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// $Id: fe_q.h 21191 2010-06-10 11:50:17Z bangerth $
// Version: $Name$
//
// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 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_q_h
#define __deal2__fe_q_h
#include <base/config.h>
#include <base/tensor_product_polynomials.h>
#include <fe/fe_poly.h>
DEAL_II_NAMESPACE_OPEN
/*!@addtogroup fe */
/*@{*/
/**
* Implementation of a scalar Lagrange finite element @p Qp that yields the
* finite element space of continuous, piecewise polynomials of degree @p p in
* each coordinate direction. This class is realized using tensor product
* polynomials based on equidistant or given support points.
*
* The standard constructor of this class takes the degree @p p of this finite
* element. Alternatively, it can take a quadrature formula @p points defining
* the support points of the Lagrange interpolation in one coordinate direction.
*
* For more information about the <tt>spacedim</tt> template parameter
* check the documentation of FiniteElement or the one of
* Triangulation.
*
* <h3>Implementation</h3>
*
* The constructor creates a TensorProductPolynomials object that includes the
* tensor product of @p LagrangeEquidistant polynomials of degree @p p. This
* @p TensorProductPolynomials object provides all values and derivatives of
* the shape functions. In case a quadrature rule is given, the constructor
* creates a TensorProductPolynomials object that includes the tensor product
* of @p Lagrange polynomials with the support points from @p points.
*
* Furthermore the constructor fills the @p interface_constraints, the
* @p prolongation (embedding) and the @p restriction matrices. These
* are implemented only up to a certain degree and may not be
* available for very high polynomial degree.
*
*
* <h3>Numbering of the degrees of freedom (DoFs)</h3>
*
* The original ordering of the shape functions represented by the
* TensorProductPolynomials is a tensor product
* numbering. However, the shape functions on a cell are renumbered
* beginning with the shape functions whose support points are at the
* vertices, then on the line, on the quads, and finally (for 3d) on
* the hexes. To be explicit, these numberings are listed in the
* following:
*
* <h4>Q1 elements</h4>
* <ul>
* <li> 1D case:
* @verbatim
* 0-------1
* @endverbatim
*
* <li> 2D case:
* @verbatim
* 2-------3
* | |
* | |
* | |
* 0-------1
* @endverbatim
*
* <li> 3D case:
* @verbatim
* 6-------7 6-------7
* /| | / /|
* / | | / / |
* / | | / / |
* 4 | | 4-------5 |
* | 2-------3 | | 3
* | / / | | /
* | / / | | /
* |/ / | |/
* 0-------1 0-------1
* @endverbatim
*
* The respective coordinate values of the support points of the degrees
* of freedom are as follows:
* <ul>
* <li> Index 0: <tt>[0, 0, 0]</tt>;
* <li> Index 1: <tt>[1, 0, 0]</tt>;
* <li> Index 2: <tt>[0, 1, 0]</tt>;
* <li> Index 3: <tt>[1, 1, 0]</tt>;
* <li> Index 4: <tt>[0, 0, 1]</tt>;
* <li> Index 5: <tt>[1, 0, 1]</tt>;
* <li> Index 6: <tt>[0, 1, 1]</tt>;
* <li> Index 7: <tt>[1, 1, 1]</tt>;
* </ul>
* </ul>
* <h4>Q2 elements</h4>
* <ul>
* <li> 1D case:
* @verbatim
* 0---2---1
* @endverbatim
*
* <li> 2D case:
* @verbatim
* 2---7---3
* | |
* 4 8 5
* | |
* 0---6---1
* @endverbatim
*
* <li> 3D case:
* @verbatim
* 6--15---7 6--15---7
* /| | / /|
* 12 | 19 12 1319
* / 18 | / / |
* 4 | | 4---14--5 |
* | 2---11--3 | | 3
* | / / | 17 /
* 16 8 9 16 | 9
* |/ / | |/
* 0---10--1 0---8---1
*
* *-------* *-------*
* /| | / /|
* / | 23 | / 25 / |
* / | | / / |
* * | | *-------* |
* |20 *-------* | |21 *
* | / / | 22 | /
* | / 24 / | | /
* |/ / | |/
* *-------* *-------*
* @endverbatim
* The center vertex has number 26.
*
* The respective coordinate values of the support points of the degrees
* of freedom are as follows:
* <ul>
* <li> Index 0: <tt>[0, 0, 0]</tt>;
* <li> Index 1: <tt>[1, 0, 0]</tt>;
* <li> Index 2: <tt>[0, 1, 0]</tt>;
* <li> Index 3: <tt>[1, 1, 0]</tt>;
* <li> Index 4: <tt>[0, 0, 1]</tt>;
* <li> Index 5: <tt>[1, 0, 1]</tt>;
* <li> Index 6: <tt>[0, 1, 1]</tt>;
* <li> Index 7: <tt>[1, 1, 1]</tt>;
* <li> Index 8: <tt>[0, 1/2, 0]</tt>;
* <li> Index 9: <tt>[1, 1/2, 0]</tt>;
* <li> Index 10: <tt>[1/2, 0, 0]</tt>;
* <li> Index 11: <tt>[1/2, 1, 0]</tt>;
* <li> Index 12: <tt>[0, 1/2, 1]</tt>;
* <li> Index 13: <tt>[1, 1/2, 1]</tt>;
* <li> Index 14: <tt>[1/2, 0, 1]</tt>;
* <li> Index 15: <tt>[1/2, 1, 1]</tt>;
* <li> Index 16: <tt>[0, 0, 1/2]</tt>;
* <li> Index 17: <tt>[1, 0, 1/2]</tt>;
* <li> Index 18: <tt>[0, 1, 1/2]</tt>;
* <li> Index 19: <tt>[1, 1, 1/2]</tt>;
* <li> Index 20: <tt>[0, 1/2, 1/2]</tt>;
* <li> Index 21: <tt>[1, 1/2, 1/2]</tt>;
* <li> Index 22: <tt>[1/2, 0, 1/2]</tt>;
* <li> Index 23: <tt>[1/2, 1, 1/2]</tt>;
* <li> Index 24: <tt>[1/2, 1/2, 0]</tt>;
* <li> Index 25: <tt>[1/2, 1/2, 1]</tt>;
* <li> Index 26: <tt>[1/2, 1/2, 1/2]</tt>;
* </ul>
* </ul>
* <h4>Q3 elements</h4>
* <ul>
* <li> 1D case:
* @verbatim
* 0--2--3--1
* @endverbatim
*
* <li> 2D case:
* @verbatim
* 2--10-11-3
* | |
* 5 14 15 7
* | |
* 4 12 13 6
* | |
* 0--8--9--1
* @endverbatim
* </ul>
* <h4>Q4 elements</h4>
* <ul>
* <li> 1D case:
* @verbatim
* 0--2--3--4--1
* @endverbatim
*
* <li> 2D case:
* @verbatim
* 2--13-14-15-3
* | |
* 6 22 23 24 9
* | |
* 5 19 20 21 8
* | |
* 4 16 17 18 7
* | |
* 0--10-11-12-1
* @endverbatim
* </ul>
*
* @author Wolfgang Bangerth, 1998, 2003; Guido Kanschat, 2001; Ralf Hartmann, 2001, 2004, 2005; Oliver Kayser-Herold, 2004; Katharina Kormann, 2008; Martin Kronbichler, 2008
*/
template <int dim, int spacedim=dim>
class FE_Q : public FE_Poly<TensorProductPolynomials<dim>,dim,spacedim>
{
public:
/**
* Constructor for tensor product
* polynomials of degree @p p.
*/
FE_Q (const unsigned int p);
/**
* Constructor for tensor product
* polynomials with support points @p
* points based on a one-dimensional
* quadrature formula. The degree of the
* finite element is
* <tt>points.size()-1</tt>. Note that
* the first point has to be 0 and the
* last one 1.
*/
FE_Q (const Quadrature<1> &points);
/**
* Return a string that uniquely
* identifies a finite
* element. This class returns
* <tt>FE_Q<dim>(degree)</tt>, with
* @p dim and @p degree
* replaced by appropriate
* values.
*/
virtual std::string get_name () const;
/**
* Return the matrix
* interpolating from the given
* finite element to the present
* one. The size of the matrix is
* then @p dofs_per_cell times
* <tt>source.dofs_per_cell</tt>.
*
* These matrices are only
* available if the source
* element is also a @p FE_Q
* element. Otherwise, an
* exception of type
* FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented
* is thrown.
*/
virtual void
get_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
FullMatrix<double> &matrix) const;
/**
* 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
* FiniteElement<dim,spacedim>::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 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
* FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented.
*/
virtual void
get_subface_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
const unsigned int subface,
FullMatrix<double> &matrix) const;
/**
* Check for non-zero values on a face.
*
* This function returns
* @p true, if the shape
* function @p shape_index has
* non-zero values on the face
* @p face_index.
*
* Implementation of the
* interface in
* FiniteElement
*/
virtual bool has_support_on_face (const unsigned int shape_index,
const unsigned int face_index) const;
/**
* @name Functions to support hp
* @{
*/
/**
* Return whether this element
* implements its hanging node
* constraints in the new way,
* which has to be used to make
* elements "hp compatible".
*
* For the FE_Q class the result is
* always true (independent of the degree
* of the element), as it implements the
* complete set of functions necessary
* for hp capability.
*/
virtual bool hp_constraints_are_implemented () 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;
//@}
/**
* 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;
protected:
/**
* @p clone function instead of
* a copy constructor.
*
* This function is needed by the
* constructors of @p FESystem.
*/
virtual FiniteElement<dim,spacedim> * clone() const;
private:
/**
* Only for internal use. Its
* full name is
* @p get_dofs_per_object_vector
* function and it creates the
* @p dofs_per_object vector that is
* needed within the constructor to
* be passed to the constructor of
* @p FiniteElementData.
*/
static std::vector<unsigned int> get_dpo_vector(const unsigned int degree);
/**
* This is an analogon to the
* FETools::lexicographic_to_hierarchic_numbering
* function, but working on
* faces. Called from the
* constructor.
*/
static
std::vector<unsigned int>
face_lexicographic_to_hierarchic_numbering (const unsigned int degree);
/**
* Initialize the hanging node
* constraints matrices. Called
* from the constructor.
*/
void initialize_constraints ();
/**
* Initialize the embedding
* matrices. Called from the
* constructor.
*/
void initialize_embedding ();
/**
* Initialize the restriction
* matrices. Called from the
* constructor.
*/
void initialize_restriction ();
/**
* Initialize the
* @p unit_support_points field
* of the FiniteElement
* class. Called from the
* constructor.
*/
void initialize_unit_support_points ();
/**
* Initialize the @p unit_support_points
* field of the FiniteElement
* class. Called from the constructor in
* case the finite element is based on
* quadrature points.
*/
void initialize_unit_support_points (const Quadrature<1> &points);
/**
* Initialize the
* @p unit_face_support_points field
* of the FiniteElement
* class. Called from the
* constructor.
*/
void initialize_unit_face_support_points ();
/**
* Initialize the @p
* unit_face_support_points field of the
* FiniteElement class. Called from the
* constructor in case the finite element
* is based on quadrature points.
*/
void initialize_unit_face_support_points (const Quadrature<1> &points);
/**
* Initialize the
* @p adjust_quad_dof_index_for_face_orientation_table field
* of the FiniteElement
* class. Called from the
* constructor.
*/
void initialize_quad_dof_index_permutation ();
/**
* Mapping from hierarchic to
* lexicographic numbering on
* first face. Hierarchic is the
* numbering of the shape
* functions.
*/
const std::vector<unsigned int> face_index_map;
/**
* Forward declaration of a class
* into which we put significant
* parts of the implementation.
*
* See the .cc file for more
* information.
*/
struct Implementation;
/**
* Allow access from other
* dimensions. We need this since
* we want to call the functions
* @p get_dpo_vector and
* @p lexicographic_to_hierarchic_numbering
* for the faces of the finite
* element of dimension dim+1.
*/
template <int, int> friend class FE_Q;
friend class FE_Q<dim,spacedim>::Implementation;
};
/*@}*/
/* -------------- declaration of explicit specializations ------------- */
template <>
void FE_Q<1>::initialize_unit_face_support_points ();
template <>
std::vector<unsigned int>
FE_Q<1>::face_lexicographic_to_hierarchic_numbering (const unsigned int);
#if deal_II_dimension != 3
template <>
void FE_Q<1,2>::initialize_unit_face_support_points ();
template <>
std::vector<unsigned int>
FE_Q<1,2>::face_lexicographic_to_hierarchic_numbering (const unsigned int);
#endif
DEAL_II_NAMESPACE_CLOSE
#endif
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