/usr/include/deal.II/base/polynomials

// ---------------------------------------------------------------------
// $Id: polynomials_nedelec.h 31791 2013-11-25 10:36:38Z felix.gruber $
//
// Copyright (C) 2010 - 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__polynomials_nedelec_h
#define __deal2__polynomials_nedelec_h


#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/tensor.h>
#include <deal.II/base/point.h>
#include <deal.II/base/polynomial.h>
#include <deal.II/base/polynomial_space.h>
#include <deal.II/base/tensor_product_polynomials.h>
#include <deal.II/base/table.h>

#include <vector>

DEAL_II_NAMESPACE_OPEN

/**
 * This class implements the first family <i>H<sup>curl</sup></i>-conforming,
 * vector-valued polynomials, proposed by J.-C. Nédélec in 1980
 * (Numer. Math. 35).
 *
 * The Nédélec polynomials are constructed such that the curl
 * 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 two directions,
 * yielding the polynomial spaces <i>(Q<sub>k,k+1</sub>,
 * Q<sub>k+1,k</sub>)</i> and <i>(Q<sub>k,k+1,k+1</sub>,
 * Q<sub>k+1,k,k+1</sub>, Q<sub>k+1,k+1,k</sub>)</i> in 2D and 3D, resp.
 *
 * @ingroup Polynomials
 * @author Markus Bürg
 * @date 2009, 2010
 */
template <int dim>
class PolynomialsNedelec
{
public:
  /**
   * Constructor. Creates all basis
   * functions for Nédélec polynomials
   * of given degree.
   *
   * @arg k: the degree of the
   * Nédélec space, which is the degree
   * of the largest tensor product
   * polynomial space
   * <i>Q<sub>k</sub></i> contained.
   */
  PolynomialsNedelec (const unsigned int k);

  /**
   * Computes the value and the
   * first and second derivatives
   * of each Nédélec
   * 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 Nédélec
     * polynomials.
   */
  unsigned int n () const;

  /**
   * Returns the degree of the Nédélec
   * space, which is one less than
   * the highest polynomial degree.
   */
  unsigned int degree () const;

  /**
   * Return the name of the space,
   * which is <tt>Nedelec</tt>.
   */
  std::string name () const;

  /**
   * Return the number of
   * polynomials in the space
   * <TT>N(degree)</tt> without
   * requiring to build an object
   * of PolynomialsNedelec. 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 Nédélec 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 PolynomialsNedelec<dim>::n () const
{
  return n_pols;
}


template <int dim>
inline unsigned int PolynomialsNedelec<dim>::degree () const
{
  return my_degree;
}


template <int dim>
inline std::string
PolynomialsNedelec<dim>::name() const
{
  return "Nedelec";
}


DEAL_II_NAMESPACE_CLOSE

#endif
libdeal.ii-dev 8.1.0-4 / usr / include / deal.II / base / polynomials_nedelec.h
