pml_fourier_decomposed_helmholtz_elements.h

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// LIC// ====================================================================
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// LIC// multi-physics finite-element library, available
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// LIC//
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// Header file for Fourier-decomposed Helmholtz elements
#ifndef OOMPH_PML_FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENTS_HEADER
#define OOMPH_PML_FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENTS_HEADER
 
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
#include "math.h"
#include <complex>
 
 
// OOMPH-LIB headers
#include "../generic/projection.h"
#include "../generic/nodes.h"
#include "../generic/Qelements.h"
#include "../generic/oomph_utilities.h"
#include "../generic/pml_meshes.h"
#include "../generic/projection.h"
#include "../generic/oomph_definitions.h"
 
 
namespace oomph
{
  //========================================================================
  /// Helper namespace for functions required for Helmholtz computations
  //========================================================================
  namespace Legendre_functions_helper
  {
    /// Factorial
    extern double factorial(const unsigned& l);
 
    /// Legendre polynomials depending on one parameter
    extern double plgndr1(const unsigned& n, const double& x);
 
    /// Legendre polynomials depending on two parameters
    extern double plgndr2(const unsigned& l,
                          const unsigned& m,
                          const double& x);
 
  } // namespace Legendre_functions_helper
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
  //=======================================================================
  /// Class to hold the mapping function for the PML
  ///
  //=======================================================================
  class PMLMappingAndTransformedCoordinate
  {
  public:
    /// Default constructor (empty)
    PMLMappingAndTransformedCoordinate(){};
 
    /// Pure virtual to return PML mapping gamma, where gamma is the
    /// \f$d\tilde x / d x\f$ as  function of \f$\nu\f$ where \f$\nu = x - h\f$
    /// where h is the vector from the origin to the start of the PML
    virtual std::complex<double> gamma(const double& nu_i,
                                       const double& pml_width_i,
                                       const double& k_squared) = 0;
 
    /// Pure virtual to return PML transformed coordinate, also known as
    /// \f$d\tilde x \f$ as  function of \f$\nu\f$ where \f$\nu = x - h\f$ where
    /// h is the vector from the origin to the start of the PML
    virtual std::complex<double> transformed_coordinate(
      const double& nu_i,
      const double& pml_width_i,
      const double& pml_inner_boundary,
      const double& k_squared) = 0;
  };
 
 
  //=======================================================================
  /// The mapping function propsed by Bermudez et al, appears to be the best
  /// and so this will be the default mapping (see definition of
  /// PMLHelmholtzEquations)
  //=======================================================================
  class BermudezPMLMappingAndTransformedCoordinate
    : public PMLMappingAndTransformedCoordinate
  {
  public:
    /// Default constructor (empty)
    BermudezPMLMappingAndTransformedCoordinate(){};
 
    /// Overwrite the pure PML mapping coefficient function to return the
    /// mapping function proposed by Bermudez et al
    std::complex<double> gamma(const double& nu_i,
                               const double& pml_width_i,
                               const double& k_squared)
    {
      /// return \f$\gamma=1 + (1/k)(i/|outer_boundary - x|)\f$ or more
      /// abstractly \f$\gamma = 1 + \frac i {k\delta_{pml}}(1/|1-\bar\nu|)\f$
      return 1.0 +
             std::complex<double>(1.0 / sqrt(k_squared), 0) *
               std::complex<double>(0.0, 1.0 / (std::fabs(pml_width_i - nu_i)));
    }
 
    /// Overwrite the pure PML mapping coefficient function to return the
    /// transformed coordinate proposed by Bermudez et al
    std::complex<double> transformed_coordinate(
      const double& nu_i,
      const double& pml_width_i,
      const double& pml_inner_boundary,
      const double& k_squared)
    {
      /// return \f$\tilde x = h + \nu + \log(1-|\nu / \delta|)\f$
      double log_arg = 1.0 - std::fabs(nu_i / pml_width_i);
      return std::complex<double>(pml_inner_boundary + nu_i,
                                  -log(log_arg) / sqrt(k_squared));
    }
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //=============================================================
  /// A class for all isoparametric elements that solve the
  /// Helmholtz equations with pml capabilities.
  /// in Fourier decomposed form (cylindrical polars):
  /// \f[ U(r,\varphi,z) = \Re( u^{(n)}(r,z) \exp(-i n \varphi)) \f]
  /// We are solving for \f$ u^{(n)}(r,z)\f$ for given parameters
  /// \f$ k^2 \f$ and \f$ n \f$ .
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  //=============================================================
  class PMLFourierDecomposedHelmholtzEquations
    : public virtual PMLElementBase<2>,
      public virtual FiniteElement
  {
  public:
    /// Function pointer to source function fct(x,f(x)) --
    /// x is a Vector!
    typedef void (*PMLFourierDecomposedHelmholtzSourceFctPt)(
      const Vector<double>& x, std::complex<double>& f);
 
    /// Constructor
    PMLFourierDecomposedHelmholtzEquations()
      : Source_fct_pt(0), K_squared_pt(0), N_pml_fourier_pt(0)
    {
      // Provide pointer to static method (Save memory)
      Pml_mapping_and_transformed_coordinate_pt =
        &PMLFourierDecomposedHelmholtzEquations::
          Default_pml_mapping_and_transformed_coordinate;
 
      Alpha_pt = &Default_Physical_Constant_Value;
    }
 
 
    /// Broken copy constructor
    PMLFourierDecomposedHelmholtzEquations(
      const PMLFourierDecomposedHelmholtzEquations& dummy) = delete;
 
    /// Broken assignment operator
    // Commented out broken assignment operator because this can lead to a
    // conflict warning when used in the virtual inheritence hierarchy.
    // Essentially the compiler doesn't realise that two separate
    // implementations of the broken function are the same and so, quite
    // rightly, it shouts.
    /*void operator=(const PMLFourierDecomposedHelmholtzEquations&) = delete;*/
 
    /// Return the index at which the unknown value
    /// is stored: Real/imag part of index contains (real) index of
    /// real/imag part.
    virtual inline std::complex<unsigned> u_index_pml_fourier_decomposed_helmholtz()
      const
    {
      return std::complex<unsigned>(0, 1);
    }
 
 
    /// Get pointer to frequency
    double*& k_squared_pt()
    {
      return K_squared_pt;
    }
 
 
    /// Get k squared
    double k_squared()
    {
#ifdef PARANOID
      if (K_squared_pt == 0)
      {
        throw OomphLibError(
          "Please set pointer to k_squared using access fct to pointer!",
          OOMPH_CURRENT_FUNCTION,
          OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return *K_squared_pt;
    }
 
    /// Get pointer to complex shift
    double*& alpha_pt()
    {
      return Alpha_pt;
    }
 
 
    /// Get complex shift
    double alpha()
    {
      return *Alpha_pt;
    }
 
    /// Get pointer to Fourier wavenumber
    int*& pml_fourier_wavenumber_pt()
    {
      return N_pml_fourier_pt;
    }
 
    /// Get the Fourier wavenumber
    int pml_fourier_wavenumber()
    {
      if (N_pml_fourier_pt == 0)
      {
        return 0;
      }
      else
      {
        return *N_pml_fourier_pt;
      }
    }
 
 
    /// Output with default number of plot points
    void output(std::ostream& outfile)
    {
      const unsigned n_plot = 5;
      output(outfile, n_plot);
    }
 
    /// Output FE representation of soln: x,y,u_re,u_im or
    /// x,y,z,u_re,u_im at  n_plot^2 plot points
    void output(std::ostream& outfile, const unsigned& n_plot);
 
    /// 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.
    /// r,z,u at n_plot plot points in each coordinate
    /// direction
    void output_real(std::ostream& outfile,
                     const double& phi,
                     const unsigned& n_plot);
 
    /// C_style output with default number of plot points
    void output(FILE* file_pt)
    {
      const unsigned n_plot = 5;
      output(file_pt, n_plot);
    }
 
    /// C-style output FE representation of soln: r,z,u_re,u_im or
    /// at n_plot^2 plot points
    void output(FILE* file_pt, const unsigned& n_plot);
 
    /// Output exact soln: r,z,u_re_exact,u_im_exact
    /// at n_plot^2 plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output exact soln: (dummy time-dependent version to
    /// keep intel compiler happy)
    virtual void output_fct(
      std::ostream& outfile,
      const unsigned& n_plot,
      const double& time,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      throw OomphLibError("There is no time-dependent output_fct() for "
                          "PMLFourierDecomposedHelmholtz elements ",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// 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.
    /// r,z,u at n_plot plot points in each coordinate
    /// direction
    void output_real_fct(std::ostream& outfile,
                         const double& phi,
                         const unsigned& n_plot,
                         FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
 
    /// Get error against and norm of exact solution
    void compute_error(std::ostream& outfile,
                       FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
 
    /// Dummy, time dependent error checker
    void compute_error(std::ostream& outfile,
                       FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
                       const double& time,
                       double& error,
                       double& norm)
    {
      throw OomphLibError("There is no time-dependent compute_error() for "
                          "PMLFourierDecomposedHelmholtz elements",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Compute norm of fe solution
    void compute_norm(double& norm);
 
 
    /// Access function: Pointer to source function
    PMLFourierDecomposedHelmholtzSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to source function. Const version
    PMLFourierDecomposedHelmholtzSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
 
    /// Get source term at (Eulerian) position x. This function is
    /// virtual to allow overloading in multi-physics problems where
    /// the strength of the source function might be determined by
    /// another system of equations.
    inline virtual void get_source_pml_fourier_decomposed_helmholtz(
      const unsigned& ipt,
      const Vector<double>& x,
      std::complex<double>& source) const
    {
      // If no source function has been set, return zero
      if (Source_fct_pt == 0)
      {
        source = std::complex<double>(0.0, 0.0);
      }
      else
      {
        // Get source strength
        (*Source_fct_pt)(x, source);
      }
    }
 
    /// Pure virtual function in which we specify the
    /// values to be pinned (and set to zero) on the outer edge of
    /// the pml layer. All of them! Vector is resized internally.
    void values_to_be_pinned_on_outer_pml_boundary(
      Vector<unsigned>& values_to_pin)
    {
      values_to_pin.resize(2);
      for (unsigned j = 0; j < 2; j++)
      {
        values_to_pin[j] = j;
      }
    }
 
 
    /// Get flux: flux[i] = du/dx_i for real and imag part
    void get_flux(const Vector<double>& s,
                  Vector<std::complex<double>>& flux) const
    {
      // Find out how many nodes there are in the element
      const unsigned n_node = nnode();
 
      // Set up memory for the shape and test functions
      Shape psi(n_node);
      DShape dpsidx(n_node, 2);
 
      // Call the derivatives of the shape and test functions
      dshape_eulerian(s, psi, dpsidx);
 
      // Initialise to zero
      const std::complex<double> zero(0.0, 0.0);
      for (unsigned j = 0; j < 2; j++)
      {
        flux[j] = zero;
      }
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Cache the complex value of the unknown
        const std::complex<double> u_value(
          this->nodal_value(l,
                            u_index_pml_fourier_decomposed_helmholtz().real()),
          this->nodal_value(l,
                            u_index_pml_fourier_decomposed_helmholtz().imag()));
 
        // Loop over derivative directions
        for (unsigned j = 0; j < 2; j++)
        {
          flux[j] += u_value * dpsidx(l, j);
        }
      }
    }
 
 
    /// Add the element's contribution to its residual vector (wrapper)
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Call the generic residuals function with flag set to 0
      // using a dummy matrix argument
      fill_in_generic_residual_contribution_pml_fourier_decomposed_helmholtz(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
 
    /// Add the element's contribution to its residual vector and
    /// element Jacobian matrix (wrapper)
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      fill_in_generic_residual_contribution_pml_fourier_decomposed_helmholtz(
        residuals, jacobian, 1);
    }
 
 
    /// Return FE representation of function value u(s)
    /// at local coordinate s
    inline std::complex<double> interpolated_u_pml_fourier_decomposed_helmholtz(
      const Vector<double>& s) const
    {
      // Find number of nodes
      const unsigned n_node = nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      std::complex<double> interpolated_u(0.0, 0.0);
 
      // Get the index at which the helmholtz unknown is stored
      const unsigned u_nodal_index_real =
        u_index_pml_fourier_decomposed_helmholtz().real();
      const unsigned u_nodal_index_imag =
        u_index_pml_fourier_decomposed_helmholtz().imag();
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        // Make a temporary complex number from the stored data
        const std::complex<double> u_value(
          this->nodal_value(l, u_nodal_index_real),
          this->nodal_value(l, u_nodal_index_imag));
        // Add to the interpolated value
        interpolated_u += u_value * psi[l];
      }
      return interpolated_u;
    }
 
 
    /// Self-test: Return 0 for OK
    unsigned self_test();
 
 
  protected:
    /// Compute pml coefficients at position x and integration point ipt.
    /// pml_laplace_factor is used in the residual contribution from the laplace
    /// operator, similarly pml_k_squared_factor is used in the contribution
    /// from the k^2 of the Helmholtz operator.
    void compute_pml_coefficients(
      const unsigned& ipt,
      const Vector<double>& x,
      Vector<std::complex<double>>& pml_laplace_factor,
      std::complex<double>& pml_k_squared_factor)
    {
      /// Vector which points from the inner boundary to x
      Vector<double> nu(2);
      for (unsigned k = 0; k < 2; k++)
      {
        nu[k] = x[k] - this->Pml_inner_boundary[k];
      }
 
      /// Vector which points from the inner boundary to the edge of the
      /// boundary
      Vector<double> pml_width(2);
      for (unsigned k = 0; k < 2; k++)
      {
        pml_width[k] =
          this->Pml_outer_boundary[k] - this->Pml_inner_boundary[k];
      }
 
      // Declare gamma_i vectors of complex numbers for PML weights
      Vector<std::complex<double>> pml_gamma(2);
 
      if (this->Pml_is_enabled)
      {
        // Cache k_squared to pass into mapping function
        double k_squared_local = k_squared();
 
        for (unsigned k = 0; k < 2; k++)
        {
          // If PML is enabled in the respective direction
          if (this->Pml_direction_active[k])
          {
            pml_gamma[k] = Pml_mapping_and_transformed_coordinate_pt->gamma(
              nu[k], pml_width[k], k_squared_local);
          }
          else
          {
            pml_gamma[k] = 1.0;
          }
        }
 
        ///  for 2D, in order:
        /// g_y/g_x, g_x/g_y for Laplace bit and g_x*g_y for Helmholtz bit
        /// for 3D, in order: g_y*g_x/g_x, g*x*g_z/g_y, g_x*g_y/g_z for Laplace
        /// bit and g_x*g_y*g_z for Helmholtz factor
        pml_laplace_factor[0] = pml_gamma[1] / pml_gamma[0];
        pml_laplace_factor[1] = pml_gamma[0] / pml_gamma[1];
        pml_k_squared_factor = pml_gamma[0] * pml_gamma[1];
      }
      else
      {
        /// The weights all default to 1.0 as if the propagation
        /// medium is the physical domain
        for (unsigned k = 0; k < 2; k++)
        {
          pml_laplace_factor[k] = std::complex<double>(1.0, 0.0);
        }
 
        pml_k_squared_factor = std::complex<double>(1.0, 0.0);
      }
    }
 
 
    /// Compute complex variable r at position x[0] and
    /// integration point ipt
    void compute_complex_r(const unsigned& ipt,
                           const Vector<double>& x,
                           std::complex<double>& complex_r)
    {
      // Cache current position r
      double r = x[0];
 
      /// The complex r variable is only imaginary on two
      /// conditions: First, the decaying nature of the
      /// pml layers is active.  Secondly, the
      /// integration point is contained in the right pml
      /// layer or the two corner pml layers.
 
      // If the complex r variable is imaginary
      if (this->Pml_is_enabled && (this->Pml_direction_active[0]))
      {
        double nu = x[0] - Pml_inner_boundary[0];
        double pml_width = Pml_outer_boundary[0] - Pml_inner_boundary[0];
        double k_squared_local = k_squared();
 
        // Determine the complex r variable
        complex_r =
          Pml_mapping_and_transformed_coordinate_pt->transformed_coordinate(
            nu, pml_width, Pml_inner_boundary[0], k_squared_local);
      }
      else
      {
        // The complex r variable is infact purely real, and
        // is equal to x[0]
        complex_r = std::complex<double>(r, 0.0);
      }
 
    } // end of compute_complex_r
 
    /// Return a pointer to the PML Mapping object
    PMLMappingAndTransformedCoordinate*& pml_mapping_and_transformed_coordinate_pt()
    {
      return Pml_mapping_and_transformed_coordinate_pt;
    }
 
    /// Return a pointer to the PML Mapping object (const version)
    PMLMappingAndTransformedCoordinate* const& pml_mapping_and_transformed_coordinate_pt()
      const
    {
      return Pml_mapping_and_transformed_coordinate_pt;
    }
 
    /// Static so that the class doesn't need to instantiate a new default
    /// everytime it uses it
    static BermudezPMLMappingAndTransformedCoordinate
      Default_pml_mapping_and_transformed_coordinate;
 
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// local coord. s; return  Jacobian of mapping
    virtual double dshape_and_dtest_eulerian_pml_fourier_decomposed_helmholtz(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// integration point ipt; return  Jacobian of mapping
    virtual double dshape_and_dtest_eulerian_at_knot_pml_fourier_decomposed_helmholtz(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Compute element residual Vector only (if flag=and/or element
    /// Jacobian matrix
    virtual void fill_in_generic_residual_contribution_pml_fourier_decomposed_helmholtz(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag);
 
    /// Pointer to source function:
    PMLFourierDecomposedHelmholtzSourceFctPt Source_fct_pt;
 
    /// Pointer to k^2 (wavenumber squared)
    double* K_squared_pt;
 
    /// Pointer to class which holds the pml mapping function (also known
    /// as gamma) and the associated transformed coordinate
    PMLMappingAndTransformedCoordinate*
      Pml_mapping_and_transformed_coordinate_pt;
 
    /// Pointer to wavenumber complex shift
    double* Alpha_pt;
 
    /// Static default value for the physical constants (initialised to zero)
    static double Default_Physical_Constant_Value;
 
    /// Pointer to Fourier wave number
    int* N_pml_fourier_pt;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// QPMLFourierDecomposedHelmholtzElement elements are
  /// linear/quadrilateral/brick-shaped PMLFourierDecomposedHelmholtz
  /// elements with isoparametric interpolation for the function.
  //======================================================================
  template<unsigned NNODE_1D>
  class QPMLFourierDecomposedHelmholtzElement
    : public virtual QElement<2, NNODE_1D>,
      public virtual PMLFourierDecomposedHelmholtzEquations
  {
  private:
    /// Static int that holds the number of variables at
    /// nodes: always the same
    static const unsigned Initial_Nvalue;
 
  public:
    /// Constructor: Call constructors for QElement and
    /// PMLFourierDecomposedHelmholtz equations
    QPMLFourierDecomposedHelmholtzElement()
      : QElement<2, NNODE_1D>(), PMLFourierDecomposedHelmholtzEquations()
    {
    }
 
    /// Broken copy constructor
    QPMLFourierDecomposedHelmholtzElement(
      const QPMLFourierDecomposedHelmholtzElement<NNODE_1D>& dummy) = delete;
 
    /// Broken assignment operator
    /*void operator=(const
                   QPMLFourierDecomposedHelmholtzElement<NNODE_1D>&) = delete;*/
 
 
    ///  Required  # of `values' (pinned or dofs)
    /// at node n
    inline unsigned required_nvalue(const unsigned& n) const
    {
      return Initial_Nvalue;
    }
 
    /// Output function: r,z,u
    void output(std::ostream& outfile)
    {
      PMLFourierDecomposedHelmholtzEquations::output(outfile);
    }
 
    ///  Output function:
    ///   r,z,u at n_plot^2 plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      PMLFourierDecomposedHelmholtzEquations::output(outfile, n_plot);
    }
 
    /// 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.
    /// r,z,u  at n_plot plot points in each coordinate
    /// direction
    void output_real(std::ostream& outfile,
                     const double& phi,
                     const unsigned& n_plot)
    {
      PMLFourierDecomposedHelmholtzEquations::output_real(outfile, phi, n_plot);
    }
 
    /// C-style output function:  r,z,u
    void output(FILE* file_pt)
    {
      PMLFourierDecomposedHelmholtzEquations::output(file_pt);
    }
 
    ///  C-style output function:
    ///   r,z,u  at n_plot^2 plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      PMLFourierDecomposedHelmholtzEquations::output(file_pt, n_plot);
    }
 
    /// Output function for an exact solution:
    /// r,z,u_exact at n_plot^2 plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLFourierDecomposedHelmholtzEquations::output_fct(
        outfile, n_plot, exact_soln_pt);
    }
 
    /// 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.
    /// r,z,u  at n_plot plot points in each coordinate
    /// direction
    void output_real_fct(std::ostream& outfile,
                         const double& phi,
                         const unsigned& n_plot,
                         FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLFourierDecomposedHelmholtzEquations::output_real_fct(
        outfile, phi, n_plot, exact_soln_pt);
    }
 
 
    /// Output function for a time-dependent exact solution.
    ///  r,z,u_exact at n_plot^2 plot points
    /// (Calls the steady version)
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLFourierDecomposedHelmholtzEquations::output_fct(
        outfile, n_plot, time, exact_soln_pt);
    }
 
  protected:
    /// Shape, test functions & derivs. w.r.t. to global coords.
    /// Return Jacobian.
    inline double dshape_and_dtest_eulerian_pml_fourier_decomposed_helmholtz(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
 
    /// Shape, test functions & derivs. w.r.t. to global coords. at
    /// integration point ipt. Return Jacobian.
    inline double dshape_and_dtest_eulerian_at_knot_pml_fourier_decomposed_helmholtz(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
  };
 
 
  // Inline functions:
 
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. global coordinates and return Jacobian of mapping.
  ///
  /// Galerkin: Test functions = shape functions
  //======================================================================
  template<unsigned NNODE_1D>
  double QPMLFourierDecomposedHelmholtzElement<NNODE_1D>::
    dshape_and_dtest_eulerian_pml_fourier_decomposed_helmholtz(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian(s, psi, dpsidx);
 
    // Set the test functions equal to the shape functions
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. global coordinates and return Jacobian of mapping.
  ///
  /// Galerkin: Test functions = shape functions
  //======================================================================
  template<unsigned NNODE_1D>
  double QPMLFourierDecomposedHelmholtzElement<NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_pml_fourier_decomposed_helmholtz(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
 
    // Set the pointers of the test functions
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the QPMLFourierDecomposedHelmholtzElement
  /// elements:
  /// The spatial dimension of the face elements is one lower than that of the
  /// bulk element but they have the same number of points
  /// along their 1D edges.
  //=======================================================================
  template<unsigned NNODE_1D>
  class FaceGeometry<QPMLFourierDecomposedHelmholtzElement<NNODE_1D>>
    : public virtual QElement<1, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : QElement<1, NNODE_1D>() {}
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //==========================================================
  /// Fourier decomposed Helmholtz upgraded to become projectable
  //==========================================================
  template<class FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENT>
  class ProjectablePMLFourierDecomposedHelmholtzElement
    : public virtual ProjectableElement<FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENT>
  {
  public:
    /// Constructor [this was only required explicitly
    /// from gcc 4.5.2 onwards...]
    ProjectablePMLFourierDecomposedHelmholtzElement() {}
 
    /// Specify the values associated with field fld.
    /// The information is returned in a vector of pairs which comprise
    /// the Data object and the value within it, that correspond to field fld.
    Vector<std::pair<Data*, unsigned>> data_values_of_field(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Fourier decomposed Helmholtz elements only store 2 "
                        "fields so fld = "
                     << fld << " is illegal \n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Create the vector
      unsigned nnod = this->nnode();
      Vector<std::pair<Data*, unsigned>> data_values(nnod);
 
      // Loop over all nodes
      for (unsigned j = 0; j < nnod; j++)
      {
        // Add the data value associated field: The node itself
        data_values[j] = std::make_pair(this->node_pt(j), fld);
      }
 
      // Return the vector
      return data_values;
    }
 
    /// Number of fields to be projected: 2 (real and imag part)
    unsigned nfields_for_projection()
    {
      return 2;
    }
 
    /// Number of history values to be stored for fld-th field.
    /// (Note: count includes current value!)
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return this->node_pt(0)->ntstorage();
    }
 
    /// Number of positional history values
    /// (Note: count includes current value!)
    unsigned nhistory_values_for_coordinate_projection()
    {
      return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
    }
 
    /// Return Jacobian of mapping and shape functions of field fld
    /// at local coordinate s
    double jacobian_and_shape_of_field(const unsigned& fld,
                                       const Vector<double>& s,
                                       Shape& psi)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal.\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      unsigned n_dim = this->dim();
      unsigned n_node = this->nnode();
      Shape test(n_node);
      DShape dpsidx(n_node, n_dim), dtestdx(n_node, n_dim);
      double J =
        this->dshape_and_dtest_eulerian_pml_fourier_decomposed_helmholtz(
          s, psi, dpsidx, test, dtestdx);
      return J;
    }
 
 
    /// Return interpolated field fld at local coordinate s, at time
    /// level t (t=0: present; t>0: history values)
    double get_field(const unsigned& t,
                     const unsigned& fld,
                     const Vector<double>& s)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Find the index at which the variable is stored
      std::complex<unsigned> complex_u_nodal_index =
        this->u_index_pml_fourier_decomposed_helmholtz();
      unsigned u_nodal_index = 0;
      if (fld == 0)
      {
        u_nodal_index = complex_u_nodal_index.real();
      }
      else
      {
        u_nodal_index = complex_u_nodal_index.imag();
      }
 
 
      // Local shape function
      unsigned n_node = this->nnode();
      Shape psi(n_node);
 
      // Find values of shape function
      this->shape(s, psi);
 
      // Initialise value of u
      double interpolated_u = 0.0;
 
      // Sum over the local nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_u += this->nodal_value(t, l, u_nodal_index) * psi[l];
      }
      return interpolated_u;
    }
 
 
    /// Return number of values in field fld: One per node
    unsigned nvalue_of_field(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return this->nnode();
    }
 
 
    /// Return local equation number of value j in field fld.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      std::complex<unsigned> complex_u_nodal_index =
        this->u_index_pml_fourier_decomposed_helmholtz();
      unsigned u_nodal_index = 0;
      if (fld == 0)
      {
        u_nodal_index = complex_u_nodal_index.real();
      }
      else
      {
        u_nodal_index = complex_u_nodal_index.imag();
      }
      return this->nodal_local_eqn(j, u_nodal_index);
    }
 
 
    /// Output FE representation of soln: x,y,u or x,y,z,u at
    /// n_plot^DIM plot points
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENT::output(outfile, nplot);
    }
  };
 
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<ProjectablePMLFourierDecomposedHelmholtzElement<ELEMENT>>
    : public virtual FaceGeometry<ELEMENT>
  {
  public:
    FaceGeometry() : FaceGeometry<ELEMENT>() {}
  };
 
 
  //=======================================================================
  /// Face geometry of the Face Geometry for element is the same as
  /// that for the underlying wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<
    FaceGeometry<ProjectablePMLFourierDecomposedHelmholtzElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
  //=======================================================================
  /// Policy class defining the elements to be used in the actual
  /// PML layers. Same!
  //=======================================================================
  template<unsigned NNODE_1D>
  class PMLLayerElement<QPMLFourierDecomposedHelmholtzElement<NNODE_1D>>
    : public virtual QPMLFourierDecomposedHelmholtzElement<NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate QElement
    PMLLayerElement() : QPMLFourierDecomposedHelmholtzElement<NNODE_1D>() {}
  };
 
} // namespace oomph
 
#endif