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// $Id: fe_dgp_monomial.h 19046 2009-07-08 19:30:23Z bangerth $
// Version: $Name$
//
// Copyright (C) 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_dgp_monomial_h
#define __deal2__fe_dgp_monomial_h
#include <base/config.h>
#include <base/polynomials_p.h>
#include <fe/fe_poly.h>
DEAL_II_NAMESPACE_OPEN
template <int dim, int spacedim> class MappingQ;
/*!@addtogroup fe */
/*@{*/
/**
* Discontinuous finite elements based on monomials.
*
* This finite element implements complete polynomial spaces, that is,
* dim-dimensional polynomials of degree p. For example, in 2d the
* element FE_DGP(1) would represent the span of the functions
* $\{1,\hat x,\hat y\}$, which is in contrast to the element FE_DGQ(1)
* that is formed by the span of $\{1,\hat x,\hat y,\hat x\hat y\}$. Since the
* DGP space has only three unknowns for each quadrilateral, it is
* immediately clear that this element can not be continuous.
*
* The basis functions for this element are chosen to be the monomials
* listed above. Note that this is the main difference to the FE_DGP
* class that uses a set of polynomials of complete degree
* <code>p</code> that form a Legendre basis on the unit square. Thus,
* there, the mass matrix is diagonal, if the grid cells are
* parallelograms. The basis here does not have this property;
* however, it is simpler to compute. On the other hand, this element
* has the additional disadvantage that the local cell matrices
* usually have a worse condition number than the ones originating
* from the FE_DGP element.
*
* This class is not implemented for the codimension one case
* (<tt>spacedim != dim</tt>).
*
* <h3>Transformation properties</h3>
*
* It is worth noting that under a (bi-, tri-)linear mapping, the
* space described by this element does not contain $P(k)$, even if we
* use a basis of polynomials of degree $k$. Consequently, for
* example, on meshes with non-affine cells, a linear function can not
* be exactly represented by elements of type FE_DGP(1) or
* FE_DGPMonomial(1).
*
* This can be understood by the following 2-d example: consider the
* cell with vertices at $(0,0),(1,0),(0,1),(s,s)$:
* @image html dgp_doesnt_contain_p.png
*
* For this cell, a bilinear transformation $F$ produces the relations
* $x=\hat x+\hat x\hat y$ and $y=\hat y+\hat x\hat y$ that correlate
* reference coordinates $\hat x,\hat y$ and coordinates in real space
* $x,y$. Under this mapping, the constant function is clearly mapped
* onto itself, but the two other shape functions of the $P_1$ space,
* namely $\phi_1(\hat x,\hat y)=\hat x$ and $\phi_2(\hat x,\hat
* y)=\hat y$ are mapped onto
* $\phi_1(x,y)=\frac{x-t}{t(s-1)},\phi_2(x,y)=t$ where
* $t=\frac{y}{s-x+sx+y-sy}$.
*
* For the simple case that $s=1$, i.e. if the real cell is the unit
* square, the expressions can be simplified to $t=y$ and
* $\phi_1(x,y)=x,\phi_2(x,y)=y$. However, for all other cases, the
* functions $\phi_1(x,y),\phi_2(x,y)$ are not linear any more, and
* neither is any linear combincation of them. Consequently, the
* linear functions are not within the range of the mapped $P_1$
* polynomials.
*
*
* @author Ralf Hartmann, 2004
*/
template <int dim>
class FE_DGPMonomial : public FE_Poly<PolynomialsP<dim>,dim>
{
public:
/**
* Constructor for the polynomial
* space of degree <tt>p</tt>.
*/
FE_DGPMonomial (const unsigned int p);
/**
* Return a string that uniquely
* identifies a finite
* element. This class returns
* <tt>FE_DGPMonomial<dim>(degree)</tt>,
* with <tt>dim</tt> and
* <tt>p</tt> replaced by
* appropriate values.
*/
virtual std::string get_name () const;
/**
* @name Functions to support hp
* @{
*/
/**
* 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.
*
* This being a discontinuous element,
* the set of such constraints is of
* course empty.
*/
virtual
std::vector<std::pair<unsigned int, unsigned int> >
hp_vertex_dof_identities (const FiniteElement<dim> &fe_other) const;
/**
* Same as
* hp_vertex_dof_indices(),
* except that the function
* treats degrees of freedom on
* lines.
*
* This being a discontinuous element,
* the set of such constraints is of
* course empty.
*/
virtual
std::vector<std::pair<unsigned int, unsigned int> >
hp_line_dof_identities (const FiniteElement<dim> &fe_other) const;
/**
* Same as
* hp_vertex_dof_indices(),
* except that the function
* treats degrees of freedom on
* quads.
*
* This being a discontinuous element,
* the set of such constraints is of
* course empty.
*/
virtual
std::vector<std::pair<unsigned int, unsigned int> >
hp_quad_dof_identities (const FiniteElement<dim> &fe_other) const;
/**
* 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_DGPMonomial class the
* result is always true (independent of
* the degree of the element), as it has
* no hanging nodes (being a
* discontinuous element).
*/
virtual bool hp_constraints_are_implemented () 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> &fe_other) 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>::ExcInterpolationNotImplemented
* is thrown.
*/
virtual void
get_interpolation_matrix (const FiniteElement<dim> &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 @p dofs_per_face times
* <tt>source.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>::ExcInterpolationNotImplemented.
*/
virtual void
get_face_interpolation_matrix (const FiniteElement<dim> &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 @p dofs_per_face times
* <tt>source.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>::ExcInterpolationNotImplemented.
*/
virtual void
get_subface_interpolation_matrix (const FiniteElement<dim> &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;
/**
* 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> *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);
/**
* Initialize the embedding
* matrices. Called from the
* constructor.
*/
void initialize_embedding ();
/**
* Initialize the restriction
* matrices. Called from the
* constructor.
*/
void initialize_restriction ();
};
/*@}*/
DEAL_II_NAMESPACE_CLOSE
#endif
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