pml_helmholtz_elements.cc

Go to the documentation of this file.

// 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// Copyright (C) 2006-2025 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
// LIC// License along with this library; if not, write to the Free Software
// LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
// LIC// 02110-1301  USA.
// 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>
  const unsigned QPMLHelmholtzElement<DIM, NNODE_1D>::Initial_Nvalue = 2;
 
  /// PML Helmholtz equations static data, so that by default we can point to a
  /// 0
  template<unsigned DIM>
  double PMLHelmholtzEquations<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>
  void PMLHelmholtzEquations<DIM>::
    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 = 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 = 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] += raw_nodal_position(l, j) * psi(l);
        }
 
        // Get the nodal value of the helmholtz unknown
        const std::complex<double> u_value(
          raw_nodal_value(l, u_index_helmholtz().real()),
          raw_nodal_value(l, 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);
      get_source_helmholtz(ipt, interpolated_x, source);
 
 
      // Declare a vector of complex numbers for pml weights on the Laplace bit
      Vector<std::complex<double>> pml_laplace_factor(DIM);
      // Declare a complex number for pml weights on the mass matrix bit
      std::complex<double> pml_k_squared_factor =
        std::complex<double>(1.0, 0.0);
 
      // All the PML weights that participate in the assemby process
      // are computed here. pml_laplace_factor will contain the entries
      // for the Laplace bit, while pml_k_squared_factor contains the
      // contributions to the Helmholtz bit. Both default to 1.0, should the PML
      // not be enabled via enable_pml.
      compute_pml_coefficients(
        ipt, interpolated_x, pml_laplace_factor, 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() - alpha() * pml_k_squared_factor.imag(),
        alpha() * pml_k_squared_factor.real() + pml_k_squared_factor.imag());
 
 
      //  std::complex<double> alpha_pml_k_squared_factor
      //  if(alpha_pt() == 0)
      //  {
      //  std::complex<double> alpha_pml_k_squared_factor =
      //  std::complex<double>(
      //    pml_k_squared_factor.real() -  alpha() *
      //    pml_k_squared_factor.imag(), 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 = nodal_local_eqn(l, u_index_helmholtz().real());
        local_eqn_imag = nodal_local_eqn(l, 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() * k_squared() *
                                interpolated_u.real() -
                              alpha_pml_k_squared_factor.imag() * k_squared() *
                                interpolated_u.imag())) *
            test(l) * W;
 
          // The Laplace bit
          for (unsigned k = 0; k < DIM; k++)
          {
            residuals[local_eqn_real] +=
              (pml_laplace_factor[k].real() * interpolated_dudx[k].real() -
               pml_laplace_factor[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 =
                nodal_local_eqn(l2, u_index_helmholtz().real());
              local_unknown_imag =
                nodal_local_eqn(l2, 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) +=
                    pml_laplace_factor[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() * 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) -=
                    pml_laplace_factor[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() * k_squared() * psi(l2) *
                  test(l) * W;
              }
            }
          }
        }
 
        // Second, compute the imaginary part contribution
        //------------------------------------------------
 
        // Get the local equation
        local_eqn_imag = nodal_local_eqn(l, u_index_helmholtz().imag());
        local_eqn_real = nodal_local_eqn(l, 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() * k_squared() *
                                interpolated_u.real() +
                              alpha_pml_k_squared_factor.real() * k_squared() *
                                interpolated_u.imag())) *
            test(l) * W;
 
          // The Laplace bit
          for (unsigned k = 0; k < DIM; k++)
          {
            residuals[local_eqn_imag] +=
              (pml_laplace_factor[k].imag() * interpolated_dudx[k].real() +
               pml_laplace_factor[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 =
                nodal_local_eqn(l2, u_index_helmholtz().imag());
              local_unknown_real =
                nodal_local_eqn(l2, 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) +=
                    pml_laplace_factor[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() * 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) +=
                    pml_laplace_factor[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() * k_squared() * psi(l2) *
                  test(l) * W;
              }
            }
          }
        }
      }
    } // End of loop over integration points
  }
 
 
  //======================================================================
  /// Self-test:  Return 0 for OK
  //======================================================================
  template<unsigned DIM>
  unsigned PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 PMLHelmholtzEquations<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 class PMLHelmholtzEquations<1>;
  template class PMLHelmholtzEquations<2>;
  template class PMLHelmholtzEquations<3>;
 
  template<unsigned DIM>
  BermudezPMLMapping PMLHelmholtzEquations<DIM>::Default_pml_mapping;
 
  template class QPMLHelmholtzElement<1, 2>;
  template class QPMLHelmholtzElement<1, 3>;
  template class QPMLHelmholtzElement<1, 4>;
 
  template class QPMLHelmholtzElement<2, 2>;
  template class QPMLHelmholtzElement<2, 3>;
  template class QPMLHelmholtzElement<2, 4>;
 
  template class QPMLHelmholtzElement<3, 2>;
  template class QPMLHelmholtzElement<3, 3>;
  template class QPMLHelmholtzElement<3, 4>;
 
} // namespace oomph