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// $Id: polynomials_raviart_thomas.h 17196 2008-10-13 18:09:20Z bangerth $
// Version: $Name$
//
// Copyright (C) 2004, 2005, 2006, 2007, 2008 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__polynomials_raviart_thomas_h
#define __deal2__polynomials_raviart_thomas_h
#include <base/config.h>
#include <base/exceptions.h>
#include <base/tensor.h>
#include <base/point.h>
#include <base/polynomial.h>
#include <base/polynomial_space.h>
#include <base/tensor_product_polynomials.h>
#include <base/table.h>
#include <vector>
DEAL_II_NAMESPACE_OPEN
/**
* @addtogroup Polynomials
* @{
*/
/**
* This class implements the <i>H<sup>div</sup></i>-conforming,
* vector-valued Raviart-Thomas polynomials as described in the
* book by Brezzi and Fortin.
*
* The Raviart-Thomas polynomials are constructed such that the
* divergence is in the tensor product polynomial space
* <i>Q<sub>k</sub></i>. Therefore, the polynomial order of each
* component must be one order higher in the corresponding direction,
* yielding the polynomial spaces <i>(Q<sub>k+1,k</sub>,
* Q<sub>k,k+1</sub>)</i> and <i>(Q<sub>k+1,k,k</sub>,
* Q<sub>k,k+1,k</sub>, Q<sub>k,k,k+1</sub>)</i> in 2D and 3D, resp.
*
* @author Guido Kanschat, 2005
*/
template <int dim>
class PolynomialsRaviartThomas
{
public:
/**
* Constructor. Creates all basis
* functions for Raviart-Thomas polynomials
* of given degree.
*
* @arg k: the degree of the
* Raviart-Thomas-space, which is the degree
* of the largest tensor product
* polynomial space
* <i>Q<sub>k</sub></i> contained.
*/
PolynomialsRaviartThomas (const unsigned int k);
/**
* Computes the value and the
* first and second derivatives
* of each Raviart-Thomas
* polynomial at @p unit_point.
*
* The size of the vectors must
* either be zero or equal
* <tt>n()</tt>. In the
* first case, the function will
* not compute these values.
*
* If you need values or
* derivatives of all tensor
* product polynomials then use
* this function, rather than
* using any of the
* <tt>compute_value</tt>,
* <tt>compute_grad</tt> or
* <tt>compute_grad_grad</tt>
* functions, see below, in a
* loop over all tensor product
* polynomials.
*/
void compute (const Point<dim> &unit_point,
std::vector<Tensor<1,dim> > &values,
std::vector<Tensor<2,dim> > &grads,
std::vector<Tensor<3,dim> > &grad_grads) const;
/**
* Returns the number of Raviart-Thomas polynomials.
*/
unsigned int n () const;
/**
* Returns the degree of the Raviart-Thomas
* space, which is one less than
* the highest polynomial degree.
*/
unsigned int degree () const;
/**
* Return the number of
* polynomials in the space
* <TT>RT(degree)</tt> without
* requiring to build an object
* of PolynomialsRaviartThomas. This is
* required by the FiniteElement
* classes.
*/
static unsigned int compute_n_pols(unsigned int degree);
private:
/**
* The degree of this object as
* given to the constructor.
*/
const unsigned int my_degree;
/**
* An object representing the
* polynomial space for a single
* component. We can re-use it by
* rotating the coordinates of
* the evaluation point.
*/
const AnisotropicPolynomials<dim> polynomial_space;
/**
* Number of Raviart-Thomas
* polynomials.
*/
const unsigned int n_pols;
/**
* A static member function that
* creates the polynomial space
* we use to initialize the
* #polynomial_space member
* variable.
*/
static
std::vector<std::vector< Polynomials::Polynomial< double > > >
create_polynomials (const unsigned int k);
};
/** @} */
template <int dim>
inline unsigned int
PolynomialsRaviartThomas<dim>::n() const
{
return n_pols;
}
template <int dim>
inline unsigned int
PolynomialsRaviartThomas<dim>::degree() const
{
return my_degree;
}
DEAL_II_NAMESPACE_CLOSE
#endif
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