advection_diffusion_elements.h

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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// Copyright (C) 2006-2025 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
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// LIC// License as published by the Free Software Foundation; either
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// LIC//
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// LIC//====================================================================
// Header file for Advection Diffusion elements
#ifndef OOMPH_ADV_DIFF_ELEMENTS_HEADER
#define OOMPH_ADV_DIFF_ELEMENTS_HEADER
 
 
// Config header
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// OOMPH-LIB headers
#include "generic/nodes.h"
#include "generic/Qelements.h"
#include "generic/oomph_utilities.h"
 
namespace oomph
{
  //=============================================================
  /// A class for all elements that solve the Advection
  /// Diffusion equations using isoparametric elements.
  /// \f[ \frac{\partial^2 u}{\partial x_i^2} = Pe w_i(x_k) \frac{\partial u}{\partial x_i} + f(x_j) \f]
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  //=============================================================
  template<unsigned DIM>
  class AdvectionDiffusionEquations : public virtual FiniteElement
  {
  public:
    /// Function pointer to source function fct(x,f(x)) --
    /// x is a Vector!
    typedef void (*AdvectionDiffusionSourceFctPt)(const Vector<double>& x,
                                                  double& f);
 
 
    /// Function pointer to wind function fct(x,w(x)) --
    /// x is a Vector!
    typedef void (*AdvectionDiffusionWindFctPt)(const Vector<double>& x,
                                                Vector<double>& wind);
 
 
    /// Constructor: Initialise the Source_fct_pt and Wind_fct_pt
    /// to null and set (pointer to) Peclet number to default
    AdvectionDiffusionEquations()
      : Source_fct_pt(0), Wind_fct_pt(0), ALE_is_disabled(false)
    {
      // Set Peclet number to default
      Pe_pt = &Default_peclet_number;
      // Set Peclet Strouhal number to default
      PeSt_pt = &Default_peclet_number;
    }
 
    /// Broken copy constructor
    AdvectionDiffusionEquations(const AdvectionDiffusionEquations& dummy) =
      delete;
 
    /// Broken assignment operator
    void operator=(const AdvectionDiffusionEquations&) = delete;
 
    /// Return the index at which the unknown value
    /// is stored. The default value, 0, is appropriate for single-physics
    /// problems, when there is only one variable, the value that satisfies
    /// the advection-diffusion equation.
    /// In derived multi-physics elements, this function should be overloaded
    /// to reflect the chosen storage scheme. Note that these equations require
    /// that the unknown is always stored at the same index at each node.
    virtual inline unsigned u_index_adv_diff() const
    {
      return 0;
    }
 
    /// du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double du_dt_adv_diff(const unsigned& n) const
    {
      // Get the data's timestepper
      TimeStepper* time_stepper_pt = this->node_pt(n)->time_stepper_pt();
 
      // Initialise dudt
      double dudt = 0.0;
      // Loop over the timesteps, if there is a non Steady timestepper
      if (!time_stepper_pt->is_steady())
      {
        // Find the index at which the variable is stored
        const unsigned u_nodal_index = u_index_adv_diff();
 
        // Number of timsteps (past & present)
        const unsigned n_time = time_stepper_pt->ntstorage();
 
        for (unsigned t = 0; t < n_time; t++)
        {
          dudt +=
            time_stepper_pt->weight(1, t) * nodal_value(t, n, u_nodal_index);
        }
      }
      return dudt;
    }
 
    /// Disable ALE, i.e. assert the mesh is not moving -- you do this
    /// at your own risk!
    void disable_ALE()
    {
      ALE_is_disabled = true;
    }
 
 
    /// (Re-)enable ALE, i.e. take possible mesh motion into account
    /// when evaluating the time-derivative. Note: By default, ALE is
    /// enabled, at the expense of possibly creating unnecessary work
    /// in problems where the mesh is, in fact, stationary.
    void enable_ALE()
    {
      ALE_is_disabled = false;
    }
 
    /// Number of scalars/fields output by this element. Reimplements
    /// broken virtual function in base class.
    unsigned nscalar_paraview() const
    {
      return DIM + 1;
    }
 
    /// Write values of the i-th scalar field at the plot points. Needs
    /// to be implemented for each new specific element type.
    void scalar_value_paraview(std::ofstream& file_out,
                               const unsigned& i,
                               const unsigned& nplot) const
    {
      // Vector of local coordinates
      Vector<double> s(DIM);
 
      // Loop over plot points
      unsigned num_plot_points = nplot_points_paraview(nplot);
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        get_s_plot(iplot, nplot, s);
 
        // Get Eulerian coordinate of plot point
        Vector<double> x(DIM);
        interpolated_x(s, x);
 
        if (i < DIM)
        {
          // Get the wind
          Vector<double> wind(DIM);
 
          // Dummy ipt argument needed... ?
          unsigned ipt = 0;
          get_wind_adv_diff(ipt, s, x, wind);
 
          file_out << wind[i] << std::endl;
        }
        // Advection Diffusion
        else if (i == DIM)
        {
          file_out << interpolated_u_adv_diff(s) << std::endl;
        }
        // Never get here
        else
        {
          std::stringstream error_stream;
          error_stream << "Advection Diffusion Elements only store " << DIM + 1
                       << " fields " << std::endl;
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
    }
 
    /// Name of the i-th scalar field. Default implementation
    /// returns V1 for the first one, V2 for the second etc. Can (should!) be
    /// overloaded with more meaningful names in specific elements.
    std::string scalar_name_paraview(const unsigned& i) const
    {
      // Winds
      if (i < DIM)
      {
        return "Wind " + StringConversion::to_string(i);
      }
      // Advection Diffusion field
      else if (i == DIM)
      {
        return "Advection Diffusion";
      }
      // Never get here
      else
      {
        std::stringstream error_stream;
        error_stream << "Advection Diffusion Elements only store " << DIM + 1
                     << " fields" << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
        // Dummy return
        return " ";
      }
    }
 
    /// Output with default number of plot points
    void output(std::ostream& outfile)
    {
      unsigned nplot = 5;
      output(outfile, nplot);
    }
 
    /// Output FE representation of soln: x,y,u or x,y,z,u at
    /// nplot^DIM plot points
    void output(std::ostream& outfile, const unsigned& nplot);
 
 
    /// C_style output with default number of plot points
    void output(FILE* file_pt)
    {
      unsigned n_plot = 5;
      output(file_pt, n_plot);
    }
 
    /// C-style output FE representation of soln: x,y,u or x,y,z,u at
    /// n_plot^DIM plot points
    void output(FILE* file_pt, const unsigned& n_plot);
 
 
    /// Output exact soln: x,y,u_exact or x,y,z,u_exact at nplot^DIM plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output exact soln: x,y,u_exact or x,y,z,u_exact at
    /// nplot^DIM plot points (dummy time-dependent version to
    /// keep intel compiler happy)
    virtual void output_fct(
      std::ostream& outfile,
      const unsigned& nplot,
      const double& time,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      throw OomphLibError("There is no time-dependent output_fct() for "
                          "Advection Diffusion elements",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// 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(
        "No time-dependent compute_error() for Advection Diffusion elements",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// Access function: Pointer to source function
    AdvectionDiffusionSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to source function. Const version
    AdvectionDiffusionSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to wind function
    AdvectionDiffusionWindFctPt& wind_fct_pt()
    {
      return Wind_fct_pt;
    }
 
 
    /// Access function: Pointer to wind function. Const version
    AdvectionDiffusionWindFctPt wind_fct_pt() const
    {
      return Wind_fct_pt;
    }
 
    /// Peclet number
    const double& pe() const
    {
      return *Pe_pt;
    }
 
    /// Pointer to Peclet number
    double*& pe_pt()
    {
      return Pe_pt;
    }
 
    /// Peclet number multiplied by Strouhal number
    const double& pe_st() const
    {
      return *PeSt_pt;
    }
 
    /// Pointer to Peclet number multipled by Strouha number
    double*& pe_st_pt()
    {
      return PeSt_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_adv_diff(const unsigned& ipt,
                                            const Vector<double>& x,
                                            double& source) const
    {
      // If no source function has been set, return zero
      if (Source_fct_pt == 0)
      {
        source = 0.0;
      }
      else
      {
        // Get source strength
        (*Source_fct_pt)(x, source);
      }
    }
 
    /// Get wind at (Eulerian) position x and/or local coordinate s.
    /// This function is
    /// virtual to allow overloading in multi-physics problems where
    /// the wind function might be determined by
    /// another system of equations
    inline virtual void get_wind_adv_diff(const unsigned& ipt,
                                          const Vector<double>& s,
                                          const Vector<double>& x,
                                          Vector<double>& wind) const
    {
      // If no wind function has been set, return zero
      if (Wind_fct_pt == 0)
      {
        for (unsigned i = 0; i < DIM; i++)
        {
          wind[i] = 0.0;
        }
      }
      else
      {
        // Get wind
        (*Wind_fct_pt)(x, wind);
      }
    }
 
    /// Get flux: \f$\mbox{flux}[i] = \mbox{d}u / \mbox{d}x_i \f$
    void get_flux(const Vector<double>& s, Vector<double>& flux) const
    {
      // Find out how many nodes there are in the element
      unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      unsigned u_nodal_index = u_index_adv_diff();
 
      // Set up memory for the shape and test functions
      Shape psi(n_node);
      DShape dpsidx(n_node, DIM);
 
      // Call the derivatives of the shape and test functions
      dshape_eulerian(s, psi, dpsidx);
 
      // Initialise to zero
      for (unsigned j = 0; j < DIM; j++)
      {
        flux[j] = 0.0;
      }
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over derivative directions
        for (unsigned j = 0; j < DIM; j++)
        {
          flux[j] += nodal_value(l, u_nodal_index) * 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 and using
      // a dummy matrix
      fill_in_generic_residual_contribution_adv_diff(
        residuals,
        GeneralisedElement::Dummy_matrix,
        GeneralisedElement::Dummy_matrix,
        0);
    }
 
 
    /// Add the element's contribution to its residual vector and
    /// the 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_adv_diff(
        residuals, jacobian, GeneralisedElement::Dummy_matrix, 1);
    }
 
 
    /// Add the element's contribution to its residuals vector,
    /// jacobian matrix and mass matrix
    void fill_in_contribution_to_jacobian_and_mass_matrix(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix)
    {
      // Call the generic routine with the flag set to 2
      fill_in_generic_residual_contribution_adv_diff(
        residuals, jacobian, mass_matrix, 2);
    }
 
 
    /// Return FE representation of function value u(s) at local coordinate s
    inline double interpolated_u_adv_diff(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      unsigned u_nodal_index = u_index_adv_diff();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      double interpolated_u = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_u += nodal_value(l, u_nodal_index) * psi[l];
      }
 
      return (interpolated_u);
    }
 
 
    /// Return derivative of u at point s with respect to all data
    /// that can affect its value.
    /// In addition, return the global equation numbers corresponding to the
    /// data. This is virtual so that it can be overloaded in the
    /// refineable version
    virtual void dinterpolated_u_adv_diff_ddata(
      const Vector<double>& s,
      Vector<double>& du_ddata,
      Vector<unsigned>& global_eqn_number)
    {
      // Find number of nodes
      const unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      const unsigned u_nodal_index = u_index_adv_diff();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Find the number of dofs associated with interpolated u
      unsigned n_u_dof = 0;
      for (unsigned l = 0; l < n_node; l++)
      {
        int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
        // If it's positive add to the count
        if (global_eqn >= 0)
        {
          ++n_u_dof;
        }
      }
 
      // Now resize the storage schemes
      du_ddata.resize(n_u_dof, 0.0);
      global_eqn_number.resize(n_u_dof, 0);
 
      // Loop over the nodes again and set the derivatives
      unsigned count = 0;
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the global equation number
        int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
        // If it's positive
        if (global_eqn >= 0)
        {
          // Set the global equation number
          global_eqn_number[count] = global_eqn;
          // Set the derivative with respect to the unknown
          du_ddata[count] = psi[l];
          // Increase the counter
          ++count;
        }
      }
    }
 
 
    /// Self-test: Return 0 for OK
    unsigned self_test();
 
  protected:
    /// 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_adv_diff(
      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_adv_diff(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Add the element's contribution to its residual vector only
    /// (if flag=and/or element  Jacobian matrix
    virtual void fill_in_generic_residual_contribution_adv_diff(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
    /// Pointer to global Peclet number
    double* Pe_pt;
 
    /// Pointer to global Peclet number multiplied by Strouhal number
    double* PeSt_pt;
 
    /// Pointer to source function:
    AdvectionDiffusionSourceFctPt Source_fct_pt;
 
    /// Pointer to wind function:
    AdvectionDiffusionWindFctPt Wind_fct_pt;
 
    /// Boolean flag to indicate if ALE formulation is disabled when
    /// time-derivatives are computed. Only set to true if you're sure
    /// that the mesh is stationary.
    bool ALE_is_disabled;
 
  private:
    /// Static default value for the Peclet number
    static double Default_peclet_number;
  };
 
 
  ///////////////////////////////////////////////////////////////////////////
  ///////////////////////////////////////////////////////////////////////////
  ///////////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// QAdvectionDiffusionElement elements are
  /// linear/quadrilateral/brick-shaped Advection Diffusion elements with
  /// isoparametric interpolation for the function.
  //======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  class QAdvectionDiffusionElement
    : public virtual QElement<DIM, NNODE_1D>,
      public virtual AdvectionDiffusionEquations<DIM>
  {
  private:
    /// Static array of ints to hold number of variables at
    /// nodes: Initial_Nvalue[n]
    static const unsigned Initial_Nvalue;
 
  public:
    /// Constructor: Call constructors for QElement and
    /// Advection Diffusion equations
    QAdvectionDiffusionElement()
      : QElement<DIM, NNODE_1D>(), AdvectionDiffusionEquations<DIM>()
    {
    }
 
    /// Broken copy constructor
    QAdvectionDiffusionElement(
      const QAdvectionDiffusionElement<DIM, NNODE_1D>& dummy) = delete;
 
    /// Broken assignment operator
    void operator=(const QAdvectionDiffusionElement<DIM, 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:
    ///  x,y,u   or    x,y,z,u
    void output(std::ostream& outfile)
    {
      AdvectionDiffusionEquations<DIM>::output(outfile);
    }
 
    /// Output function:
    ///  x,y,u   or    x,y,z,u at n_plot^DIM plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      AdvectionDiffusionEquations<DIM>::output(outfile, n_plot);
    }
 
 
    /// C-style output function:
    ///  x,y,u   or    x,y,z,u
    void output(FILE* file_pt)
    {
      AdvectionDiffusionEquations<DIM>::output(file_pt);
    }
 
    ///  C-style output function:
    ///   x,y,u   or    x,y,z,u at n_plot^DIM plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      AdvectionDiffusionEquations<DIM>::output(file_pt, n_plot);
    }
 
    /// Output function for an exact solution:
    ///  x,y,u_exact   or    x,y,z,u_exact at n_plot^DIM plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      AdvectionDiffusionEquations<DIM>::output_fct(
        outfile, n_plot, exact_soln_pt);
    }
 
 
    /// Output function for a time-dependent exact solution.
    ///  x,y,u_exact   or    x,y,z,u_exact at n_plot^DIM plot points
    /// (Calls the steady version)
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      AdvectionDiffusionEquations<DIM>::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_adv_diff(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_adv_diff(
      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 DIM, unsigned NNODE_1D>
  double QAdvectionDiffusionElement<DIM, NNODE_1D>::
    dshape_and_dtest_eulerian_adv_diff(const Vector<double>& s,
                                       Shape& psi,
                                       DShape& dpsidx,
                                       Shape& test,
                                       DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian(s, psi, dpsidx);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < NNODE_1D; i++)
    {
      test[i] = psi[i];
      for (unsigned j = 0; j < DIM; j++)
      {
        dtestdx(i, j) = dpsidx(i, j);
      }
    }
 
    // 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 DIM, unsigned NNODE_1D>
  double QAdvectionDiffusionElement<DIM, NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_adv_diff(const unsigned& ipt,
                                               Shape& psi,
                                               DShape& dpsidx,
                                               Shape& test,
                                               DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
 
    // Set the test functions equal to the shape functions (pointer copy)
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
 
  ////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the QAdvectionDiffusionElement 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 DIM, unsigned NNODE_1D>
  class FaceGeometry<QAdvectionDiffusionElement<DIM, NNODE_1D>>
    : public virtual QElement<DIM - 1, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : QElement<DIM - 1, NNODE_1D>() {}
  };
 
 
  ////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the 1D QAdvectionDiffusion elements: Point elements
  //=======================================================================
  template<unsigned NNODE_1D>
  class FaceGeometry<QAdvectionDiffusionElement<1, NNODE_1D>>
    : public virtual PointElement
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : PointElement() {}
  };
 
} // namespace oomph
 
#endif