pml_helmholtz_elements.cc

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//LIC// ====================================================================
//LIC// This file forms part of oomph-lib, the object-oriented, 
//LIC// multi-physics finite-element library, available 
//LIC// at http://www.oomph-lib.org.
//LIC// 
//LIC//    Version 1.0; svn revision $LastChangedRevision$
//LIC//
//LIC// $LastChangedDate$
//LIC// 
//LIC// Copyright (C) 2006-2016 Matthias Heil and Andrew Hazel
//LIC// 
//LIC// This library is free software; you can redistribute it and/or
//LIC// modify it under the terms of the GNU Lesser General Public
//LIC// License as published by the Free Software Foundation; either
//LIC// version 2.1 of the License, or (at your option) any later version.
//LIC// 
//LIC// This library is distributed in the hope that it will be useful,
//LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
//LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
//LIC// Lesser General Public License for more details.
//LIC// 
//LIC// You should have received a copy of the GNU Lesser General Public
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//LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
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//LIC// 
//LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
//LIC// 
//LIC//====================================================================
//Non-inline functions for gen Helmholtz elements
#include "pml_helmholtz_elements.h"



namespace oomph
{

//======================================================================
/// Set the data for the number of Variables at each node, always two
/// (real and imag part) in every case
//======================================================================
 template<unsigned DIM, unsigned NNODE_1D, class PML_ELEMENT>
 const unsigned QPMLHelmholtzElement<DIM,NNODE_1D,PML_ELEMENT>::Initial_Nvalue = 2;

 /// PML Helmholtz equations static data, so that by default we can point to a 0
 template <unsigned DIM>
 double PMLHelmholtzEquationsBase<DIM>::Default_Physical_Constant_Value = 0.0; // faire: why is this not const (wasn't in navier_stokes.cc), but it is above

//======================================================================
/// Compute element residual Vector and/or element Jacobian matrix
///
/// flag=1: compute both
/// flag=0: compute only residual Vector
///
/// Pure version without hanging nodes
//======================================================================
template <unsigned DIM, class PML_ELEMENT>
void  PMLHelmholtzEquations<DIM,PML_ELEMENT>::
fill_in_generic_residual_contribution_helmholtz(Vector<double> &residuals,
                                                DenseMatrix<double> &jacobian,
                                                const unsigned& flag)
{
 //Find out how many nodes there are
 const unsigned n_node = this->nnode();

 //Set up memory for the shape and test functions
 Shape psi(n_node), test(n_node);
 DShape dpsidx(n_node,DIM), dtestdx(n_node,DIM);

 //Set the value of n_intpt
 const unsigned n_intpt = integral_pt()->nweight();

 //Integers to store the local equation and unknown numbers
 int local_eqn_real=0, local_unknown_real=0;
 int local_eqn_imag=0, local_unknown_imag=0;

 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {
   //Get the integral weight
   double w = integral_pt()->weight(ipt);

   //Call the derivatives of the shape and test functions
   double J = this->dshape_and_dtest_eulerian_at_knot_helmholtz(
     ipt, psi, dpsidx, test, dtestdx);

   //Premultiply the weights and the Jacobian
   double W = w*J;

   //Calculate local values of unknown
   //Allocate and initialise to zero
   std::complex<double> interpolated_u(0.0,0.0);
   Vector<double> interpolated_x(DIM,0.0);
   Vector< std::complex<double> > interpolated_dudx(DIM);

   //Calculate function value and derivatives:
   //-----------------------------------------
   // Loop over nodes
   for(unsigned l=0;l<n_node;l++)
    {
     // Loop over directions
     for(unsigned j=0;j<DIM;j++)
      {
       interpolated_x[j] += this->raw_nodal_position(l,j)*psi(l);
      }

     //Get the nodal value of the helmholtz unknown
     const std::complex<double>
      u_value(this->raw_nodal_value(l, this->u_index_helmholtz().real()),
              this->raw_nodal_value(l, this->u_index_helmholtz().imag()));

     //Add to the interpolated value
     interpolated_u += u_value*psi(l);

     // Loop over directions
     for(unsigned j=0;j<DIM;j++)
      {
       interpolated_dudx[j] += u_value*dpsidx(l,j);
      }
    }

   //Get source function
   //-------------------
   std::complex<double> source(0.0,0.0);
   this->get_source_helmholtz(ipt,interpolated_x,source);



   // All the PML weights that participate in the assemby process
   // are computed here. 
   // The Jacobian and Laplace matrix (and therefore the det) default to the
   // identity if the PML is disabled
   Vector<double> s(DIM);
   DiagonalComplexMatrix pml_jacobian(DIM);
   this->pml_transformation_jacobian(ipt, s, interpolated_x, pml_jacobian);
  
   // Declare a complex number for pml weights on the mass matrix bit
   // We will set this equal to the determinant of the pml Jacobian
   std::complex<double> pml_k_squared_factor = std::complex<double>(1.0,0.0);

   // Declare a diagonal matrix for pml weights on the Laplace bit
   DiagonalComplexMatrix laplace_matrix(DIM);

   this->compute_laplace_matrix_and_det(pml_jacobian, laplace_matrix,
                                        pml_k_squared_factor);

   // Alpha adjusts the pml factors, the imaginary part produces cross terms
   std::complex<double> alpha_pml_k_squared_factor = std::complex<double>(
     pml_k_squared_factor.real() - this->alpha() * pml_k_squared_factor.imag(),
     this->alpha() * pml_k_squared_factor.real() + pml_k_squared_factor.imag());

   // Assemble residuals and Jacobian
   //--------------------------------
   // Loop over the test functions
   for(unsigned l=0;l<n_node;l++)
    {

     // first, compute the real part contribution
     //-------------------------------------------

     //Get the local equation
     local_eqn_real = this->nodal_local_eqn(l,this->u_index_helmholtz().real());
     local_eqn_imag = this->nodal_local_eqn(l,this->u_index_helmholtz().imag());

     /*IF it's not a boundary condition*/
     if(local_eqn_real >= 0)
      {
       // Add body force/source term and Helmholtz bit
       residuals[local_eqn_real] +=
        ( source.real() -
          (
           alpha_pml_k_squared_factor.real() * this->k_squared() * interpolated_u.real()
          -alpha_pml_k_squared_factor.imag() * this->k_squared() * interpolated_u.imag()
          )
        )*test(l)*W;

       // The Laplace bit
       for(unsigned k=0;k<DIM;k++)
        {
         residuals[local_eqn_real] +=
         (
          laplace_matrix(k,k).real() * interpolated_dudx[k].real()
         -laplace_matrix(k,k).imag() * interpolated_dudx[k].imag()
         )*dtestdx(l,k)*W;
        }
 
       // Calculate the jacobian
       //-----------------------
       if(flag)
        {
         //Loop over the velocity shape functions again
         for(unsigned l2=0;l2<n_node;l2++)
          {
           local_unknown_real = this->nodal_local_eqn(l2,this->u_index_helmholtz().real());
           local_unknown_imag = this->nodal_local_eqn(l2,this->u_index_helmholtz().imag());

           //If at a non-zero degree of freedom add in the entry
           if(local_unknown_real >= 0)
            {
             //Add contribution to Elemental Matrix
             for(unsigned i=0;i<DIM;i++)
              {
               jacobian(local_eqn_real,local_unknown_real)
                += laplace_matrix(i,i).real() * dpsidx(l2,i)*dtestdx(l,i)*W;
              }
             // Add the helmholtz contribution
             jacobian(local_eqn_real,local_unknown_real)
              += -alpha_pml_k_squared_factor.real() * this->k_squared()*psi(l2)*test(l)*W;
            }
           //If at a non-zero degree of freedom add in the entry
           if(local_unknown_imag >= 0)
            {
             //Add contribution to Elemental Matrix
             for(unsigned i=0;i<DIM;i++)
              {
               jacobian(local_eqn_real,local_unknown_imag)
                -= laplace_matrix(i,i).imag() * dpsidx(l2,i)*dtestdx(l,i)*W;

              }
             // Add the helmholtz contribution
             jacobian(local_eqn_real,local_unknown_imag)
              += alpha_pml_k_squared_factor.imag() * this->k_squared()*psi(l2)*test(l)*W;
            }
          }
        }
      }

     // Second, compute the imaginary part contribution
     //------------------------------------------------

     //Get the local equation
     local_eqn_imag = this->nodal_local_eqn(l,this->u_index_helmholtz().imag());
     local_eqn_real = this->nodal_local_eqn(l,this->u_index_helmholtz().real());

     /*IF it's not a boundary condition*/
     if(local_eqn_imag >= 0)
      {
       // Add body force/source term and Helmholtz bit
       residuals[local_eqn_imag] +=
        ( source.imag() -
         (
           alpha_pml_k_squared_factor.imag() * this->k_squared()*interpolated_u.real()
         + alpha_pml_k_squared_factor.real() * this->k_squared()*interpolated_u.imag()
         )
        )*test(l)*W;

       // The Laplace bit
       for(unsigned k=0;k<DIM;k++)
        {
         residuals[local_eqn_imag] += (
               laplace_matrix(k,k).imag() * interpolated_dudx[k].real()
              +laplace_matrix(k,k).real() * interpolated_dudx[k].imag()
         )*dtestdx(l,k)*W;
        }

       // Calculate the jacobian
       //-----------------------
       if(flag)
        {
         //Loop over the velocity shape functions again
         for(unsigned l2=0;l2<n_node;l2++)
          {
           local_unknown_imag = this->nodal_local_eqn(l2,this->u_index_helmholtz().imag());
           local_unknown_real = this->nodal_local_eqn(l2,this->u_index_helmholtz().real());

           //If at a non-zero degree of freedom add in the entry
           if(local_unknown_imag >= 0)
            {
             //Add contribution to Elemental Matrix
             for(unsigned i=0;i<DIM;i++)
              {
               jacobian(local_eqn_imag,local_unknown_imag)
                += laplace_matrix(i,i).real() * dpsidx(l2,i)*dtestdx(l,i)*W;
              }
             // Add the helmholtz contribution
             jacobian(local_eqn_imag,local_unknown_imag)
              += -alpha_pml_k_squared_factor.real()*this->k_squared() * psi(l2)*test(l)*W;
            }
           if(local_unknown_real >= 0)
            {
             //Add contribution to Elemental Matrix
             for(unsigned i=0;i<DIM;i++)
              {
               jacobian(local_eqn_imag,local_unknown_real)
                +=laplace_matrix(i,i).imag() * dpsidx(l2,i)*dtestdx(l,i)*W;
              }
             // Add the helmholtz contribution
             jacobian(local_eqn_imag,local_unknown_real)
              += -alpha_pml_k_squared_factor.imag()*this->k_squared() * psi(l2)*test(l)*W;
            }
          }
        }
      }
    }
  } // End of loop over integration points
}


//======================================================================
/// Self-test:  Return 0 for OK
//======================================================================
template <unsigned DIM>
unsigned  PMLHelmholtzEquationsBase<DIM>::self_test()
{

 bool passed=true;

 // Check lower-level stuff
 if (FiniteElement::self_test()!=0)
  {
   passed=false;
  }

 // Return verdict
 if (passed)
  {
   return 0;
  }
 else
  {
   return 1;
  }

}


//======================================================================
/// Output function:
///
///   x,y,u_re,u_imag   or    x,y,z,u_re,u_imag
///
/// nplot points in each coordinate direction
//======================================================================
template <unsigned DIM>
void PMLHelmholtzEquationsBase<DIM>::output(std::ostream &outfile,
                                            const unsigned &nplot)
{

 //Vector of local coordinates
 Vector<double> s(DIM);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {

   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
   std::complex<double> u(interpolated_u_pml_helmholtz(s));
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << interpolated_x(s,i) << " ";
    }
   outfile << u.real() << " " << u.imag() << std::endl;

  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);

}




//======================================================================
/// Output function for real part of full time-dependent solution
///
///  u = Re( (u_r +i u_i) exp(-i omega t)
///
/// at phase angle omega t = phi.
///
///   x,y,u   or    x,y,z,u
///
/// Output at nplot points in each coordinate direction
//======================================================================
template <unsigned DIM>
void PMLHelmholtzEquationsBase<DIM>::output_real(std::ostream &outfile,
                                                 const double& phi,
                                                 const unsigned &nplot)
{
 // Vector of local coordinates
 Vector<double> s(DIM);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
   std::complex<double> u(interpolated_u_pml_helmholtz(s));
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << interpolated_x(s,i) << " ";
    }
   outfile << u.real()*cos(phi)+u.imag()*sin(phi) << std::endl;
  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);

}

//======================================================================
/// Output function for real part of full time-dependent solution
/// constructed by adding the scattered field
///
///  u = Re( (u_r +i u_i) exp(-i omega t)
///
/// at phase angle omega t = phi computed here, to the corresponding
/// incoming wave specified via the function pointer.
///
///   x,y,u   or    x,y,z,u
///
/// Output at nplot points in each coordinate direction
//======================================================================
template <unsigned DIM>
void PMLHelmholtzEquationsBase<DIM>::output_total_real(
 std::ostream &outfile,
 FiniteElement::SteadyExactSolutionFctPt incoming_wave_fct_pt,
 const double& phi,
 const unsigned &nplot)
{
 // Vector of local coordinates
 Vector<double> s(DIM);

 // Vector for coordintes
 Vector<double> x(DIM);

 // Real and imag part of incoming wave
 Vector<double> incoming_soln(2);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
   std::complex<double> u(interpolated_u_pml_helmholtz(s));

   // Get x position as Vector
   interpolated_x(s,x);

   // Get exact solution at this point
   (*incoming_wave_fct_pt)(x,incoming_soln);

   for(unsigned i=0;i<DIM;i++)
    {
     outfile << interpolated_x(s,i) << " ";
    }

   outfile << (u.real()+incoming_soln[0])*cos(phi)+
    (u.imag()+incoming_soln[1])*sin(phi) << std::endl;
  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);

}

//======================================================================
/// Output function for imaginary part of full time-dependent solution
///
///  u = Im( (u_r +i u_i) exp(-i omega t))
///
/// at phase angle omega t = phi.
///
///   x,y,u   or    x,y,z,u
///
/// Output at nplot points in each coordinate direction
//======================================================================
template <unsigned DIM>
void  PMLHelmholtzEquationsBase<DIM>::output_imag(std::ostream &outfile,
                                           const double& phi,
                                           const unsigned &nplot)
{
 // Vector of local coordinates
 Vector<double> s(DIM);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
   std::complex<double> u(interpolated_u_pml_helmholtz(s));
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << interpolated_x(s,i) << " ";
    }
   outfile << u.imag()*cos(phi)-u.real()*sin(phi) << std::endl;
  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);

}


//======================================================================
/// C-style output function:
///
///   x,y,u   or    x,y,z,u
///
/// nplot points in each coordinate direction
//======================================================================
template <unsigned DIM>
void PMLHelmholtzEquationsBase<DIM>::output(FILE* file_pt,
                                            const unsigned &nplot)
{
 // Vector of local coordinates
 Vector<double> s(DIM);

 // Tecplot header info
 fprintf(file_pt,"%s",tecplot_zone_string(nplot).c_str());

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
   std::complex<double> u(interpolated_u_pml_helmholtz(s));

   for(unsigned i=0;i<DIM;i++)
    {
     fprintf(file_pt,"%g ",interpolated_x(s,i));
    }

   for(unsigned i=0;i<DIM;i++)
    {
     fprintf(file_pt,"%g ",interpolated_x(s,i));
    }
   fprintf(file_pt,"%g ",u.real());
   fprintf(file_pt,"%g \n",u.imag());

  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(file_pt,nplot);
}



//======================================================================
 /// Output exact solution
 ///
 /// Solution is provided via function pointer.
 /// Plot at a given number of plot points.
 ///
 ///   x,y,u_exact    or    x,y,z,u_exact
//======================================================================
template <unsigned DIM>
void PMLHelmholtzEquationsBase<DIM>::
output_fct(std::ostream &outfile,const unsigned &nplot,
           FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
{
 // Vector of local coordinates
 Vector<double> s(DIM);

 // Vector for coordinates
 Vector<double> x(DIM);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);

 // Exact solution Vector
 Vector<double> exact_soln(2);

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);

   // Get x position as Vector
   interpolated_x(s,x);

   // Get exact solution at this point
   (*exact_soln_pt)(x,exact_soln);

   // Output x,y,...,u_exact
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << x[i] << " ";
    }
   outfile << exact_soln[0] << " " << exact_soln[1] << std::endl;
  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);
}



//======================================================================
/// Output function for real part of full time-dependent fct
///
///  u = Re( (u_r +i u_i) exp(-i omega t)
///
/// at phase angle omega t = phi.
///
///   x,y,u   or    x,y,z,u
///
/// Output at nplot points in each coordinate direction
//======================================================================
template <unsigned DIM>
void PMLHelmholtzEquationsBase<DIM>::output_real_fct(
 std::ostream &outfile,
 const double& phi,
 const unsigned &nplot,
 FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
{
 // Vector of local coordinates
 Vector<double> s(DIM);

 // Vector for coordinates
 Vector<double> x(DIM);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);

 // Exact solution Vector
 Vector<double> exact_soln(2);

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);

   // Get x position as Vector
   interpolated_x(s,x);

   // Get exact solution at this point
   (*exact_soln_pt)(x,exact_soln);

   // Output x,y,...,u_exact
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << x[i] << " ";
    }
   outfile << exact_soln[0]*cos(phi)+ exact_soln[1]*sin(phi) << std::endl;
  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);
}

//======================================================================
/// Output function for imaginary part of full time-dependent fct
///
///  u = Im( (u_r +i u_i) exp(-i omega t))
///
/// at phase angle omega t = phi.
///
///   x,y,u   or    x,y,z,u
///
/// Output at nplot points in each coordinate direction
//======================================================================
template <unsigned DIM>
void PMLHelmholtzEquationsBase<DIM>::output_imag_fct(
 std::ostream &outfile,
 const double& phi,
 const unsigned &nplot,
 FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
{
 //Vector of local coordinates
 Vector<double> s(DIM);

  // Vector for coordintes
  Vector<double> x(DIM);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);

 // Exact solution Vector
 Vector<double> exact_soln(2);

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {

   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);

   // Get x position as Vector
   interpolated_x(s,x);

   // Get exact solution at this point
   (*exact_soln_pt)(x,exact_soln);

   //Output x,y,...,u_exact
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << x[i] << " ";
    }
   outfile << exact_soln[1]*cos(phi) - exact_soln[0]*sin(phi) << std::endl;
  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);
}




//======================================================================
 /// Validate against exact solution
 ///
 /// Solution is provided via function pointer.
 /// Plot error at a given number of plot points.
 ///
//======================================================================
template <unsigned DIM>
void PMLHelmholtzEquationsBase<DIM>::compute_error(
  std::ostream &outfile,
  FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
  double& error,
  double& norm)
{

 // Initialise
 error=0.0;
 norm=0.0;

 //Vector of local coordinates
 Vector<double> s(DIM);

 // Vector for coordintes
 Vector<double> x(DIM);

 //Find out how many nodes there are in the element
 unsigned n_node = nnode();

 Shape psi(n_node);

 //Set the value of n_intpt
 unsigned n_intpt = integral_pt()->nweight();

 // Tecplot
 outfile << "ZONE" << std::endl;

 // Exact solution Vector
 Vector<double> exact_soln(2);

 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {

   //Assign values of s
   for(unsigned i=0;i<DIM;i++)
    {
     s[i] = integral_pt()->knot(ipt,i);
    }

   //Get the integral weight
   double w = integral_pt()->weight(ipt);

   // Get jacobian of mapping
   double J=J_eulerian(s);

   //Premultiply the weights and the Jacobian
   double W = w*J;

   // Get x position as Vector
   interpolated_x(s,x);

   // Get FE function value
   std::complex<double> u_fe=interpolated_u_pml_helmholtz(s);

   // Get exact solution at this point
   (*exact_soln_pt)(x,exact_soln);

   //Output x,y,...,error
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << x[i] << " ";
    }
   outfile << exact_soln[0] << " "  << exact_soln[1] << " "
           << exact_soln[0]-u_fe.real() << " " << exact_soln[1]-u_fe.imag()
           << std::endl;

   // Add to error and norm
   norm+=(exact_soln[0]*exact_soln[0]+exact_soln[1]*exact_soln[1])*W;
   error+=( (exact_soln[0]-u_fe.real())*(exact_soln[0]-u_fe.real())+
            (exact_soln[1]-u_fe.imag())*(exact_soln[1]-u_fe.imag()) )*W;

  }
}




//======================================================================
 /// Compute norm of fe solution
//======================================================================
template <unsigned DIM>
void PMLHelmholtzEquationsBase<DIM>::compute_norm(double& norm)
{

 // Initialise
 norm=0.0;

 //Vector of local coordinates
 Vector<double> s(2);

 // Vector for coordintes
 Vector<double> x(2);

 //Find out how many nodes there are in the element
  unsigned n_node = nnode();

  Shape psi(n_node);

  //Set the value of n_intpt
  unsigned n_intpt = integral_pt()->nweight();

  //Loop over the integration points
  for(unsigned ipt=0;ipt<n_intpt;ipt++)
   {

    //Assign values of s
    for(unsigned i=0;i<2;i++)
     {
      s[i] = integral_pt()->knot(ipt,i);
     }

    //Get the integral weight
    double w = integral_pt()->weight(ipt);

    // Get jacobian of mapping
    double J=J_eulerian(s);

    //Premultiply the weights and the Jacobian
    double W = w*J;

    // Get FE function value
    std::complex<double> u_fe=interpolated_u_pml_helmholtz(s);

    // Add to  norm
    norm+=(u_fe.real()*u_fe.real()+u_fe.imag()*u_fe.imag())*W;

   }
 }

//====================================================================
// Force build of templates
//====================================================================
template<unsigned DIM, class PML_ELEMENT>
BermudezPMLMapping PMLHelmholtzEquations<DIM,PML_ELEMENT>::Default_pml_mapping;

template class PMLHelmholtzEquationsBase<1>;
template class PMLHelmholtzEquationsBase<2>;
template class PMLHelmholtzEquationsBase<3>;

template class PMLHelmholtzEquations<1,AxisAlignedPMLElement<1>>;
template class PMLHelmholtzEquations<2,AxisAlignedPMLElement<2>>;
template class PMLHelmholtzEquations<3,AxisAlignedPMLElement<3>>;

template class QPMLHelmholtzElement<1,2,AxisAlignedPMLElement<1>>;
template class QPMLHelmholtzElement<1,3,AxisAlignedPMLElement<1>>;
template class QPMLHelmholtzElement<1,4,AxisAlignedPMLElement<1>>;

template class QPMLHelmholtzElement<2,2,AxisAlignedPMLElement<2>>;
template class QPMLHelmholtzElement<2,3,AxisAlignedPMLElement<2>>;
template class QPMLHelmholtzElement<2,4,AxisAlignedPMLElement<2>>;

template class QPMLHelmholtzElement<3,2,AxisAlignedPMLElement<3>>;
template class QPMLHelmholtzElement<3,3,AxisAlignedPMLElement<3>>;
template class QPMLHelmholtzElement<3,4,AxisAlignedPMLElement<3>>;

}