discontinuous_galerkin_space_time_unsteady_heat_mixed_order_elements.h

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// Header file for SpaceTimeUnsteadyHeatMixedOrder elements
#ifndef OOMPH_DISCONTINUOUS_GALERKIN_SPACE_TIME_UNSTEADY_HEAT_MIXED_ORDER_ELEMENTS_HEADER
#define OOMPH_DISCONTINUOUS_GALERKIN_SPACE_TIME_UNSTEADY_HEAT_MIXED_ORDER_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// Oomph-lib headers
#include "generic/Qelements.h"
#include "generic/shape.h"
#include "generic/projection.h"
 
/// //////////////////////////////////////////////////////////////////////
/// //////////////////////////////////////////////////////////////////////
/// //////////////////////////////////////////////////////////////////////
 
namespace oomph
{
  //============================================================================
  /// Base class so that we don't need to know the dimension just to
  /// set the source function!
  //============================================================================
  class SpaceTimeUnsteadyHeatEquationsBase : public virtual FiniteElement
  {
  public:
    /// Function pointer to source function fct(t,x,f(x,t)) -- x
    /// is a Vector!
    typedef void (*SpaceTimeUnsteadyHeatSourceFctPt)(const double& time,
                                                     const Vector<double>& x,
                                                     double& u);
 
    /// Access function: Pointer to source function
    virtual SpaceTimeUnsteadyHeatSourceFctPt& source_fct_pt() = 0;
  };
 
  //============================================================================
  /// A class for all isoparametric elements that solve the
  /// SpaceTimeUnsteadyHeatMixedOrder equations.
  /// \f[ \frac{\partial^2 u}{\partial x_i^2}=\frac{\partial u}{\partial t}+f(t,x_j) \f]
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  /// Note that this class assumes an isoparametric formulation, i.e. that
  /// the scalar unknown is interpolated using the same shape funcitons
  /// as the position.
  //============================================================================
  template<unsigned SPATIAL_DIM>
  class SpaceTimeUnsteadyHeatMixedOrderEquations
    : public virtual SpaceTimeUnsteadyHeatEquationsBase
  {
  public:
    /// Constructor: Initialises the Source_fct_pt to null and sets
    /// flag to use ALE formulation of the equations. Also, set Alpha (thermal
    /// inertia) and Beta (thermal conductivity) parameters to defaults (both
    /// one for natural scaling).
    SpaceTimeUnsteadyHeatMixedOrderEquations()
      : Source_fct_pt(0), ALE_is_disabled(false)
    {
      // Set Alpha parameter to default (one for natural scaling)
      Alpha_pt = &Default_alpha_parameter;
 
      // Set Beta parameter to default (one for natural scaling)
      Beta_pt = &Default_beta_parameter;
    } // End of SpaceTimeUnsteadyHeatMixedOrderEquations
 
 
    /// Broken copy constructor
    SpaceTimeUnsteadyHeatMixedOrderEquations(
      const SpaceTimeUnsteadyHeatMixedOrderEquations& dummy) = delete;
 
    /// Disable ALE, i.e. assert the mesh is not moving -- you do this
    /// at your own risk!
    void disable_ALE()
    {
      // Set the flag to true
      ALE_is_disabled = true;
    } // End of disable_ALE
 
 
    /// (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()
    {
      // Set the flag to false
      ALE_is_disabled = false;
    } // End of enable_ALE
 
 
    /// Compute norm of FE solution
    void compute_norm(double& norm);
 
 
    /// Output with default number of plot points
    void output(std::ostream& outfile)
    {
      // Number of plot points
      unsigned nplot = 5;
 
      // Output the solution
      output(outfile, nplot);
    } // End of output
 
 
    /// Output FE representation of soln: x,y,u or x,y,z,u at
    /// nplot^SPATIAL_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)
    {
      // Number of plot points
      unsigned nplot = 5;
 
      // Output the solution
      output(file_pt, nplot);
    } // End of output
 
 
    /// C-style output FE representation of soln: x,y,u or x,y,z,u at
    /// nplot^SPATIAL_DIM plot points
    void output(FILE* file_pt, const unsigned& nplot);
 
 
    /// Output exact soln: x,y,u_exact or x,y,z,u_exact at nplot^SPATIAL_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^SPATIAL_DIM plot points (time-dependent version)
    virtual void output_fct(
      std::ostream& outfile,
      const unsigned& nplot,
      const double& time,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt);
 
 
    /// Get error and norm against exact solution
    void compute_error(std::ostream& outfile,
                       FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
 
    /// Get error and norm against exact solution
    void compute_error(std::ostream& outfile,
                       FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
                       const double& time,
                       double& error,
                       double& norm);
 
 
    /// C-style output FE representation of soln: x,y,u or x,y,z,u at
    /// nplot^SPATIAL_DIM plot points
    void output_element_paraview(std::ofstream& outfile, const unsigned& nplot);
 
 
    /// Number of scalars/fields output by this element. Reimplements
    /// broken virtual function in base class.
    unsigned nscalar_paraview() const
    {
      // Only one field to output
      return 1;
    } // End of nscalar_paraview
 
 
    /// 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
    {
#ifdef PARANOID
      if (i != 0)
      {
        std::stringstream error_stream;
        error_stream << "Space-time unsteady heat elements only store a single "
                     << "field so i must be 0 rather than " << i << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Get the number of plot points
      unsigned local_loop = this->nplot_points_paraview(nplot);
 
      // Loop over the plot points
      for (unsigned j = 0; j < local_loop; j++)
      {
        // Storage for the local coordinates
        Vector<double> s(SPATIAL_DIM + 1);
 
        // Get the local coordinate of the required plot point
        this->get_s_plot(j, nplot, s);
 
        // Output the interpolated solution value
        file_out << this->interpolated_u_ust_heat(s) << std::endl;
      }
    } // End of scalar_value_paraview
 
 
    /// 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_fct_paraview(
      std::ofstream& file_out,
      const unsigned& i,
      const unsigned& nplot,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt) const
    {
#ifdef PARANOID
      if (i != 0)
      {
        std::stringstream error_stream;
        error_stream << "Space-time unsteady heat elements only store a single "
                     << "field so i must be 0 rather than " << i << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Get the number of plot points
      unsigned local_loop = this->nplot_points_paraview(nplot);
 
      // Loop over the plot points
      for (unsigned j = 0; j < local_loop; j++)
      {
        // Storage for the local coordinates
        Vector<double> s(SPATIAL_DIM + 1);
 
        // Storage for the global coordinates
        Vector<double> spatial_coordinates(SPATIAL_DIM);
 
        // Get the local coordinate of the required plot point
        this->get_s_plot(j, nplot, s);
 
        // Loop over the spatial coordinates
        for (unsigned i = 0; i < SPATIAL_DIM; i++)
        {
          // Assign the i-th spatial coordinate
          spatial_coordinates[i] = interpolated_x(s, i);
        }
 
        // Exact solution vector (here it's simply a scalar)
        Vector<double> exact_soln(1, 0.0);
 
        // Get the exact solution at this point
        (*exact_soln_pt)(spatial_coordinates, exact_soln);
 
        // Output the interpolated solution value
        file_out << exact_soln[0] << std::endl;
      } // for (unsigned j=0;j<local_loop;j++)
    } // End of scalar_value_fct_paraview
 
 
    /// 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_fct_paraview(
      std::ofstream& file_out,
      const unsigned& i,
      const unsigned& nplot,
      const double& time,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt) const
    {
#ifdef PARANOID
      if (i != 0)
      {
        std::stringstream error_stream;
        error_stream << "Space-time unsteady heat elements only store a single "
                     << "field so i must be 0 rather than " << i << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Get the number of plot points
      unsigned local_loop = this->nplot_points_paraview(nplot);
 
      // Loop over the plot points
      for (unsigned j = 0; j < local_loop; j++)
      {
        // Storage for the local coordinates
        Vector<double> s(SPATIAL_DIM + 1);
 
        // Storage for the time value
        double interpolated_t = 0.0;
 
        // Storage for the global coordinates
        Vector<double> spatial_coordinates(SPATIAL_DIM);
 
        // Get the local coordinate of the required plot point
        this->get_s_plot(j, nplot, s);
 
        // Loop over the spatial coordinates
        for (unsigned i = 0; i < SPATIAL_DIM; i++)
        {
          // Assign the i-th spatial coordinate
          spatial_coordinates[i] = interpolated_x(s, i);
        }
 
        // Get the time value
        interpolated_t = interpolated_x(s, SPATIAL_DIM);
 
        // Exact solution vector (here it's simply a scalar)
        Vector<double> exact_soln(1, 0.0);
 
        // Get the exact solution at this point
        (*exact_soln_pt)(interpolated_t, spatial_coordinates, exact_soln);
 
        // Output the interpolated solution value
        file_out << exact_soln[0] << std::endl;
      } // for (unsigned j=0;j<local_loop;j++)
    } // End of scalar_value_fct_paraview
 
 
    /// Name of the i-th scalar field. Default implementation
    /// returns V1 for the first one, V2 for the second etc.
    std::string scalar_name_paraview(const unsigned& i) const
    {
      // If we're outputting the solution
      if (i == 0)
      {
        // There's only one field to output
        return "U";
      }
      // Never get here
      else
      {
        std::stringstream error_stream;
        error_stream << "These unsteady heat elements only store 1 field, \n"
                     << "but i is currently  " << i << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
 
        // Dummy return
        return " ";
      }
    } // End of scalar_name_paraview
 
 
    /// Access function: Pointer to source function
    SpaceTimeUnsteadyHeatSourceFctPt& source_fct_pt()
    {
      // Return the source function pointer
      return Source_fct_pt;
    } // End of source_fct_pt
 
 
    /// Access function: Pointer to source function. Const version
    SpaceTimeUnsteadyHeatSourceFctPt source_fct_pt() const
    {
      // Return the source function pointer
      return Source_fct_pt;
    }
 
 
    /// Get source term at continous time t and (Eulerian) position x.
    /// Virtual so it can be overloaded in derived multi-physics elements.
    virtual inline void get_source_ust_heat(const double& t,
                                            const unsigned& ipt,
                                            const Vector<double>& x,
                                            double& source) const
    {
      // If no source function has been set, return zero
      if (Source_fct_pt == 0)
      {
        // Set the source term value to zero
        source = 0.0;
      }
      // Otherwise return the appropriate value
      else
      {
        // Get source strength
        (*Source_fct_pt)(t, x, source);
      }
    } // End of get_source_ust_heat
 
 
    /// Alpha parameter (thermal inertia)
    const double& alpha() const
    {
      // Return the value of Alpha
      return *Alpha_pt;
    } // End of alpha
 
 
    /// Pointer to Alpha parameter (thermal inertia)
    double*& alpha_pt()
    {
      // Return the pointer to Alpha
      return Alpha_pt;
    } // End of alpha_pt
 
 
    /// Beta parameter (thermal conductivity)
    const double& beta() const
    {
      // Return the pointer to Beta
      return *Beta_pt;
    } // End of beta
 
 
    /// Pointer to Beta parameter (thermal conductivity)
    double*& beta_pt()
    {
      // Return the pointer to Beta
      return Beta_pt;
    } // End of beta_pt
 
 
    /// Get flux: flux[i]=du/dx_i
    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();
 
      // Find the index at which the variable is stored
      unsigned u_nodal_index = u_index_ust_heat();
 
      // Set up memory for the shape and test functions
      Shape psi(n_node);
 
      // Set up memory for the derivatives of the shape functions
      DShape dpsidx(n_node, SPATIAL_DIM + 1);
 
      //------------dshape_eulerian(s,psi,dpsidx)----------------------------
      // Find the element dimension
      const unsigned el_dim = this->dim();
 
      // Get the values of the shape functions and their local derivatives;
      // temporarily stored in dpsi
      dshape_local_ust_heat(s, psi, dpsidx);
 
      // Allocate memory for the inverse jacobian
      DenseMatrix<double> inverse_jacobian(el_dim);
 
      // Now calculate the inverse jacobian
      local_to_eulerian_mapping(dpsidx, inverse_jacobian);
 
      // Now set the values of the derivatives to be dpsidx
      transform_derivatives(inverse_jacobian, dpsidx);
      //------------dshape_eulerian(s,psi,dpsidx)----------------------------
 
      // Loop over the entries of the flux vector
      for (unsigned j = 0; j < SPATIAL_DIM; j++)
      {
        // Initialise j-th flux entry to zero
        flux[j] = 0.0;
      }
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over derivative directions
        for (unsigned j = 0; j < SPATIAL_DIM; j++)
        {
          // Update the flux value
          flux[j] += nodal_value(l, u_nodal_index) * dpsidx(l, j);
        }
      } // for (unsigned l=0;l<n_node;l++)
    } // End of get_flux
 
 
    /// Compute element 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_ust_heat(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    } // End of fill_in_contribution_to_residuals
 
 
    /// Compute element 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_ust_heat(residuals, jacobian, 1);
    } // End of fill_in_contribution_to_jacobian
 
 
    /// Return FE representation of function value u(s) at local coordinate s
    inline double interpolated_u_ust_heat(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Find the index at which the variable is stored
      unsigned u_nodal_index = u_index_ust_heat();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of the shape functions at local coordinate s
      shape_ust_heat(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++)
      {
        // Update the interpolated u value
        interpolated_u += nodal_value(l, u_nodal_index) * psi[l];
      }
 
      // Return the interpolated u value
      return interpolated_u;
    } // End of interpolated_u_ust_heat
 
 
    /// 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
    /// unsteady heat 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_ust_heat() const
    {
      // Return the default value
      return 0;
    } // End of u_index_ust_heat
 
 
    /// Return FE representation of function value u(s) at local
    /// coordinate s at previous time t (t=0: present)
    /// DRAIG: This needs to be broken; doesn't make sense in space-time
    /// elements!
    inline double interpolated_u_ust_heat(const unsigned& t,
                                          const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Find the index at which the variable is stored
      unsigned u_nodal_index = u_index_ust_heat();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape_ust_heat(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++)
      {
        // Update the interpolated u value
        interpolated_u += nodal_value(t, l, u_nodal_index) * psi[l];
      }
 
      // Return the interpolated u value
      return interpolated_u;
    } // End of interpolated_u_ust_heat
 
 
    /// Calculate du/dt at the n-th local node. Uses suitably
    /// interpolated value for hanging nodes.
    double du_dt_ust_heat(const unsigned& n) const
    {
      // Storage for the local coordinates
      Vector<double> s(SPATIAL_DIM + 1, 0.0);
 
      // Get the local coordinate at the n-th node
      local_coordinate_of_node(n, s);
 
      // Return the interpolated du/dt value
      return interpolated_du_dt_ust_heat(s);
    } // End of du_dt_ust_heat
 
 
    /// Return FE representation of function value du/dt(s) at local coordinate
    /// s
    inline double interpolated_du_dt_ust_heat(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Find the index at which the variable is stored
      unsigned u_nodal_index = u_index_ust_heat();
 
      // Local shape function
      Shape psi(n_node);
 
      // Allocate space for the derivatives of the shape functions
      DShape dpsidx(n_node, SPATIAL_DIM + 1);
 
      //------------dshape_eulerian(s,psi,dpsidx)----------------------------
      // Find the element dimension
      const unsigned el_dim = this->dim();
 
      // Get the values of the shape functions and their local derivatives;
      // temporarily stored in dpsi
      dshape_local_ust_heat(s, psi, dpsidx);
 
      // Allocate memory for the inverse jacobian
      DenseMatrix<double> inverse_jacobian(el_dim);
 
      // Now calculate the inverse jacobian
      local_to_eulerian_mapping(dpsidx, inverse_jacobian);
 
      // Now set the values of the derivatives to be dpsidx
      transform_derivatives(inverse_jacobian, dpsidx);
      //------------dshape_eulerian(s,psi,dpsidx)----------------------------
 
      // Initialise value of du/dt
      double interpolated_dudt = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        // Update the interpolated du/dt value
        interpolated_dudt +=
          nodal_value(l, u_nodal_index) * dpsidx(l, SPATIAL_DIM);
      }
 
      // Return the interpolated du/dt value
      return interpolated_dudt;
    } // End of interpolated_du_dt_ust_heat
 
 
    /// Self-test: Return 0 for OK
    unsigned self_test();
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// local coordinate s; return Jacobian of mapping
    virtual double dshape_and_dtest_eulerian_ust_heat(
      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_ust_heat(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
  protected:
    /// Shape functions w.r.t. to local coords
    virtual void shape_ust_heat(const Vector<double>& s, Shape& psi) const = 0;
 
 
    /// Shape functions & derivs. w.r.t. to local coords
    virtual void dshape_local_ust_heat(const Vector<double>& s,
                                       Shape& psi,
                                       DShape& dpsidx) const = 0;
 
 
    /// Test functions & derivs. w.r.t. to local coords
    virtual void dtest_local_ust_heat(const Vector<double>& s,
                                      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_ust_heat(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag);
 
    /// Pointer to source function:
    SpaceTimeUnsteadyHeatSourceFctPt Source_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;
 
    /// Pointer to Alpha parameter (thermal inertia)
    double* Alpha_pt;
 
    /// Pointer to Beta parameter (thermal conductivity)
    double* Beta_pt;
 
  private:
    /// Static default value for the Alpha parameter (thermal inertia):
    /// One for natural scaling
    static double Default_alpha_parameter;
 
    /// Static default value for the Beta parameter (thermal
    /// conductivity): One for natural scaling
    static double Default_beta_parameter;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
 
  //=========================================================================
  /// QUnsteadyHeatMixedOrderSpaceTimeElement elements are quadrilateral/brick-
  /// shaped UnsteadyHeatMixedOrder elements with isoparametric interpolation
  /// for the function.
  //=========================================================================
  template<unsigned SPATIAL_DIM, unsigned NNODE_1D>
  class QUnsteadyHeatMixedOrderSpaceTimeElement
    : public virtual QElement<SPATIAL_DIM + 1, NNODE_1D>,
      public virtual SpaceTimeUnsteadyHeatMixedOrderEquations<SPATIAL_DIM>
  {
  public:
    /// Constructor: Call constructors for QElement and
    /// SpaceTimeUnsteadyHeatMixedOrder equations
    QUnsteadyHeatMixedOrderSpaceTimeElement()
      : QElement<SPATIAL_DIM + 1, NNODE_1D>(),
        SpaceTimeUnsteadyHeatMixedOrderEquations<SPATIAL_DIM>()
    {
    }
 
    /// Broken copy constructor
    QUnsteadyHeatMixedOrderSpaceTimeElement(
      const QUnsteadyHeatMixedOrderSpaceTimeElement<SPATIAL_DIM, NNODE_1D>&
        dummy) = delete;
 
    /// Required number of 'values' (pinned or dofs) at node n
    inline unsigned required_nvalue(const unsigned& n) const
    {
      // Return the appropriate value
      return Initial_Nvalue;
    } // End of required_nvalue
 
 
    /// Output function:
    /// x,t,u   or   x,y,t,u
    void output(std::ostream& outfile)
    {
      // Call the function in the base class
      SpaceTimeUnsteadyHeatMixedOrderEquations<SPATIAL_DIM>::output(outfile);
    } // End of output
 
 
    /// Output function:
    /// x,t,u   or   x,y,t,u at n_plot^(SPATIAL_DIM+1) plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      // Call the function in the base class
      SpaceTimeUnsteadyHeatMixedOrderEquations<SPATIAL_DIM>::output(outfile,
                                                                    n_plot);
    } // End of output
 
 
    /// C-style output function:
    /// x,t,u   or   x,y,t,u
    void output(FILE* file_pt)
    {
      // Call the function in the base class
      SpaceTimeUnsteadyHeatMixedOrderEquations<SPATIAL_DIM>::output(file_pt);
    } // End of output
 
 
    /// C-style output function:
    /// x,t,u   or   x,y,t,u at n_plot^(SPATIAL_DIM+1) plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      // Call the function in the base class
      SpaceTimeUnsteadyHeatMixedOrderEquations<SPATIAL_DIM>::output(file_pt,
                                                                    n_plot);
    } // End of output
 
 
    /// Output function for an exact solution:
    /// x,t,u_exact   or   x,y,t,u_exact at n_plot^(SPATIAL_DIM+1) plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      // Call the function in the base class
      SpaceTimeUnsteadyHeatMixedOrderEquations<SPATIAL_DIM>::output_fct(
        outfile, n_plot, exact_soln_pt);
    } // End of output_fct
 
 
    /// Output function for a time-dependent exact solution.
    ///  x,t,u_exact   or    x,y,t,u_exact at n_plot^(SPATIAL_DIM+1) plot points
    /// (Calls the unsteady version)
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      // Call the function in the base class
      SpaceTimeUnsteadyHeatMixedOrderEquations<SPATIAL_DIM>::output_fct(
        outfile, n_plot, time, exact_soln_pt);
    } // End of output_fct
 
 
    /// Shape/test functions & derivs. w.r.t. to global coords. Return Jacobian.
    inline double dshape_and_dtest_eulerian_ust_heat(const Vector<double>& s,
                                                     Shape& psi,
                                                     DShape& dpsidx,
                                                     Shape& test,
                                                     DShape& dtestdx) const;
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// integration point ipt; return Jacobian of mapping
    inline double dshape_and_dtest_eulerian_at_knot_ust_heat(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
  protected:
    /// Shape functions w.r.t. to local coords
    inline void shape_ust_heat(const Vector<double>& s, Shape& psi) const;
 
 
    /// Shape functions & derivs. w.r.t. to local coords
    inline void dshape_local_ust_heat(const Vector<double>& s,
                                      Shape& psi,
                                      DShape& dpsidx) const;
 
 
    /// Test functions & derivs. w.r.t. to local coords
    inline void dtest_local_ust_heat(const Vector<double>& s,
                                     Shape& test,
                                     DShape& dtestdx) const;
 
 
  private:
    /// Static array of ints to hold number of variables at nodes:
    /// Initial_Nvalue[n]
    static const unsigned Initial_Nvalue;
  };
 
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. local coordinates
  ///
  /// Galerkin: Test functions=shape functions
  //======================================================================
  template<unsigned SPATIAL_DIM, unsigned NNODE_1D>
  inline void QUnsteadyHeatMixedOrderSpaceTimeElement<SPATIAL_DIM, NNODE_1D>::
    shape_ust_heat(const Vector<double>& s, Shape& psi) const
  {
    // Local storage (for 3D = 2D space + 1D time)
    double psi_values[3][NNODE_1D];
 
    // Index of the total shape function
    unsigned index = 0;
 
    // Call the 1D shape functions and derivatives
    OneDimLagrange::shape<NNODE_1D>(s[0], psi_values[0]);
    OneDimLagrange::shape<NNODE_1D>(s[1], psi_values[1]);
 
    // Set the time discretisation
    OneDimDiscontinuousGalerkinMixedOrderBasis::shape<NNODE_1D>(s[2],
                                                                psi_values[2]);
 
    // Loop over the nodes in the third local coordinate direction
    for (unsigned k = 0; k < NNODE_1D; k++)
    {
      // Loop over the nodes in the second local coordinate direction
      for (unsigned j = 0; j < NNODE_1D; j++)
      {
        // Loop over the nodes in the first local coordinate direction
        for (unsigned i = 0; i < NNODE_1D; i++)
        {
          // Calculate the index-th entry of psi
          psi[index] = psi_values[0][i] * psi_values[1][j] * psi_values[2][k];
 
          // Increment the index
          index++;
        }
      } // for (unsigned j=0;j<NNODE_1D;j++)
    } // for (unsigned k=0;k<NNODE_1D;k++)
  } // End of shape_ust_heat
 
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. local coordinates
  ///
  /// Galerkin: Test functions=shape functions
  //======================================================================
  template<unsigned SPATIAL_DIM, unsigned NNODE_1D>
  inline void QUnsteadyHeatMixedOrderSpaceTimeElement<SPATIAL_DIM, NNODE_1D>::
    dshape_local_ust_heat(const Vector<double>& s,
                          Shape& psi,
                          DShape& dpsidx) const
  {
    // Local storage (for 3D = 2D space + 1D time)
    double psi_values[3][NNODE_1D];
    double dpsi_values[3][NNODE_1D];
 
    // Index of the total shape function
    unsigned index = 0;
 
    // Call the 1D shape functions and derivatives
    OneDimLagrange::shape<NNODE_1D>(s[0], psi_values[0]);
    OneDimLagrange::shape<NNODE_1D>(s[1], psi_values[1]);
    OneDimLagrange::dshape<NNODE_1D>(s[0], dpsi_values[0]);
    OneDimLagrange::dshape<NNODE_1D>(s[1], dpsi_values[1]);
 
    // Set the time discretisation
    OneDimDiscontinuousGalerkinMixedOrderBasis::shape<NNODE_1D>(s[2],
                                                                psi_values[2]);
    OneDimDiscontinuousGalerkinMixedOrderBasis::dshape<NNODE_1D>(
      s[2], dpsi_values[2]);
 
    // Loop over the nodes in the third local coordinate direction
    for (unsigned k = 0; k < NNODE_1D; k++)
    {
      // Loop over the nodes in the second local coordinate direction
      for (unsigned j = 0; j < NNODE_1D; j++)
      {
        // Loop over the nodes in the first local coordinate direction
        for (unsigned i = 0; i < NNODE_1D; i++)
        {
          // Calculate dpsi/ds_0
          dpsidx(index, 0) =
            dpsi_values[0][i] * psi_values[1][j] * psi_values[2][k];
 
          // Calculate dpsi/ds_1
          dpsidx(index, 1) =
            psi_values[0][i] * dpsi_values[1][j] * psi_values[2][k];
 
          // Calculate dpsi/ds_2
          dpsidx(index, 2) =
            psi_values[0][i] * psi_values[1][j] * dpsi_values[2][k];
 
          // Calculate the index-th entry of psi
          psi[index] = psi_values[0][i] * psi_values[1][j] * psi_values[2][k];
 
          // Increment the index
          index++;
        }
      } // for (unsigned j=0;j<NNODE_1D;j++)
    } // for (unsigned k=0;k<NNODE_1D;k++)
  } // End of dshape_local_ust_heat
 
 
  //======================================================================
  /// Define the test functions and derivatives w.r.t. local coordinates
  //======================================================================
  template<unsigned SPATIAL_DIM, unsigned NNODE_1D>
  inline void QUnsteadyHeatMixedOrderSpaceTimeElement<SPATIAL_DIM, NNODE_1D>::
    dtest_local_ust_heat(const Vector<double>& s,
                         Shape& test,
                         DShape& dtestdx) const
  {
    // Local storage (for 3D = 2D space + 1D time)
    double test_values[3][NNODE_1D];
    double dtest_values[3][NNODE_1D];
 
    // Index of the total shape function
    unsigned index = 0;
 
    // Call the 1D shape functions and derivatives
    OneDimLagrange::shape<NNODE_1D>(s[0], test_values[0]);
    OneDimLagrange::shape<NNODE_1D>(s[1], test_values[1]);
    OneDimLagrange::dshape<NNODE_1D>(s[0], dtest_values[0]);
    OneDimLagrange::dshape<NNODE_1D>(s[1], dtest_values[1]);
 
    // Set the time discretisation
    OneDimDiscontinuousGalerkinMixedOrderTest::shape<NNODE_1D>(s[2],
                                                               test_values[2]);
    OneDimDiscontinuousGalerkinMixedOrderTest::dshape<NNODE_1D>(
      s[2], dtest_values[2]);
 
    // Loop over the nodes in the third local coordinate direction
    for (unsigned k = 0; k < NNODE_1D; k++)
    {
      // Loop over the nodes in the second local coordinate direction
      for (unsigned j = 0; j < NNODE_1D; j++)
      {
        // Loop over the nodes in the first local coordinate direction
        for (unsigned i = 0; i < NNODE_1D; i++)
        {
          // Calculate dtest/ds_0
          dtestdx(index, 0) =
            dtest_values[0][i] * test_values[1][j] * test_values[2][k];
 
          // Calculate dtest/ds_1
          dtestdx(index, 1) =
            test_values[0][i] * dtest_values[1][j] * test_values[2][k];
 
          // Calculate dtest/ds_2
          dtestdx(index, 2) =
            test_values[0][i] * test_values[1][j] * dtest_values[2][k];
 
          // Calculate the index-th entry of test
          test[index] =
            test_values[0][i] * test_values[1][j] * test_values[2][k];
 
          // Increment the index
          index++;
        }
      } // for (unsigned j=0;j<NNODE_1D;j++)
    } // for (unsigned k=0;k<NNODE_1D;k++)
  } // End of dtest_local_ust_heat
 
 
  //======================================================================
  /// 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 SPATIAL_DIM, unsigned NNODE_1D>
  inline double QUnsteadyHeatMixedOrderSpaceTimeElement<SPATIAL_DIM, NNODE_1D>::
    dshape_and_dtest_eulerian_ust_heat(const Vector<double>& s,
                                       Shape& psi,
                                       DShape& dpsidx,
                                       Shape& test,
                                       DShape& dtestdx) const
  {
    //--------------------------
    // Call the shape functions:
    //--------------------------
    // Find the element dimension
    const unsigned el_dim = this->dim();
 
    // Make sure we're using 3D space-time elements
    if (el_dim != 3)
    {
      // Create an output stream
      std::ostringstream error_message_stream;
 
      // Create an error message
      error_message_stream << "Need 3D space-time elements for this to work!"
                           << std::endl;
 
      // Throw the error message
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    // Compute the geometric shape functions and also first derivatives
    // w.r.t. local coordinates at local coordinate s
    dshape_local_ust_heat(s, psi, dpsidx);
 
    // Allocate memory for the inverse jacobian
    DenseMatrix<double> inverse_jacobian(el_dim);
 
    // Now calculate the inverse jacobian
    const double det =
      this->local_to_eulerian_mapping(dpsidx, inverse_jacobian);
 
    // Now set the values of the derivatives to be dpsidx
    this->transform_derivatives(inverse_jacobian, dpsidx);
 
    //-------------------------
    // Call the test functions:
    //-------------------------
    // Compute the geometric test functions and also first derivatives
    // w.r.t. local coordinates at local coordinate s
    dtest_local_ust_heat(s, test, dtestdx);
 
    // Transform derivatives from dtest/ds to dtest/dx
    this->transform_derivatives(inverse_jacobian, dtestdx);
 
    // Return the determinant value
    return det;
  } // End of dshape_and_dtest_eulerian_ust_heat
 
 
  //======================================================================
  /// 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 SPATIAL_DIM, unsigned NNODE_1D>
  inline double QUnsteadyHeatMixedOrderSpaceTimeElement<SPATIAL_DIM, NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_ust_heat(const unsigned& ipt,
                                               Shape& psi,
                                               DShape& dpsidx,
                                               Shape& test,
                                               DShape& dtestdx) const
  {
    // Find the element dimension
    const unsigned el_dim = SPATIAL_DIM + 1;
 
    // Storage for the local coordinates of the integration point
    Vector<double> s(el_dim, 0.0);
 
    // Set the local coordinate
    for (unsigned i = 0; i < el_dim; i++)
    {
      // Calculate the i-th local coordinate at the ipt-th knot point
      s[i] = this->integral_pt()->knot(ipt, i);
    }
 
    // Return the Jacobian of the geometrical shape functions and derivatives
    return dshape_and_dtest_eulerian_ust_heat(s, psi, dpsidx, test, dtestdx);
  } // End of dshape_and_dtest_eulerian_at_knot_ust_heat
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the QUnsteadyHeatMixedOrderSpaceTimeElement 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 SPATIAL_DIM, unsigned NNODE_1D>
  class FaceGeometry<
    QUnsteadyHeatMixedOrderSpaceTimeElement<SPATIAL_DIM, NNODE_1D>>
    : public virtual QElement<SPATIAL_DIM, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the appropriate
    /// lower-dimensional QElement
    FaceGeometry() : QElement<SPATIAL_DIM, NNODE_1D>() {}
  };
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
  //=======================================================================
  /// Face geometry for the 1D QUnsteadyHeatMixedOrderSpaceTimeElement
  /// elements: Point elements
  //=======================================================================
  template<unsigned NNODE_1D>
  class FaceGeometry<QUnsteadyHeatMixedOrderSpaceTimeElement<1, NNODE_1D>>
    : public virtual PointElement
  {
  public:
    /// Constructor: Call the constructor for the appropriate
    /// lower-dimensional QElement
    FaceGeometry() : PointElement() {}
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //==========================================================
  /// SpaceTimeUnsteadyHeatMixedOrder upgraded to become projectable
  //==========================================================
  template<class UNSTEADY_HEAT_ELEMENT>
  class ProjectableUnsteadyHeatMixedOrderSpaceTimeElement
    : public virtual ProjectableElement<UNSTEADY_HEAT_ELEMENT>
  {
  public:
    /// Constructor [this was only required explicitly
    /// from gcc 4.5.2 onwards...]
    ProjectableUnsteadyHeatMixedOrderSpaceTimeElement() {}
 
 
    /// 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 we're not dealing with the first field
      if (fld != 0)
      {
        // Create a stringstream object to create an error message
        std::stringstream error_stream;
 
        // Create the error string
        error_stream
          << "SpaceTimeUnsteadyHeatMixedOrder elements only store a single "
          << "field so fld must be 0 rather than " << fld << std::endl;
 
        // Throw an error
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // The number of nodes in this element
      unsigned nnod = this->nnode();
 
      // Storage for the pairs
      Vector<std::pair<Data*, unsigned>> data_values(nnod);
 
      // Loop over all nodes
      for (unsigned j = 0; j < nnod; j++)
      {
        // Add the data value and associated field: The node itself
        data_values[j] = std::make_pair(this->node_pt(j), fld);
      }
 
      // Return the vector
      return data_values;
    } // End of data_values_of_field
 
 
    /// Number of fields to be projected: Just one
    unsigned nfields_for_projection()
    {
      // Return the appropriate value
      return 1;
    } // End of nfields_for_projection
 
 
    /// 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 we're not dealing with the first field
      if (fld != 0)
      {
        // Create a stringstream object to create an error message
        std::stringstream error_stream;
 
        // Create the error string
        error_stream
          << "SpaceTimeUnsteadyHeatMixedOrder elements only store a single "
          << "field so fld must be 0 rather than " << fld << std::endl;
 
        // Throw an error
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Return the number of stored values
      return this->node_pt(0)->ntstorage();
    } // End of nhistory_values_for_projection
 
 
    /// Number of positional history values (Note: count includes
    /// current value!)
    unsigned nhistory_values_for_coordinate_projection()
    {
      // Return the number of history values stored by the position timestepper
      return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
    } // End of nhistory_values_for_coordinate_projection
 
 
    /// 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 we're not dealing with the first field
      if (fld != 0)
      {
        // Create a stringstream object to create an error message
        std::stringstream error_stream;
 
        // Create the error string
        error_stream
          << "SpaceTimeUnsteadyHeatMixedOrder elements only store a single "
          << "field so fld must be 0 rather than " << fld << std::endl;
 
        // Throw an error
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Get the number of dimensions in the element
      unsigned n_dim = this->dim();
 
      // Get the number of nodes in the element
      unsigned n_node = this->nnode();
 
      // Allocate space for the test functions
      Shape test(n_node);
 
      // Allocate space for the derivatives of the shape functions
      DShape dpsidx(n_node, n_dim);
 
      // Allocate space for the derivatives of the test functions
      DShape dtestdx(n_node, n_dim);
 
      // Calculate the shape functions and their derivatives at the local
      // coordinate s (and the same for the test functions). On top of this
      // calculate the determinant of the Jacobian
      double J =
        this->dshape_and_dtest_eulerian_ust_heat(s, psi, dpsidx, test, dtestdx);
 
      // Return the determinant of the Jacobian
      return J;
    } // End of jacobian_and_shape_of_field
 
 
    /// 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 we're not dealing with the first field
      if (fld != 0)
      {
        // Create a stringstream object to create an error message
        std::stringstream error_stream;
 
        // Create the error string
        error_stream
          << "SpaceTimeUnsteadyHeatMixedOrder elements only store a single "
          << "field so fld must be 0 rather than " << fld << std::endl;
 
        // Throw an error
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Find the index at which the variable is stored
      unsigned u_nodal_index = this->u_index_ust_heat();
 
      // Get the number of nodes in the element
      unsigned n_node = this->nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      this->shape(s, psi);
 
      // Initialise value of u
      double interpolated_u = 0.0;
 
      // Loop over the local nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Update the interpolated solution value
        interpolated_u += this->nodal_value(t, l, u_nodal_index) * psi[l];
      }
 
      // Return the interpolated solution value
      return interpolated_u;
    } // End of get_field
 
 
    /// Return number of values in field fld: One per node
    unsigned nvalue_of_field(const unsigned& fld)
    {
#ifdef PARANOID
      // If we're not dealing with the first field
      if (fld != 0)
      {
        // Create a stringstream object to create an error message
        std::stringstream error_stream;
 
        // Create the error string
        error_stream
          << "SpaceTimeUnsteadyHeatMixedOrder elements only store a single "
          << "field so fld must be 0 rather than " << fld << std::endl;
 
        // Throw an error
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Return the number of nodes in the element
      return this->nnode();
    } // End of nvalue_of_field
 
 
    /// Return local equation number of value j in field fld.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
#ifdef PARANOID
      // If we're not dealing with the first field
      if (fld != 0)
      {
        // Create a stringstream object to create an error message
        std::stringstream error_stream;
 
        // Create the error string
        error_stream << "SpaceTimeUnsteadyHeatMixedOrder elements only store a "
                     << "single field so fld must be 0 rather than " << fld
                     << std::endl;
 
        // Throw an error
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Get the nodal index of the unknown
      const unsigned u_nodal_index = this->u_index_ust_heat();
 
      // Output the local equation number
      return this->nodal_local_eqn(j, u_nodal_index);
    } // End of local_equation
 
 
    /// Output FE representation of soln: x,t,u or x,y,t,u
    /// at n_plot^(SPATIAL_DIM+1) plot points
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      // Get the dimension of the element
      unsigned el_dim = this->dim();
 
      // Vector of local coordinates
      Vector<double> s(el_dim, 0.0);
 
      // Tecplot header info
      outfile << this->tecplot_zone_string(nplot);
 
      // Get the number of plot points
      unsigned num_plot_points = this->nplot_points(nplot);
 
      // Loop over plot points
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        this->get_s_plot(iplot, nplot, s);
 
        // Loop over the coordinate directions
        for (unsigned i = 0; i < el_dim; i++)
        {
          // Output the interpolated coordinates
          outfile << this->interpolated_x(s, i) << " ";
        }
 
        // Output the interpolated value of u(s)
        outfile << this->interpolated_u_ust_heat(s) << " ";
 
        // Output the interpolated value of du/dt(s)
        outfile << this->interpolated_du_dt_ust_heat(s) << " ";
 
        // History values of coordinates
        unsigned n_prev =
          this->node_pt(0)->position_time_stepper_pt()->ntstorage();
 
        // Loop over the previous timesteps
        for (unsigned t = 1; t < n_prev; t++)
        {
          // Loop over the coordinate directions
          for (unsigned i = 0; i < el_dim; i++)
          {
            // Output the coordinates
            outfile << this->interpolated_x(t, s, i) << " ";
          }
        } // for (unsigned t=1;t<n_prev;t++)
 
        // Number of history values of velocities
        n_prev = this->node_pt(0)->time_stepper_pt()->ntstorage();
 
        // Loop over the previous timesteps
        for (unsigned t = 1; t < n_prev; t++)
        {
          // Output the solution
          outfile << this->interpolated_u_ust_heat(t, s) << " ";
        }
 
        // Finish the line
        outfile << std::endl;
      } // for (unsigned iplot=0;iplot<num_plot_points;iplot++)
 
      // Write tecplot footer (e.g. FE connectivity lists)
      this->write_tecplot_zone_footer(outfile, nplot);
    } // End of output
  };
 
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<ProjectableUnsteadyHeatMixedOrderSpaceTimeElement<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<ProjectableUnsteadyHeatMixedOrderSpaceTimeElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
} // End of namespace oomph
 
#endif