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// $Id: polynomials_bdm.h 20467 2010-01-28 16:28:25Z bangerth $
// Version: $Name$
//
// Copyright (C) 2004, 2005, 2006, 2009, 2010 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_BDM_h
#define __deal2__polynomials_BDM_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/table.h>
#include <base/thread_management.h>
#include <vector>
DEAL_II_NAMESPACE_OPEN
/**
* @addtogroup Polynomials
* @{
*/
/**
* This class implements the <i>H<sup>div</sup></i>-conforming,
* vector-valued Brezzi-Douglas-Marini polynomials as described in the
* book by Brezzi and Fortin.
*
* These polynomial spaces are based on the space
* <i>P<sub>k</sub></i>, realized by a PolynomialSpace constructed
* with Legendre polynomials. Since these shape functions are not
* sufficient, additional functions are added. These are the following
* vector valued polynomials:
*
* <dl>
* <dt> In 2D:
* <dd> The 2D-curl of the functions <i>x<sup>k+1</sup>y</i>
* and <i>xy<sup>k+1</sup></i>.
* <dt>In 3D:
* <dd> For any <i>i=0,...,k</i> the curls of
* <i>(0,0,xy<sup>i+1</sup>z<sup>k-i</sup>)</i>,
* <i>(x<sup>k-i</sup>yz<sup>i+1</sup>,0,0)</i> and
* <i>(0,x<sup>i+1</sup>y<sup>k-i</sup>z,0)</i>
* </dl>
*
* @todo Second derivatives in 3D are missing.
*
* @author Guido Kanschat, 2003, 2005, 2009
*/
template <int dim>
class PolynomialsBDM
{
public:
/**
* Constructor. Creates all basis
* functions for BDM polynomials
* of given degree.
*
* @arg k: the degree of the
* BDM-space, which is the degree
* of the largest complete
* polynomial space
* <i>P<sub>k</sub></i> contained
* in the BDM-space.
*/
PolynomialsBDM (const unsigned int k);
/**
* Computes the value and the
* first and second derivatives
* of each BDM
* 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 BDM polynomials.
*/
unsigned int n () const;
/**
* Returns the degree of the BDM
* space, which is one less than
* the highest polynomial degree.
*/
unsigned int degree () const;
/**
* Return the number of
* polynomials in the space
* <TT>BDM(degree)</tt> without
* requiring to build an object
* of PolynomialsBDM. This is
* required by the FiniteElement
* classes.
*/
static unsigned int compute_n_pols(unsigned int degree);
private:
/**
* An object representing the
* polynomial space used
* here. The constructor fills
* this with the monomial basis.
*/
const PolynomialSpace<dim> polynomial_space;
/**
* Storage for monomials. In 2D,
* this is just the polynomial of
* order <i>k</i>. In 3D, we
* need all polynomials from
* degree zero to <i>k</i>.
*/
std::vector<Polynomials::Polynomial<double> > monomials;
/**
* Number of BDM
* polynomials.
*/
unsigned int n_pols;
/**
* A mutex that guards the
* following scratch arrays.
*/
mutable Threads::Mutex mutex;
/**
* Auxiliary memory.
*/
mutable std::vector<double> p_values;
/**
* Auxiliary memory.
*/
mutable std::vector<Tensor<1,dim> > p_grads;
/**
* Auxiliary memory.
*/
mutable std::vector<Tensor<2,dim> > p_grad_grads;
};
/** @} */
template <int dim>
inline unsigned int
PolynomialsBDM<dim>::n() const
{
return n_pols;
}
template <int dim>
inline unsigned int
PolynomialsBDM<dim>::degree() const
{
return polynomial_space.degree();
}
DEAL_II_NAMESPACE_CLOSE
#endif
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