true in
* the result of the
* get_nonzero_components()
* function.
*
* For most finite element
* spaces, the result will be
* equal to one. It is not equal
* to one only for those ansatz
* spaces for which vector-valued
* shape functions couple the
* individual components, for
* example in order to make them
* divergence-free.
*/
unsigned int
n_nonzero_components (const unsigned int i) const;
/**
* Return whether the @p ith
* shape function is primitive in
* the sense that the shape
* function is non-zero in only
* one vector
* component. Non-primitive shape
* functions would then, for
* example, be those of
* divergence free ansatz spaces,
* in which the individual vector
* components are coupled.
*
* The result of the function is
* @p true if and only if the
* result of
* <tt>n_nonzero_components(i)</tt> is
* equal to one.
*/
bool
is_primitive (const unsigned int i) const;
/**
* Return whether the entire
* finite element is primitive,
* in the sense that all its
* shape functions are
* primitive. If the finite
* element is scalar, then this
* is always the case.
*
* Since this is an extremely
* common operation, the result
* is cached in the
* #cached_primitivity
* variable which is computed in
* the constructor.
*/
bool
is_primitive () const;
/**
* Number of base elements in a
* mixed discretization.
*
* Note that even for vector
* valued finite elements, the
* number of components needs not
* coincide with the number of
* base elements, since they may
* be reused. For example, if you
* create a FESystem with
* three identical finite element
* classes by using the
* constructor that takes one
* finite element and a
* multiplicity, then the number
* of base elements is still one,
* although the number of
* components of the finite
* element is equal to the
* multiplicity.
*/
virtual unsigned int n_base_elements () const = 0;
/**
* Access to base element
* objects. If the element is
* scalar, then
* <code>base_element(0)</code> is
* @p this.
*/
virtual
const FiniteElement<dim,spacedim> &
base_element (const unsigned int index) const = 0;
/**
* This index denotes how often
* the base element @p index is
* used in a composed element. If
* the element is scalar, then
* the result is always equal to
* one. See the documentation for
* the n_base_elements()
* function for more details.
*/
virtual
unsigned int
element_multiplicity (const unsigned int index) const = 0;
/**
* Return for shape function
* @p index the base element it
* belongs to, the number of the
* copy of this base element
* (which is between zero and the
* multiplicity of this element),
* and the index of this shape
* function within this base
* element.
*
* If the element is not composed of
* others, then base and instance
* are always zero, and the index
* is equal to the number of the
* shape function. If the element
* is composed of single
* instances of other elements
* (i.e. all with multiplicity
* one) all of which are scalar,
* then base values and dof
* indices within this element
* are equal to the
* #system_to_component_table. It
* differs only in case the
* element is composed of other
* elements and at least one of
* them is vector-valued itself.
*
* This function returns valid
* values also in the case of
* vector-valued
* (i.e. non-primitive) shape
* functions, in contrast to the
* system_to_component_index()
* function.
*/
std::pair<std::pair<unsigned int, unsigned int>, unsigned int>
system_to_base_index (const unsigned int index) const;
/**
* Same as
* system_to_base_index(), but
* for degrees of freedom located
* on a face. The range of allowed
* indices is therefore
* 0..#dofs_per_face.
*
* You will rarely need this
* function in application
* programs, since almost all
* application codes only need to
* deal with cell indices, not
* face indices. The function is
* mainly there for use inside
* the library.
*/
std::pair<std::pair<unsigned int, unsigned int>, unsigned int>
face_system_to_base_index (const unsigned int index) const;
/**
* Given a base element number,
* return the first block of a
* BlockVector it would generate.
*/
unsigned int first_block_of_base (const unsigned int b) const;
/**
* For each vector component,
* return which base
* element implements this
* component and which vector
* component in this base element
* this is. This information is
* only of interest for
* vector-valued finite elements
* which are composed of several
* sub-elements. In that case,
* one may want to obtain
* information about the element
* implementing a certain vector
* component, which can be done
* using this function and the
* FESystem::base_element()
* function.
*
* If this is a scalar finite
* element, then the return value
* is always equal to a pair of
* zeros.
*/
std::pair<unsigned int, unsigned int>
component_to_base_index (const unsigned int component) const;
/**
* Return the base element for
* this block and the number of
* the copy of the base element.
*/
std::pair<unsigned int,unsigned int>
block_to_base_index (const unsigned int block) const;
/**
* The vector block and the index
* inside the block for this
* shape function.
*/
std::pair<unsigned int,unsigned int>
system_to_block_index (const unsigned int component) const;
/**
* The vector block for this
* component.
*/
unsigned int
component_to_block_index (const unsigned int component) const;
//@}
/**
* @name Support points and interpolation
* @{
*/
/**
* Return the support points of
* the trial functions on the
* unit cell, if the derived
* finite element defines some.
* Finite elements that allow
* some kind of interpolation
* operation usually have support
* points. On the other hand,
* elements that define their
* degrees of freedom by, for
* example, moments on faces, or
* as derivatives, don't have
* support points. In that case,
* the returned field is empty.
*
* If the finite element defines
* support points, then their
* number equals the number of
* degrees of freedom of the
* element. The order of points
* in the array matches that
* returned by the
* <tt>cell->get_dof_indices</tt>
* function.
*
* See the class documentation
* for details on support points.
*/
const std::vector<Point<dim> > &
get_unit_support_points () const;
/**
* Return whether a finite
* element has defined support
* points. If the result is true,
* then a call to the
* get_unit_support_points()
* yields a non-empty array.
*
* The result may be false if an
* element is not defined by
* interpolating shape functions,
* for example by P-elements on
* quadrilaterals. It will
* usually only be true if the
* element constructs its shape
* functions by the requirement
* that they be one at a certain
* point and zero at all the
* points associated with the
* other shape functions.
*
* In composed elements (i.e. for
* the FESystem class, the
* result will be true if all all
* the base elements have defined
* support points.
*/
bool has_support_points () const;
/**
* Return the position of the
* support point of the
* @p indexth shape function. If
* it does not exist, raise an
* exception.
*
* The default implementation
* simply returns the respective
* element from the array you get
* from
* get_unit_support_points(),
* but derived elements may
* overload this function. In
* particular, note that the
* FESystem class overloads
* it so that it can return the
* support points of individual
* base elements, of not all the
* base elements define support
* points. In this way, you can
* still ask for certain support
* points, even if
* get_unit_support_points()
* only returns an empty array.
*/
virtual
Point<dim>
unit_support_point (const unsigned int index) const;
/**
* Return the support points of
* the trial functions on the
* unit face, if the derived
* finite element defines some.
* Finite elements that allow
* some kind of interpolation
* operation usually have support
* points. On the other hand,
* elements that define their
* degrees of freedom by, for
* example, moments on faces, or
* as derivatives, don't have
* support points. In that case,
* the returned field is empty
*
* Note that elements that have
* support points need not
* necessarily have some on the
* faces, even if the
* interpolation points are
* located physically on a
* face. For example, the
* discontinuous elements have
* interpolation points on the
* vertices, and for higher
* degree elements also on the
* faces, but they are not
* defined to be on faces since
* in that case degrees of
* freedom from both sides of a
* face (or from all adjacent
* elements to a vertex) would be
* identified with each other,
* which is not what we would
* like to have). Logically,
* these degrees of freedom are
* therefore defined to belong to
* the cell, rather than the face
* or vertex. In that case, the
* returned element would
* therefore have length zero.
*
* If the finite element defines
* support points, then their
* number equals the number of
* degrees of freedom on the face
* (#dofs_per_face). The order
* of points in the array matches
* that returned by the
* <tt>cell->get_dof_indices</tt>
* function.
*
* See the class documentation
* for details on support points.
*/
const std::vector<Point<dim-1> > &
get_unit_face_support_points () const;
/**
* Return whether a finite
* element has defined support
* points on faces. If the result
* is true, then a call to the
* get_unit_face_support_points()
* yields a non-empty array.
*
* For more information, see the
* documentation for the
* has_support_points()
* function.
*/
bool has_face_support_points () const;
/**
* The function corresponding to
* the unit_support_point()
* function, but for faces. See
* there for more information.
*/
virtual
Point<dim-1>
unit_face_support_point (const unsigned int index) const;
/**
* Return a support point vector
* for generalized interpolation.
*/
const std::vector<Point<dim> > &
get_generalized_support_points () const;
/**
* Returns <tt>true</tt> if the
* class provides nonempty
* vectors either from
* get_unit_support_points() or
* get_generalized_support_points().
*/
bool has_generalized_support_points () const;
/**
*
*/
const std::vector<Point<dim-1> > &
get_generalized_face_support_points () const;
/**
* Return whether a finite
* element has defined
* generalized support
* points on faces. If the result
* is true, then a call to the
* get_generalized_face_support_points
* yields a non-empty array.
*
* For more information, see the
* documentation for the
* has_support_points()
* function.
*/
bool has_generalized_face_support_points () const;
/**
* Interpolate a set of scalar
* values, computed in the
* generalized support points.
*
* @note This function is
* implemented in
* FiniteElement for the case
* that the element has support
* points. In this case, the
* resulting coefficients are
* just the values in the suport
* points. All other elements
* must reimplement it.
*/
virtual void interpolate(std::vector<double>& local_dofs,
const std::vector<double>& values) const;
/**
* Interpolate a set of vector
* values, computed in the
* generalized support points.
*
* Since a finite element often
* only interpolates part of a
* vector, <tt>offset</tt> is
* used to determine the first
* component of the vector to be
* interpolated. Maybe consider
* changing your data structures
* to use the next function.
*/
virtual void interpolate(std::vector<double>& local_dofs,
const std::vector<Vector<double> >& values,
unsigned int offset = 0) const;
/**
* Interpolate a set of vector
* values, computed in the
* generalized support points.
*/
virtual void interpolate(
std::vector<double>& local_dofs,
const VectorSlice<const std::vector<std::vector<double> > >& values) const;
//@}
/**
* Determine an estimate for the
* memory consumption (in bytes)
* of this object.
*
* This function is made virtual,
* since finite element objects
* are usually accessed through
* pointers to their base class,
* rather than the class itself.
*/
virtual unsigned int memory_consumption () const;
/**
* Exception
*
* @ingroup Exceptions
*/
DeclException1 (ExcShapeFunctionNotPrimitive,
int,
<< "The shape function with index " << arg1
<< " is not primitive, i.e. it is vector-valued and "
<< "has more than one non-zero vector component. This "
<< "function cannot be called for these shape functions. "
<< "Maybe you want to use the same function with the "
<< "_component suffix?");
/**
* Exception
*
* @ingroup Exceptions
*/
DeclException0 (ExcFENotPrimitive);
/**
* Exception
*
* @ingroup Exceptions
*/
DeclException0 (ExcUnitShapeValuesDoNotExist);
/**
* Attempt to access support
* points of a finite element
* which is not Lagrangian.
*
* @ingroup Exceptions
*/
DeclException0 (ExcFEHasNoSupportPoints);
/**
* Attempt to access embedding
* matrices of a finite element
* which did not implement these
* matrices.
*
* @ingroup Exceptions
*/
DeclException0 (ExcEmbeddingVoid);
/**
* Attempt to access restriction
* matrices of a finite element
* which did not implement these
* matrices.
*
* Exception
* @ingroup Exceptions
*/
DeclException0 (ExcProjectionVoid);
/**
* Attempt to access constraint
* matrices of a finite element
* which did not implement these
* matrices.
*
* Exception
* @ingroup Exceptions
*/
DeclException0 (ExcConstraintsVoid);
/**
* Exception
* @ingroup Exceptions
*/
DeclException2 (ExcWrongInterfaceMatrixSize,
int, int,
<< "The interface matrix has a size of " << arg1
<< "x" << arg2
<< ", which is not reasonable in the present dimension.");
/**
* Exception
* @ingroup Exceptions
*/
DeclException2 (ExcComponentIndexInvalid,
int, int,
<< "The component-index pair (" << arg1 << ", " << arg2
<< ") is invalid, i.e. non-existent");
/**
* Exception
* @ingroup Exceptions
*/
DeclException0 (ExcInterpolationNotImplemented);
/**
* Exception
*
* @ingroup Exceptions
*/
DeclException0 (ExcBoundaryFaceUsed);
/**
* Exception
*
* @ingroup Exceptions
*/
DeclException0 (ExcJacobiDeterminantHasWrongSign);
protected:
/**
* Reinit the vectors of
* restriction and prolongation
* matrices to the right sizes:
* For every refinement case,
* except for
* RefinementCase::no_refinement,
* and for every child of that
* refinement case the space of
* one restriction and
* prolongation matrix is
* allocated, see the
* documentation of the
* restriction and prolongation
* vectors for more detail on the
* actual vector sizes.
*
* @param
* isotropic_restriction_only:
* only the restriction matrices
* required for isotropic
* refinement are reinited to the
* right size.
* @param
* isotropic_prolongation_only:
* only the prolongation matrices
* required for isotropic
* refinement are reinited to the
* right size.
*/
void reinit_restriction_and_prolongation_matrices(const bool isotropic_restriction_only=false,
const bool isotropic_prolongation_only=false);
/**
* Store whether all shape
* functions are primitive. Since
* finding this out is a very
* common operation, we cache the
* result, i.e. compute the value
* in the constructor for simpler
* access.
*/
const bool cached_primitivity;
/**
* Vector of projection
* matrices. See
* get_restriction_matrix()
* above. The constructor
* initializes these matrices to
* zero dimensions, which can be
* changed by derived classes
* implementing them.
*
* Note, that
* <code>restriction[refinement_case-1][child]</code>
* includes the restriction
* matrix of child
* <code>child</code> for the
* RefinementCase
* <code>refinement_case</code>. Here,
* we use
* <code>refinement_case-1</code>
* instead of
* <code>refinement_case</code>
* as for
* RefinementCase::no_refinement(=0)
* there are no restriction
* matrices available.
*/
std::vector<std::vector<FullMatrix<double> > > restriction;
/**
* Vector of embedding
* matrices. See
* <tt>get_prolongation_matrix()</tt>
* above. The constructor
* initializes these matrices to
* zero dimensions, which can be
* changed by derived classes
* implementing them.
*
* Note, that
* <code>prolongation[refinement_case-1][child]</code>
* includes the prolongation
* matrix of child
* <code>child</code> for the
* RefinementCase
* <code>refinement_case</code>. Here,
* we use
* <code>refinement_case-1</code>
* instead of
* <code>refinement_case</code>
* as for
* RefinementCase::no_refinement(=0)
* there are no prolongation
* matrices available.
*/
std::vector<std::vector<FullMatrix<double> > > prolongation;
/**
* Specify the constraints which
* the dofs on the two sides of a
* cell interface underly if the
* line connects two cells of
* which one is refined once.
*
* For further details see the
* general description of the
* derived class.
*
* This field is obviously
* useless in one dimension
* and has there a zero size.
*/
FullMatrix<double> interface_constraints;
/**
* List of support points on the
* unit cell, in case the finite
* element has any. The
* constructor leaves this field
* empty, derived classes may
* write in some contents.
*
* Finite elements that allow
* some kind of interpolation
* operation usually have support
* points. On the other hand,
* elements that define their
* degrees of freedom by, for
* example, moments on faces, or
* as derivatives, don't have
* support points. In that case,
* this field remains empty.
*/
std::vector<Point<dim> > unit_support_points;
/**
* Same for the faces. See the
* description of the
* get_unit_face_support_points()
* function for a discussion of
* what contributes a face
* support point.
*/
std::vector<Point<dim-1> > unit_face_support_points;
/**
* Support points used for
* interpolation functions of
* non-Lagrangian elements.
*/
std::vector<Point<dim> > generalized_support_points;
/**
* Face support points used for
* interpolation functions of
* non-Lagrangian elements.
*/
std::vector<Point<dim-1> > generalized_face_support_points;
/**
* For faces with non-standard
* face_orientation in 3D, the dofs on
* faces (quads) have to be permuted in
* order to be combined with the correct
* shape functions. Given a local dof @p
* index on a quad, return the shift in the
* local index, if the face has
* non-standard face_orientation,
* i.e. <code>old_index + shift =
* new_index</code>. In 2D and 1D there is
* no need for permutation so the vector is
* empty. In 3D it has the size of <code>
* #dofs_per_quad * 8 </code>, where 8 is
* the number of orientations, a face can
* be in (all comibinations of the three
* bool flags face_orientation, face_flip
* and face_rotation).
*
* The standard implementation fills this
* with zeros, i.e. no permuatation at
* all. Derived finite element classes have
* to fill this Table with the correct
* values.
*/
Table<2,int> adjust_quad_dof_index_for_face_orientation_table;
/**
* For lines with non-standard
* line_orientation in 3D, the dofs on
* lines have to be permuted in
* order to be combined with the correct
* shape functions. Given a local dof @p
* index on a line, return the shift in the
* local index, if the line has
* non-standard line_orientation,
* i.e. <code>old_index + shift =
* new_index</code>. In 2D and 1D there is
* no need for permutation so the vector is
* empty. In 3D it has the size of
* #dofs_per_line.
*
* The standard implementation fills this
* with zeros, i.e. no permuatation at
* all. Derived finite element classes have
* to fill this vector with the correct
* values.
*/
std::vector<int> adjust_line_dof_index_for_line_orientation_table;
/**
* Return the size of interface
* constraint matrices. Since
* this is needed in every
* derived finite element class
* when initializing their size,
* it is placed into this
* function, to avoid having to
* recompute the
* dimension-dependent size of
* these matrices each time.
*
* Note that some elements do not
* implement the interface
* constraints for certain
* polynomial degrees. In this
* case, this function still
* returns the size these
* matrices should have when
* implemented, but the actual
* matrices are empty.
*/
TableIndices<2>
interface_constraints_size () const;
/**
* Compute second derivatives by
* finite differences of
* gradients.
*/
void compute_2nd (const Mapping<dim,spacedim> &mapping,
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int offset,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
InternalDataBase &fe_internal,
FEValuesData<dim,spacedim> &data) const;
/**
* Given the pattern of nonzero
* components for each shape
* function, compute for each
* entry how many components are
* non-zero for each shape
* function. This function is
* used in the constructor of
* this class.
*/
static
std::vector<unsigned int>
compute_n_nonzero_components (const std::vector<std::vector<bool> > &nonzero_components);
/**
* Determine the values a finite
* element should compute on
* initialization of data for
* FEValues.
*
* Given a set of flags
* indicating what quantities are
* requested from a FEValues
* object, update_once() and
* update_each() compute which
* values must really be
* computed. Then, the
* <tt>fill_*_values</tt> functions
* are called with the result of
* these.
*
* Furthermore, values must be
* computed either on the unit
* cell or on the physical
* cell. For instance, the
* function values of FE_Q do
* only depend on the quadrature
* points on the unit
* cell. Therefore, this flags
* will be returned by
* update_once(). The gradients
* require computation of the
* covariant transformation
* matrix. Therefore,
* @p update_covariant_transformation
* and @p update_gradients will
* be returned by
* update_each().
*
* For an example see the same
* function in the derived class
* FE_Q.
*/
virtual UpdateFlags update_once (const UpdateFlags flags) const = 0;
/**
* Complementary function for
* update_once().
*
* While update_once() returns
* the values to be computed on
* the unit cell for yielding the
* required data, this function
* determines the values that
* must be recomputed on each
* cell.
*
* Refer to update_once() for
* more details.
*/
virtual UpdateFlags update_each (const UpdateFlags flags) const = 0;
/**
* A sort of virtual copy
* constructor. Some places in
* the library, for example the
* constructors of FESystem as
* well as the hp::FECollection
* class, need to make copied of
* finite elements without
* knowing their exact type. They
* do so through this function.
*/
virtual FiniteElement<dim,spacedim> *clone() const = 0;
private:
/**
* Store what
* system_to_component_index()
* will return.
*/
std::vector< std::pair<unsigned int, unsigned int> > system_to_component_table;
/**
* Map between linear dofs and
* component dofs on face. This
* is filled with default values
* in the constructor, but
* derived classes will have to
* overwrite the information if
* necessary.
*
* By component, we mean the
* vector component, not the base
* element. The information thus
* makes only sense if a shape
* function is non-zero in only
* one component.
*/
std::vector< std::pair<unsigned int, unsigned int> > face_system_to_component_table;
/**
* For each shape function, store
* to which base element and
* which instance of this base
* element (in case its
* multiplicity is greater than
* one) it belongs, and its index
* within this base element. If
* the element is not composed of
* others, then base and instance
* are always zero, and the index
* is equal to the number of the
* shape function. If the element
* is composed of single
* instances of other elements
* (i.e. all with multiplicity
* one) all of which are scalar,
* then base values and dof
* indices within this element
* are equal to the
* #system_to_component_table. It
* differs only in case the
* element is composed of other
* elements and at least one of
* them is vector-valued itself.
*
* This array has valid values
* also in the case of
* vector-valued
* (i.e. non-primitive) shape
* functions, in contrast to the
* #system_to_component_table.
*/
std::vector<std::pair<std::pair<unsigned int,unsigned int>,unsigned int> >
system_to_base_table;
/**
* Likewise for the indices on
* faces.
*/
std::vector<std::pair<std::pair<unsigned int,unsigned int>,unsigned int> >
face_system_to_base_table;
/**
* For each base element, store
* the first block in a block
* vector it will generate.
*/
std::vector<unsigned int> first_block_of_base_table;
/**
* The base element establishing
* a component.
*
* For each component number
* <tt>c</tt>, the entries have
* the following meaning:
* <dl>
* <dt><tt>table[c].first.first</tt></dt>
* <dd>Number of the base element for <tt>c</tt>.</dd>
* <dt><tt>table[c].first.second</tt></dt>
* <dd>Component in the base element for <tt>c</tt>.</dd>
* <dt><tt>table[c].second</tt></dt>
* <dd>Multiple of the base element for <tt>c</tt>.</dd>
* </dl>
*
* This variable is set to the
* correct size by the
* constructor of this class, but
* needs to be initialized by
* derived classes, unless its
* size is one and the only entry
* is a zero, which is the case
* for scalar elements. In that
* case, the initialization by
* the base class is sufficient.
*/
std::vector<std::pair<std::pair<unsigned int, unsigned int>, unsigned int> >
component_to_base_table;
/**
* Projection matrices are
* concatenated or summed up.
*
* This flags decides on how the
* projection matrices of the
* children of the same father
* are put together to one
* operator. The possible modes
* are concatenation and
* summation.
*
* If the projection is defined
* by an interpolation operator,
* the child matrices are
* concatenated, i.e. values
* belonging to the same node
* functional are identified and
* enter the interpolated value
* only once. In this case, the
* flag must be @p false.
*
* For projections with respect
* to scalar products, the child
* matrices must be summed up to
* build the complete matrix. The
* flag should be @p true.
*
* For examples of use of these
* flags, see the places in the
* library where it is queried.
*
* There is one flag per shape
* function, indicating whether
* it belongs to the class of
* shape functions that are
* additive in the restriction or
* not.
*
* Note that in previous versions
* of the library, there was one
* flag per vector component of
* the element. This is based on
* the fact that all the shape
* functions that belong to the
* same vector component must
* necessarily behave in the same
* way, to make things
* reasonable. However, the
* problem is that it is
* sometimes impossible to query
* this flag in the vector-valued
* case: this used to be done
* with the
* #system_to_component_index
* function that returns which
* vector component a shape
* function is associated
* with. The point is that since
* we now support shape functions
* that are associated with more
* than one vector component (for
* example the shape functions of
* Raviart-Thomas, or Nedelec
* elements), that function can
* no more be used, so it can be
* difficult to find out which
* for vector component we would
* like to query the
* restriction-is-additive flags.
*/
const std::vector<bool> restriction_is_additive_flags;
/**
* For each shape function, give
* a vector of bools (with size
* equal to the number of vector
* components which this finite
* element has) indicating in
* which component each of these
* shape functions is non-zero.
*
* For primitive elements, there
* is only one non-zero
* component.
*/
const std::vector<std::vector<bool> > nonzero_components;
/**
* This array holds how many
* values in the respective entry
* of the #nonzero_components
* element are non-zero. The
* array is thus a short-cut to
* allow faster access to this
* information than if we had to
* count the non-zero entries
* upon each request for this
* information. The field is
* initialized in the constructor
* of this class.
*/
const std::vector<unsigned int> n_nonzero_components_table;
/**
* Second derivatives of shapes
* functions are not computed
* analytically, but by finite
* differences of the
* gradients. This static
* variable denotes the step
* length to be used for
* that. It's value is set to
* 1e-6.
*/
static const double fd_step_length;
/**
* Prepare internal data
* structures and fill in values
* independent of the
* cell. Returns a pointer to an
* object of which the caller of
* this function then has to
* assume ownership (which
* includes destruction when it
* is no more needed).
*/
virtual typename Mapping<dim,spacedim>::InternalDataBase*
get_data (const UpdateFlags flags,
const Mapping<dim,spacedim> &mapping,
const Quadrature<dim> &quadrature) const = 0;
/**
* Prepare internal data
* structure for transformation
* of faces and fill in values
* independent of the
* cell. Returns a pointer to an
* object of which the caller of
* this function then has to
* assume ownership (which
* includes destruction when it
* is no more needed).
*/
virtual typename Mapping<dim,spacedim>::InternalDataBase*
get_face_data (const UpdateFlags flags,
const Mapping<dim,spacedim> &mapping,
const Quadrature<dim-1> &quadrature) const;
/**
* Prepare internal data
* structure for transformation
* of children of faces and fill
* in values independent of the
* cell. Returns a pointer to an
* object of which the caller of
* this function then has to
* assume ownership (which
* includes destruction when it
* is no more needed).
*/
virtual typename Mapping<dim,spacedim>::InternalDataBase*
get_subface_data (const UpdateFlags flags,
const Mapping<dim,spacedim> &mapping,
const Quadrature<dim-1> &quadrature) const;
/**
* Fill the fields of
* FEValues. This function
* performs all the operations
* needed to compute the data of an
* FEValues object.
*
* The same function in
* @p mapping must have been
* called for the same cell first!
*/
virtual void
fill_fe_values (const Mapping<dim,spacedim> &mapping,
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const Quadrature<dim> &quadrature,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
typename Mapping<dim,spacedim>::InternalDataBase &fe_internal,
FEValuesData<dim,spacedim> &data,
CellSimilarity::Similarity &cell_similarity) const = 0;
/**
* Fill the fields of
* FEFaceValues. This function
* performs all the operations
* needed to compute the data of an
* FEFaceValues object.
*
* The same function in
* @p mapping must have been
* called for the same cell first!
*/
virtual void
fill_fe_face_values (const Mapping<dim,spacedim> &mapping,
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const Quadrature<dim-1> &quadrature,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
typename Mapping<dim,spacedim>::InternalDataBase &fe_internal,
FEValuesData<dim,spacedim> &data) const = 0;
/**
* Fill the fields of
* FESubfaceValues. This function
* performs all the operations
* needed to compute the data of an
* FESubfaceValues object.
*
* The same function in
* @p mapping must have been
* called for the same cell first!
*/
virtual void
fill_fe_subface_values (const Mapping<dim,spacedim> &mapping,
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const unsigned int sub_no,
const Quadrature<dim-1> &quadrature,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
typename Mapping<dim,spacedim>::InternalDataBase &fe_internal,
FEValuesData<dim,spacedim> &data) const = 0;
friend class InternalDataBase;
friend class FEValuesBase<dim,spacedim>;
friend class FEValues<dim,spacedim>;
friend class FEFaceValues<dim,spacedim>;
friend class FESubfaceValues<dim,spacedim>;
template <int, int > friend class FESystem;
template <class POLY, int dim_, int spacedim_> friend class FE_PolyTensor;
friend class hp::FECollection<dim,spacedim>;
};
//----------------------------------------------------------------------//
template <int dim, int spacedim>
inline
const FiniteElement<dim,spacedim> &
FiniteElement<dim,spacedim>::operator[] (const unsigned int fe_index) const
{
Assert (fe_index == 0,
ExcMessage ("A fe_index of zero is the only index allowed here"));
return *this;
}
template <int dim, int spacedim>
inline
std::pair<unsigned int,unsigned int>
FiniteElement<dim,spacedim>::system_to_component_index (const unsigned int index) const
{
Assert (index < system_to_component_table.size(),
ExcIndexRange(index, 0, system_to_component_table.size()));
Assert (is_primitive (index),
( typename FiniteElement<dim,spacedim>::ExcShapeFunctionNotPrimitive(index)) );
return system_to_component_table[index];
}
template <int dim, int spacedim>
inline
unsigned int
FiniteElement<dim,spacedim>::component_to_system_index (const unsigned int component,
const unsigned int index) const
{
std::vector< std::pair<unsigned int, unsigned int> >::const_iterator
it = std::find(system_to_component_table.begin(), system_to_component_table.end(),
std::pair<unsigned int, unsigned int>(component, index));
Assert(it != system_to_component_table.end(), ExcComponentIndexInvalid(component, index));
return std::distance(system_to_component_table.begin(), it);
}
template <int dim, int spacedim>
inline
std::pair<unsigned int,unsigned int>
FiniteElement<dim,spacedim>::face_system_to_component_index (const unsigned int index) const
{
Assert(index < face_system_to_component_table.size(),
ExcIndexRange(index, 0, face_system_to_component_table.size()));
// in debug mode, check whether the
// function is primitive, since
// otherwise the result may have no
// meaning
//
// since the primitivity tables are
// all geared towards cell dof
// indices, rather than face dof
// indices, we have to work a
// little bit...
//
// in 1d, the face index is equal
// to the cell index
Assert (is_primitive(this->face_to_equivalent_cell_index(index)),
(typename FiniteElement<dim,spacedim>::ExcShapeFunctionNotPrimitive(index)) );
return face_system_to_component_table[index];
}
template <int dim, int spacedim>
inline
std::pair<std::pair<unsigned int,unsigned int>,unsigned int>
FiniteElement<dim,spacedim>::system_to_base_index (const unsigned int index) const
{
Assert (index < system_to_base_table.size(),
ExcIndexRange(index, 0, system_to_base_table.size()));
return system_to_base_table[index];
}
template <int dim, int spacedim>
inline
std::pair<std::pair<unsigned int,unsigned int>,unsigned int>
FiniteElement<dim,spacedim>::face_system_to_base_index (const unsigned int index) const
{
Assert(index < face_system_to_base_table.size(),
ExcIndexRange(index, 0, face_system_to_base_table.size()));
return face_system_to_base_table[index];
}
template <int dim, int spacedim>
inline
unsigned int
FiniteElement<dim,spacedim>::first_block_of_base (const unsigned int index) const
{
Assert(index < first_block_of_base_table.size(),
ExcIndexRange(index, 0, first_block_of_base_table.size()));
return first_block_of_base_table[index];
}
template <int dim, int spacedim>
inline
std::pair<unsigned int,unsigned int>
FiniteElement<dim,spacedim>::component_to_base_index (const unsigned int index) const
{
Assert(index < component_to_base_table.size(),
ExcIndexRange(index, 0, component_to_base_table.size()));
return component_to_base_table[index].first;
}
template <int dim, int spacedim>
inline
std::pair<unsigned int,unsigned int>
FiniteElement<dim,spacedim>::block_to_base_index (const unsigned int index) const
{
Assert(index < this->n_blocks(),
ExcIndexRange(index, 0, this->n_blocks()));
for (int i=first_block_of_base_table.size()-1; i>=0; --i)
if (first_block_of_base_table[i] <= index)
return std::pair<unsigned int, unsigned int>(static_cast<unsigned int> (i),
index - first_block_of_base_table[i]);
return std::make_pair(numbers::invalid_unsigned_int,
numbers::invalid_unsigned_int);
}
template <int dim, int spacedim>
inline
std::pair<unsigned int,unsigned int>
FiniteElement<dim,spacedim>::system_to_block_index (const unsigned int index) const
{
Assert (index < this->dofs_per_cell,
ExcIndexRange(index, 0, this->dofs_per_cell));
// The block is computed simply as
// first block of this base plus
// the index within the base blocks
return std::pair<unsigned int, unsigned int>(
first_block_of_base(system_to_base_table[index].first.first)
+ system_to_base_table[index].first.second,
system_to_base_table[index].second);
}
template <int dim, int spacedim>
inline
bool
FiniteElement<dim,spacedim>::restriction_is_additive (const unsigned int index) const
{
Assert(index < this->dofs_per_cell,
ExcIndexRange(index, 0, this->dofs_per_cell));
return restriction_is_additive_flags[index];
}
template <int dim, int spacedim>
inline
const std::vector<bool> &
FiniteElement<dim,spacedim>::get_nonzero_components (const unsigned int i) const
{
Assert (i < this->dofs_per_cell, ExcIndexRange (i, 0, this->dofs_per_cell));
return nonzero_components[i];
}
template <int dim, int spacedim>
inline
unsigned int
FiniteElement<dim,spacedim>::n_nonzero_components (const unsigned int i) const
{
Assert (i < this->dofs_per_cell, ExcIndexRange (i, 0, this->dofs_per_cell));
return n_nonzero_components_table[i];
}
template <int dim, int spacedim>
inline
bool
FiniteElement<dim,spacedim>::is_primitive (const unsigned int i) const
{
Assert (i < this->dofs_per_cell, ExcIndexRange (i, 0, this->dofs_per_cell));
// return primitivity of a shape
// function by checking whether it
// has more than one non-zero
// component or not. we could cache
// this value in an array of bools,
// but accessing a bit-vector (as
// std::vector<bool> is) is
// probably more expensive than
// just comparing against 1
//
// for good measure, short circuit the test
// if the entire FE is primitive
return ((cached_primitivity == true)
||
(n_nonzero_components_table[i] == 1));
}
template <int dim, int spacedim>
inline
bool
FiniteElement<dim,spacedim>::is_primitive () const
{
return cached_primitivity;
}
DEAL_II_NAMESPACE_CLOSE
#endif
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