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// $Id: fe_raviart_thomas.h 31893 2013-12-05 03:00:41Z heister $
//
// Copyright (C) 2003 - 2013 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 __deal2__fe_raviart_thomas_h
#define __deal2__fe_raviart_thomas_h
#include <deal.II/base/config.h>
#include <deal.II/base/table.h>
#include <deal.II/base/polynomials_raviart_thomas.h>
#include <deal.II/base/polynomial.h>
#include <deal.II/base/tensor_product_polynomials.h>
#include <deal.II/base/geometry_info.h>
#include <deal.II/fe/fe.h>
#include <deal.II/fe/fe_poly_tensor.h>
#include <vector>
DEAL_II_NAMESPACE_OPEN
template <int dim, int spacedim> class MappingQ;
/*!@addtogroup fe */
/*@{*/
/**
* Implementation of Raviart-Thomas (RT) elements, conforming with the
* space H<sup>div</sup>. These elements generate vector fields with
* normal components continuous between mesh cells.
*
* We follow the usual definition of the degree of RT elements, which
* denotes the polynomial degree of the largest complete polynomial
* subspace contained in the RT space. Then, approximation order of
* the function itself is <i>degree+1</i>, as with usual polynomial
* spaces. The numbering so chosen implies the sequence
* @f[
* Q_{k+1}
* \stackrel{\text{grad}}{\rightarrow}
* \text{Nedelec}_k
* \stackrel{\text{curl}}{\rightarrow}
* \text{RaviartThomas}_k
* \stackrel{\text{div}}{\rightarrow}
* DGQ_{k}
* @f]
* The lowest order element is consequently FE_RaviartThomas(0).
*
* This class is not implemented for the codimension one case
* (<tt>spacedim != dim</tt>).
*
* @todo Even if this element is implemented for two and three space
* dimensions, the definition of the node values relies on
* consistently oriented faces in 3D. Therefore, care should be taken
* on complicated meshes.
*
* <h3>Interpolation</h3>
*
* The @ref GlossInterpolation "interpolation" operators associated
* with the RT element are constructed such that interpolation and
* computing the divergence are commuting operations. We require this
* from interpolating arbitrary functions as well as the #restriction
* matrices. It can be achieved by two interpolation schemes, the
* simplified one in FE_RaviartThomasNodal and the original one here:
*
* <h4>Node values on edges/faces</h4>
*
* On edges or faces, the @ref GlossNodes "node values" are the moments of
* the normal component of the interpolated function with respect to
* the traces of the RT polynomials. Since the normal trace of the RT
* space of degree <i>k</i> on an edge/face is the space
* <i>Q<sub>k</sub></i>, the moments are taken with respect to this
* space.
*
* <h4>Interior node values</h4>
*
* Higher order RT spaces have interior nodes. These are moments taken
* with respect to the gradient of functions in <i>Q<sub>k</sub></i>
* on the cell (this space is the matching space for RT<sub>k</sub> in
* a mixed formulation).
*
* <h4>Generalized support points</h4>
*
* The node values above rely on integrals, which will be computed by
* quadrature rules themselves. The generalized support points are a
* set of points such that this quadrature can be performed with
* sufficient accuracy. The points needed are thode of
* QGauss<sub>k+1</sub> on each face as well as QGauss<sub>k</sub> in
* the interior of the cell (or none for RT<sub>0</sub>).
*
*
* @author Guido Kanschat, 2005, based on previous Work by Wolfgang Bangerth
*/
template <int dim>
class FE_RaviartThomas
:
public FE_PolyTensor<PolynomialsRaviartThomas<dim>, dim>
{
public:
/**
* Constructor for the Raviart-Thomas
* element of degree @p p.
*/
FE_RaviartThomas (const unsigned int p);
/**
* Return a string that uniquely
* identifies a finite
* element. This class returns
* <tt>FE_RaviartThomas<dim>(degree)</tt>, with
* @p dim and @p degree
* replaced by appropriate
* values.
*/
virtual std::string get_name () const;
/**
* Check whether a shape function
* may be non-zero on a face.
*
* Right now, this is only
* implemented for RT0 in
* 1D. Otherwise, returns always
* @p true.
*/
virtual bool has_support_on_face (const unsigned int shape_index,
const unsigned int face_index) const;
virtual void interpolate(std::vector<double> &local_dofs,
const std::vector<double> &values) const;
virtual void interpolate(std::vector<double> &local_dofs,
const std::vector<Vector<double> > &values,
unsigned int offset = 0) const;
virtual void interpolate(
std::vector<double> &local_dofs,
const VectorSlice<const std::vector<std::vector<double> > > &values) const;
virtual std::size_t memory_consumption () const;
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 @p
* generalized_support_points
* field of the FiniteElement
* class and fill the tables with
* interpolation weights
* (#boundary_weights and
* #interior_weights). Called
* from the constructor.
*/
void initialize_support_points (const unsigned int rt_degree);
/**
* Initialize the interpolation
* from functions on refined mesh
* cells onto the father
* cell. According to the
* philosophy of the
* Raviart-Thomas element, this
* restriction operator preserves
* the divergence of a function
* weakly.
*/
void initialize_restriction ();
/**
* Fields of cell-independent data.
*
* For information about the
* general purpose of this class,
* see the documentation of the
* base class.
*/
class InternalData : public FiniteElement<dim>::InternalDataBase
{
public:
/**
* Array with shape function
* values in quadrature
* points. There is one row
* for each shape function,
* containing values for each
* quadrature point. Since
* the shape functions are
* vector-valued (with as
* many components as there
* are space dimensions), the
* value is a tensor.
*
* In this array, we store
* the values of the shape
* function in the quadrature
* points on the unit
* cell. The transformation
* to the real space cell is
* then simply done by
* multiplication with the
* Jacobian of the mapping.
*/
std::vector<std::vector<Tensor<1,dim> > > shape_values;
/**
* Array with shape function
* gradients in quadrature
* points. There is one
* row for each shape
* function, containing
* values for each quadrature
* point.
*
* We store the gradients in
* the quadrature points on
* the unit cell. We then
* only have to apply the
* transformation (which is a
* matrix-vector
* multiplication) when
* visiting an actual cell.
*/
std::vector<std::vector<Tensor<2,dim> > > shape_gradients;
};
/**
* These are the factors
* multiplied to a function in
* the
* #generalized_face_support_points
* when computing the
* integration. They are
* organized such that there is
* one row for each generalized
* face support point and one
* column for each degree of
* freedom on the face.
*
* See the @ref GlossGeneralizedSupport "glossary entry on generalized support points"
* for more information.
*/
Table<2, double> boundary_weights;
/**
* Precomputed factors for
* interpolation of interior
* degrees of freedom. The
* rationale for this Table is
* the same as for
* #boundary_weights. Only, this
* table has a third coordinate
* for the space direction of the
* component evaluated.
*/
Table<3, double> interior_weights;
/**
* Allow access from other
* dimensions.
*/
template <int dim1> friend class FE_RaviartThomas;
};
/**
* The Raviart-Thomas elements with node functionals defined as point
* values in Gauss points.
*
* <h3>Description of node values</h3>
*
* For this Raviart-Thomas element, the node values are not cell and
* face moments with respect to certain polynomials, but the values in
* quadrature points. Following the general scheme for numbering
* degrees of freedom, the node values on edges are first, edge by
* edge, according to the natural ordering of the edges of a cell. The
* interior degrees of freedom are last.
*
* For an RT-element of degree <i>k</i>, we choose
* <i>(k+1)<sup>d-1</sup></i> Gauss points on each face. These points
* are ordered lexicographically with respect to the orientation of
* the face. This way, the normal component which is in
* <i>Q<sub>k</sub></i> is uniquely determined. Furthermore, since
* this Gauss-formula is exact on <i>Q<sub>2k+1</sub></i>, these node
* values correspond to the exact integration of the moments of the
* RT-space.
*
* In the interior of the cells, the moments are with respect to an
* anisotropic <i>Q<sub>k</sub></i> space, where the test functions
* are one degree lower in the direction corresponding to the vector
* component under consideration. This is emulated by using an
* anisotropic Gauss formula for integration.
*
* @todo The current implementation is for Cartesian meshes
* only. You must use MappingCartesian.
*
* @todo Even if this element is implemented for two and three space
* dimensions, the definition of the node values relies on
* consistently oriented faces in 3D. Therefore, care should be taken
* on complicated meshes.
*
* @note The degree stored in the member variable
* FiniteElementData<dim>::degree is higher by one than the
* constructor argument!
*
* @author Guido Kanschat, 2005, Zhu Liang, 2008
*/
template <int dim>
class FE_RaviartThomasNodal
:
public FE_PolyTensor<PolynomialsRaviartThomas<dim>, dim>
{
public:
/**
* Constructor for the Raviart-Thomas
* element of degree @p p.
*/
FE_RaviartThomasNodal (const unsigned int p);
/**
* Return a string that uniquely
* identifies a finite
* element. This class returns
* <tt>FE_RaviartThomasNodal<dim>(degree)</tt>, with
* @p dim and @p degree
* replaced by appropriate
* values.
*/
virtual std::string get_name () const;
virtual FiniteElement<dim> *clone () const;
virtual void interpolate(std::vector<double> &local_dofs,
const std::vector<double> &values) const;
virtual void interpolate(std::vector<double> &local_dofs,
const std::vector<Vector<double> > &values,
unsigned int offset = 0) const;
virtual void interpolate(
std::vector<double> &local_dofs,
const VectorSlice<const std::vector<std::vector<double> > > &values) const;
virtual void get_face_interpolation_matrix (const FiniteElement<dim> &source,
FullMatrix<double> &matrix) const;
virtual void get_subface_interpolation_matrix (const FiniteElement<dim> &source,
const unsigned int subface,
FullMatrix<double> &matrix) const;
virtual bool hp_constraints_are_implemented () const;
virtual std::vector<std::pair<unsigned int, unsigned int> >
hp_vertex_dof_identities (const FiniteElement<dim> &fe_other) const;
virtual std::vector<std::pair<unsigned int, unsigned int> >
hp_line_dof_identities (const FiniteElement<dim> &fe_other) const;
virtual std::vector<std::pair<unsigned int, unsigned int> >
hp_quad_dof_identities (const FiniteElement<dim> &fe_other) const;
virtual FiniteElementDomination::Domination
compare_for_face_domination (const FiniteElement<dim> &fe_other) 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);
/**
* Compute the vector used for
* the
* @p restriction_is_additive
* field passed to the base
* class's constructor.
*/
static std::vector<bool>
get_ria_vector (const unsigned int degree);
/**
* Check whether a shape function
* may be non-zero on a face.
*
* Right now, this is only
* implemented for RT0 in
* 1D. Otherwise, returns always
* @p true.
*/
virtual bool has_support_on_face (const unsigned int shape_index,
const unsigned int face_index) const;
/**
* Initialize the
* FiniteElement<dim>::generalized_support_points
* and FiniteElement<dim>::generalized_face_support_points
* fields. Called from the
* constructor.
*
* See the @ref GlossGeneralizedSupport "glossary entry on generalized support points"
* for more information.
*/
void initialize_support_points (const unsigned int rt_degree);
};
/*@}*/
/* -------------- declaration of explicit specializations ------------- */
#ifndef DOXYGEN
template <>
void
FE_RaviartThomas<1>::initialize_restriction();
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#endif
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