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//
// Copyright (C) 2004 - 2015 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__fe_dgp_monomial_h
#define dealii__fe_dgp_monomial_h
#include <deal.II/base/config.h>
#include <deal.II/base/polynomials_p.h>
#include <deal.II/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 combination of them. Consequently, the linear functions are not
* within the range of the mapped $P_1$ polynomials.
*
*
* <h3>Visualization of shape functions</h3> In 2d, the shape functions of
* this element look as follows.
*
* <h4>$P_0$ element</h4>
*
* <table> <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P1/P1_DGPMonomial_shape0000.png
* </td>
*
* <td align="center"> </td> </tr> <tr> <td align="center"> $P_0$ element,
* shape function 0 </td>
*
* <td align="center"></tr> </table>
*
* <h4>$P_1$ element</h4>
*
* <table> <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P1/P1_DGPMonomial_shape0000.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P1/P1_DGPMonomial_shape0001.png
* </td> </tr> <tr> <td align="center"> $P_1$ element, shape function 0 </td>
*
* <td align="center"> $P_1$ element, shape function 1 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P1/P1_DGPMonomial_shape0002.png
* </td>
*
* <td align="center"> </td> </tr> <tr> <td align="center"> $P_1$ element,
* shape function 2 </td>
*
* <td align="center"></td> </tr> </table>
*
*
* <h4>$P_2$ element</h4>
*
* <table> <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P2/P2_DGPMonomial_shape0000.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P2/P2_DGPMonomial_shape0001.png
* </td> </tr> <tr> <td align="center"> $P_2$ element, shape function 0 </td>
*
* <td align="center"> $P_2$ element, shape function 1 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P2/P2_DGPMonomial_shape0002.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P2/P2_DGPMonomial_shape0003.png
* </td> </tr> <tr> <td align="center"> $P_2$ element, shape function 2 </td>
*
* <td align="center"> $P_2$ element, shape function 3 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P2/P2_DGPMonomial_shape0004.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P2/P2_DGPMonomial_shape0005.png
* </td> </tr> <tr> <td align="center"> $P_2$ element, shape function 4 </td>
*
* <td align="center"> $P_2$ element, shape function 5 </td> </tr> </table>
*
*
* <h4>$P_3$ element</h4>
*
* <table> <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0000.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0001.png
* </td> </tr> <tr> <td align="center"> $P_3$ element, shape function 0 </td>
*
* <td align="center"> $P_3$ element, shape function 1 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0002.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0003.png
* </td> </tr> <tr> <td align="center"> $P_3$ element, shape function 2 </td>
*
* <td align="center"> $P_3$ element, shape function 3 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0004.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0005.png
* </td> </tr> <tr> <td align="center"> $P_3$ element, shape function 4 </td>
*
* <td align="center"> $P_3$ element, shape function 5 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0006.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0007.png
* </td> </tr> <tr> <td align="center"> $P_3$ element, shape function 6 </td>
*
* <td align="center"> $P_3$ element, shape function 7 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0008.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P3/P3_DGPMonomial_shape0009.png
* </td> </tr> <tr> <td align="center"> $P_3$ element, shape function 8 </td>
*
* <td align="center"> $P_3$ element, shape function 9 </td> </tr> </table>
*
*
* <h4>$P_4$ element</h4> <table> <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0000.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0001.png
* </td> </tr> <tr> <td align="center"> $P_4$ element, shape function 0 </td>
*
* <td align="center"> $P_4$ element, shape function 1 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0002.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0003.png
* </td> </tr> <tr> <td align="center"> $P_4$ element, shape function 2 </td>
*
* <td align="center"> $P_4$ element, shape function 3 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0004.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0005.png
* </td> </tr> <tr> <td align="center"> $P_4$ element, shape function 4 </td>
*
* <td align="center"> $P_4$ element, shape function 5 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0006.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0007.png
* </td> </tr> <tr> <td align="center"> $P_4$ element, shape function 6 </td>
*
* <td align="center"> $P_4$ element, shape function 7 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0008.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0009.png
* </td> </tr> <tr> <td align="center"> $P_4$ element, shape function 8 </td>
*
* <td align="center"> $P_4$ element, shape function 9 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0010.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0011.png
* </td> </tr> <tr> <td align="center"> $P_4$ element, shape function 10 </td>
*
* <td align="center"> $P_4$ element, shape function 11 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0012.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0013.png
* </td> </tr> <tr> <td align="center"> $P_4$ element, shape function 12 </td>
*
* <td align="center"> $P_4$ element, shape function 13 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/DGPMonomial/P4/P4_DGPMonomial_shape0014.png
* </td>
*
* <td align="center"> </td> </tr> <tr> <td align="center"> $P_4$ element,
* shape function 14 </td>
*
* <td align="center"></td> </tr> </table>
*
* @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,
* both numbered between zero and the corresponding value of dofs_per_vertex
* of the two finite elements. 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;
/**
* This function returns @p true, if the shape function @p shape_index has
* non-zero function values somewhere on the face @p face_index.
*/
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 std::size_t 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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