tensor_product_polynomials_bubbles.h

// ---------------------------------------------------------------------
// @f$Id@f$
//
// Copyright (C) 2012 - 2016 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.
//
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#ifndef dealii__tensor_product_polynomials_bubbles_h
#define dealii__tensor_product_polynomials_bubbles_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/tensor_product_polynomials.h>
#include <deal.II/base/utilities.h>

#include <vector>

DEAL_II_NAMESPACE_OPEN


template <int dim>
class TensorProductPolynomialsBubbles : public TensorProductPolynomials<dim>
{
public:
  static const unsigned int dimension = dim;

  template <class Pol>
  TensorProductPolynomialsBubbles (const std::vector<Pol> &pols);

  void compute (const Point<dim>            &unit_point,
                std::vector<double>         &values,
                std::vector<Tensor<1,dim> > &grads,
                std::vector<Tensor<2,dim> > &grad_grads,
                std::vector<Tensor<3,dim> > &third_derivatives,
                std::vector<Tensor<4,dim> > &fourth_derivatives) const;

  double compute_value (const unsigned int i,
                        const Point<dim> &p) const;

  template <int order>
  Tensor<order,dim> compute_derivative (const unsigned int i,
                                        const Point<dim> &p) const;

  Tensor<1,dim> compute_grad (const unsigned int i,
                              const Point<dim> &p) const;

  Tensor<2,dim> compute_grad_grad (const unsigned int i,
                                   const Point<dim> &p) const;

  unsigned int n () const;
};

/* ---------------- template and inline functions ---------- */

#ifndef DOXYGEN

template <int dim>
template <class Pol>
inline
TensorProductPolynomialsBubbles<dim>::
TensorProductPolynomialsBubbles(const std::vector<Pol> &pols)
  :
  TensorProductPolynomials<dim>(pols)
{
  const unsigned int q_degree = this->polynomials.size()-1;
  const unsigned int n_bubbles = ((q_degree<=1)?1:dim);
  // append index for renumbering
  for (unsigned int i=0; i<n_bubbles; ++i)
    {
      this->index_map.push_back(i+this->n_tensor_pols);
      this->index_map_inverse.push_back(i+this->n_tensor_pols);
    }
}



template <int dim>
inline
unsigned int
TensorProductPolynomialsBubbles<dim>::n() const
{
  return this->n_tensor_pols+dim;
}



template <>
inline
unsigned int
TensorProductPolynomialsBubbles<0>::n() const
{
  return numbers::invalid_unsigned_int;
}

template <int dim>
template <int order>
Tensor<order,dim>
TensorProductPolynomialsBubbles<dim>::compute_derivative (const unsigned int i,
                                                          const Point<dim> &p) const
{
  const unsigned int q_degree = this->polynomials.size()-1;
  const unsigned int max_q_indices = this->n_tensor_pols;
  const unsigned int n_bubbles = ((q_degree<=1)?1:dim);
  (void)n_bubbles;
  Assert (i<max_q_indices+n_bubbles, ExcInternalError());

  // treat the regular basis functions
  if (i<max_q_indices)
    return this->TensorProductPolynomials<dim>::template compute_derivative<order>(i,p);

  const unsigned int comp = i - this->n_tensor_pols;

  Tensor<order,dim> derivative;
  switch (order)
    {
    case 1:
    {
      Tensor<1,dim> &derivative_1 = *reinterpret_cast<Tensor<1,dim>*>(&derivative);

      for (unsigned int d=0; d<dim ; ++d)
        {
          derivative_1[d] = 1.;
          //compute grad(4*\prod_{i=1}^d (x_i(1-x_i)))(p)
          for (unsigned j=0; j<dim; ++j)
            derivative_1[d] *= (d==j ? 4*(1-2*p(j)) : 4*p(j)*(1-p(j)));
          // and multiply with (2*x_i-1)^{r-1}
          for (unsigned int i=0; i<q_degree-1; ++i)
            derivative_1[d]*=2*p(comp)-1;
        }

      if (q_degree>=2)
        {
          //add \prod_{i=1}^d 4*(x_i(1-x_i))(p)
          double value=1.;
          for (unsigned int j=0; j < dim; ++j)
            value*=4*p(j)*(1-p(j));
          //and multiply with grad(2*x_i-1)^{r-1}
          double tmp=value*2*(q_degree-1);
          for (unsigned int i=0; i<q_degree-2; ++i)
            tmp*=2*p(comp)-1;
          derivative_1[comp]+=tmp;
        }

      return derivative;
    }
    case 2:
    {
      Tensor<2,dim> &derivative_2 = *reinterpret_cast<Tensor<2,dim>*>(&derivative);

      double v [dim+1][3];
      {
        for (unsigned int c=0; c<dim; ++c)
          {
            v[c][0] = 4*p(c)*(1-p(c));
            v[c][1] = 4*(1-2*p(c));
            v[c][2] = -8;
          }

        double tmp=1.;
        for (unsigned int i=0; i<q_degree-1; ++i)
          tmp *= 2*p(comp)-1;
        v[dim][0] = tmp;

        if (q_degree>=2)
          {
            double tmp = 2*(q_degree-1);
            for (unsigned int i=0; i<q_degree-2; ++i)
              tmp *= 2*p(comp)-1;
            v[dim][1] = tmp;
          }
        else
          v[dim][1] = 0.;

        if (q_degree>=3)
          {
            double tmp=4*(q_degree-2)*(q_degree-1);
            for (unsigned int i=0; i<q_degree-3; ++i)
              tmp *= 2*p(comp)-1;
            v[dim][2] = tmp;
          }
        else
          v[dim][2] = 0.;
      }

      //calculate (\partial_j \partial_k \psi) * monomial
      Tensor<2,dim> grad_grad_1;
      for (unsigned int d1=0; d1<dim; ++d1)
        for (unsigned int d2=0; d2<dim; ++d2)
          {
            grad_grad_1[d1][d2] = v[dim][0];
            for (unsigned int x=0; x<dim; ++x)
              {
                unsigned int derivative=0;
                if (d1==x || d2==x)
                  {
                    if (d1==d2)
                      derivative=2;
                    else
                      derivative=1;
                  }
                grad_grad_1[d1][d2] *= v[x][derivative];
              }
          }

      //calculate (\partial_j  \psi) *(\partial_k monomial)
      // and (\partial_k  \psi) *(\partial_j monomial)
      Tensor<2,dim> grad_grad_2;
      Tensor<2,dim> grad_grad_3;
      for (unsigned int d=0; d<dim; ++d)
        {
          grad_grad_2[d][comp] = v[dim][1];
          grad_grad_3[comp][d] = v[dim][1];
          for (unsigned int x=0; x<dim; ++x)
            {
              grad_grad_2[d][comp] *= v[x][d==x];
              grad_grad_3[comp][d] *= v[x][d==x];
            }
        }

      //calculate \psi *(\partial j \partial_k monomial) and sum
      double psi_value = 1.;
      for (unsigned int x=0; x<dim; ++x)
        psi_value *= v[x][0];

      for (unsigned int d1=0; d1<dim; ++d1)
        for (unsigned int d2=0; d2<dim; ++d2)
          derivative_2[d1][d2] = grad_grad_1[d1][d2]
                                 +grad_grad_2[d1][d2]
                                 +grad_grad_3[d1][d2];
      derivative_2[comp][comp]+=psi_value*v[dim][2];

      return derivative;
    }
    default:
    {
      Assert (false, ExcNotImplemented());
      return derivative;
    }
    }
}


#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE

#endif