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// $Id: fe_q_hierarchical.h 21181 2010-06-09 06:10:08Z buerg $
// Version: $Name$
//
// Copyright (C) 2002, 2003, 2004, 2005, 2006 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_hierarchical_h
#define __deal2__fe_q_hierarchical_h
#include <base/config.h>
#include <base/tensor_product_polynomials.h>
#include <fe/fe_poly.h>
DEAL_II_NAMESPACE_OPEN
template <int dim, int spacedim> class MappingQ;
/*!@addtogroup fe */
/*@{*/
/**
* Implementation of Hierarchical finite elements @p Qp that yield the
* finite element space of continuous, piecewise polynomials of degree
* @p p. This class is realized using tensor product polynomials
* based on a hierarchical basis @p Hierarchical of the interval
* <tt>[0,1]</tt> which is suitable for building an @p hp tensor product
* finite element, if we assume that each element has a single degree.
*
* There are not many differences between @p FE_Q_Hierarchical and
* @p FE_Q, except that we add a function @p embedding_dofs that takes
* a given integer @p q, between @p 1 and @p p, and
* returns the numbering of basis functions of the element of order
* @p q in basis of order @p p. This function is
* useful if one wants to make calculations using the hierarchical
* nature of these shape functions.
*
* The unit support points now are reduced to @p 0, @p 1, and <tt>0.5</tt> in
* one dimension, and tensor products in higher dimensions. Thus, various
* interpolation functions will only work correctly for the linear case.
* Future work will involve writing projection--interpolation operators
* that can interpolate onto the higher order bubble functions.
*
* The constructor of this class takes the degree @p p of this finite
* element.
*
* This class is not implemented for the codimension one case
* (<tt>spacedim != dim</tt>).
*
* <h3>Implementation</h3>
*
* The constructor creates a TensorProductPolynomials object
* that includes the tensor product of @p Hierarchical
* polynomials of degree @p p. This @p TensorProductPolynomials
* object provides all values and derivatives of the shape functions.
*
* <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 Brian Carnes, 2002, Ralf Hartmann 2004, 2005
*/
template <int dim>
class FE_Q_Hierarchical : public FE_Poly<TensorProductPolynomials<dim>,dim>
{
public:
/**
* Constructor for tensor product
* polynomials of degree @p p.
*/
FE_Q_Hierarchical (const unsigned int p);
/**
* Return a string that uniquely
* identifies a finite
* element. This class returns
* <tt>FE_Q_Hierarchical<dim>(degree)</tt>,
* with @p dim and @p degree
* replaced by appropriate
* values.
*/
virtual std::string get_name () 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_Hierarchical 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> &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;
/**
* For a finite element of degree
* @p sub_degree < @p degree, we
* return a vector which maps the
* numbering on an FE
* of degree @p sub_degree into the
* numbering on this element.
*/
std::vector<unsigned int> get_embedding_dofs (const unsigned int sub_degree) const;
protected:
/**
* @p clone function instead of
* a copy constructor.
*
* This function is needed by the
* constructors of @p FESystem.
*/
virtual FiniteElement<dim> * 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);
/**
* The numbering of the degrees
* of freedom in continous finite
* elements is hierarchic,
* i.e. in such a way that we
* first number the vertex dofs,
* in the order of the vertices
* as defined by the
* triangulation, then the line
* dofs in the order and
* respecting the direction of
* the lines, then the dofs on
* quads, etc.
*
* The dofs associated with 1d
* hierarchical polynomials are
* ordered with the vertices
* first ($phi_0(x)=1-x$ and
* $phi_1(x)=x$) and then the
* line dofs (the higher degree
* polynomials). The 2d and 3d
* hierarchical polynomials
* originate from the 1d
* hierarchical polynomials by
* tensor product. In the
* following, the resulting
* numbering of dofs will be
* denoted by `fe_q_hierarchical
* numbering`.
*
* This function constructs a
* table which fe_q_hierarchical
* index each degree of freedom
* in the hierarchic numbering
* would have.
*
* This function is anologous to
* the
* FETools::hierarchical_to_lexicographic_numbering()
* function. However, in contrast
* to the fe_q_hierarchical
* numbering defined above, the
* lexicographic numbering
* originates from the tensor
* products of consecutive
* numbered dofs (like for
* LagrangeEquidistant).
*
* It is assumed that the size of
* the output argument already
* matches the correct size,
* which is equal to the number
* of degrees of freedom in the
* finite element.
*/
static
std::vector<unsigned int> hierarchic_to_fe_q_hierarchical_numbering (
const FiniteElementData<dim> &fe);
/**
* This is an analogon to the
* previous function, but working
* on faces.
*/
static
std::vector<unsigned int>
face_fe_q_hierarchical_to_hierarchic_numbering (const unsigned int degree);
/**
* Initialize two auxiliary
* fields that will be used in
* setting up the various
* matrices in the constructor.
*/
void build_dofs_cell (std::vector<FullMatrix<double> > &dofs_cell,
std::vector<FullMatrix<double> > &dofs_subcell) const;
/**
* Initialize the hanging node
* constraints matrices. Called
* from the constructor.
*/
void initialize_constraints (const std::vector<FullMatrix<double> > &dofs_subcell);
/**
* Initialize the embedding
* matrices. Called from the
* constructor.
*/
void initialize_embedding_and_restriction (const std::vector<FullMatrix<double> > &dofs_cell,
const std::vector<FullMatrix<double> > &dofs_subcell);
/**
* Initialize the
* @p unit_support_points field
* of the FiniteElement
* class. Called from the
* constructor.
*/
void initialize_unit_support_points ();
/**
* Initialize the
* @p unit_face_support_points field
* of the FiniteElement
* class. Called from the
* constructor.
*/
void initialize_unit_face_support_points ();
/**
* Mapping from lexicographic to
* shape function numbering on first face.
*/
const std::vector<unsigned int> face_renumber;
/**
* 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 dim1> friend class FE_Q_Hierarchical;
};
/*@}*/
/* -------------- declaration of explicit specializations ------------- */
template <>
void FE_Q_Hierarchical<1>::initialize_unit_face_support_points ();
template <>
bool
FE_Q_Hierarchical<1>::has_support_on_face (const unsigned int,
const unsigned int) const;
template <>
std::vector<unsigned int>
FE_Q_Hierarchical<1>::face_fe_q_hierarchical_to_hierarchic_numbering (const unsigned int);
DEAL_II_NAMESPACE_CLOSE
#endif
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