advection_diffusion_reaction_elements.h

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// LIC// ====================================================================
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// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2024 Matthias Heil and Andrew Hazel
// LIC//
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// Header file for Advection Diffusion elements
#ifndef OOMPH_ADV_DIFF_REACT_ELEMENTS_HEADER
#define OOMPH_ADV_DIFF_REACT_ELEMENTS_HEADER
 
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// OOMPH-LIB headers
#include "../generic/projection.h"
#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 Reaction equations using isoparametric elements.
  /// \f[ \tau_{i} \frac{\partial C_{i}}{\partial t} + w_{j} \frac{\partial C_{i}}{\partial x_{j}} = D_{i}\frac{\partial^2 C_{i}}{\partial x_j^2} = - R_{i}(C_{i}) + f_{i} \f]
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  //=============================================================
  template<unsigned NREAGENT, unsigned DIM>
  class AdvectionDiffusionReactionEquations : public virtual FiniteElement
  {
  public:
    /// Function pointer to source function fct(x,f(x)) --
    /// x is a Vector!
    typedef void (*AdvectionDiffusionReactionSourceFctPt)(
      const Vector<double>& x, Vector<double>& f);
 
    /// Function pointer to reaction terms
    typedef void (*AdvectionDiffusionReactionReactionFctPt)(
      const Vector<double>& c, Vector<double>& R);
 
    /// Function pointer to derivative of reaction terms
    typedef void (*AdvectionDiffusionReactionReactionDerivFctPt)(
      const Vector<double>& c, DenseMatrix<double>& dRdC);
 
 
    /// Function pointer to wind function fct(x,w(x)) --
    /// x is a Vector!
    typedef void (*AdvectionDiffusionReactionWindFctPt)(const double& time,
                                                        const Vector<double>& x,
                                                        Vector<double>& wind);
 
 
    /// Constructor: Initialise the Source_fct_pt, Wind_fct_pt,
    /// Reaction_fct_pt to null and initialise the dimensionless
    /// timescale and diffusion ratios
    AdvectionDiffusionReactionEquations()
      : Source_fct_pt(0),
        Wind_fct_pt(0),
        Reaction_fct_pt(0),
        Reaction_deriv_fct_pt(0),
        ALE_is_disabled(false)
    {
      // Set diffusion coefficients to default
      Diff_pt = &Default_dimensionless_number;
      // Set timescales to default
      Tau_pt = &Default_dimensionless_number;
    }
 
    /// Broken copy constructor
    AdvectionDiffusionReactionEquations(
      const AdvectionDiffusionReactionEquations& dummy) = delete;
 
    /// Broken assignment operator
    void operator=(const AdvectionDiffusionReactionEquations&) = delete;
 
    /// Return the index at which the unknown i-th reagent
    /// is stored. The default value, i, is appropriate for single-physics
    /// problems, when there are only i variables, the values that satisfies
    /// the advection-diffusion-reaction 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 c_index_adv_diff_react(const unsigned& i) const
    {
      return i;
    }
 
    /// dc_r/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double dc_dt_adv_diff_react(const unsigned& n, const unsigned& r) 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 c_nodal_index = c_index_adv_diff_react(r);
 
        // 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, c_nodal_index);
        }
      }
      return dudt;
    }
 
 
    /// dc/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    void dc_dt_adv_diff_react(const unsigned& n, Vector<double>& dc_dt) const
    {
      // Get the data's timestepper
      TimeStepper* time_stepper_pt = this->node_pt(n)->time_stepper_pt();
 
      // Initialise to zero
      for (unsigned r = 0; r < NREAGENT; r++)
      {
        dc_dt[r] = 0.0;
      }
 
      // Loop over the timesteps, if there is a non Steady timestepper
      if (!time_stepper_pt->is_steady())
      {
        // Number of timsteps (past & present)
        const unsigned n_time = time_stepper_pt->ntstorage();
        // Local storage (cache) for the weights
        double weight[n_time];
        for (unsigned t = 0; t < n_time; t++)
        {
          weight[t] = time_stepper_pt->weight(1, t);
        }
 
        // Loop over the reagents
        for (unsigned r = 0; r < NREAGENT; r++)
        {
          // Find the index at which the variable is stored
          const unsigned c_nodal_index = c_index_adv_diff_react(r);
 
          for (unsigned t = 0; t < n_time; t++)
          {
            dc_dt[r] += weight[t] * nodal_value(t, n, c_nodal_index);
          }
        }
      }
    }
 
    /// 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;
    }
 
 
    /// 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);
    }
 
    /// Compute norm of the solution:
    /// sum of squares of the L2 norm for each reagent
    void compute_norm(double& norm);
 
    /// 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
    AdvectionDiffusionReactionSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to source function. Const version
    AdvectionDiffusionReactionSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to wind function
    AdvectionDiffusionReactionWindFctPt& wind_fct_pt()
    {
      return Wind_fct_pt;
    }
 
    /// Access function: Pointer to reaction function. Const version
    AdvectionDiffusionReactionWindFctPt wind_fct_pt() const
    {
      return Wind_fct_pt;
    }
 
    /// Access function: Pointer to reaction function
    AdvectionDiffusionReactionReactionFctPt& reaction_fct_pt()
    {
      return Reaction_fct_pt;
    }
 
    /// Access function: Pointer to reaction function. Const version
    AdvectionDiffusionReactionReactionFctPt reaction_fct_pt() const
    {
      return Reaction_fct_pt;
    }
 
    /// Access function: Pointer to reaction derivatives function
    AdvectionDiffusionReactionReactionDerivFctPt& reaction_deriv_fct_pt()
    {
      return Reaction_deriv_fct_pt;
    }
 
    /// Access function: Pointer to reaction function. Const version
    AdvectionDiffusionReactionReactionDerivFctPt reaction_deriv_fct_pt() const
    {
      return Reaction_deriv_fct_pt;
    }
 
    /// Vector of diffusion coefficients
    const Vector<double>& diff() const
    {
      return *Diff_pt;
    }
 
    /// Pointer to vector of diffusion coefficients
    Vector<double>*& diff_pt()
    {
      return Diff_pt;
    }
 
    /// Vector of dimensionless timescales
    const Vector<double>& tau() const
    {
      return *Tau_pt;
    }
 
    /// Pointer to vector of dimensionless timescales
    Vector<double>*& tau_pt()
    {
      return Tau_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_react(const unsigned& ipt,
                                                  const Vector<double>& x,
                                                  Vector<double>& source) const
    {
      // If no source function has been set, return zero
      if (Source_fct_pt == 0)
      {
        for (unsigned r = 0; r < NREAGENT; r++)
        {
          source[r] = 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_react(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 continuous time from timestepper of first node
        double time = node_pt(0)->time_stepper_pt()->time_pt()->time();
 
        // Get wind
        (*Wind_fct_pt)(time, x, wind);
      }
    }
 
 
    /// Get reaction as a function of the given reagent concentrations
    /// This function is
    /// virtual to allow overloading in multi-physics problems where
    /// the reaction function might be determined by
    /// another system of equations
    inline virtual void get_reaction_adv_diff_react(const unsigned& ipt,
                                                    const Vector<double>& C,
                                                    Vector<double>& R) const
    {
      // If no wind function has been set, return zero
      if (Reaction_fct_pt == 0)
      {
        for (unsigned r = 0; r < NREAGENT; r++)
        {
          R[r] = 0.0;
        }
      }
      else
      {
        // Get reaction terms
        (*Reaction_fct_pt)(C, R);
      }
    }
 
    /// Get the derivatives of the reaction terms with respect to the
    /// concentration variables. If no explicit function pointer is set,
    /// these will be calculated by finite differences
    virtual void get_reaction_deriv_adv_diff_react(
      const unsigned& ipt,
      const Vector<double>& C,
      DenseMatrix<double>& dRdC) const
    {
      // If no reaction pointer set, return zero
      if (Reaction_fct_pt == 0)
      {
        for (unsigned r = 0; r < NREAGENT; r++)
        {
          for (unsigned p = 0; p < NREAGENT; p++)
          {
            dRdC(r, p) = 0.0;
          }
        }
      }
      else
      {
        // If no function pointer get finite differences
        if (Reaction_deriv_fct_pt == 0)
        {
          // Local copy of the unknowns
          Vector<double> C_local = C;
          // Finite differences
          Vector<double> R(NREAGENT), R_plus(NREAGENT), R_minus(NREAGENT);
          // Get the initial reaction terms
          //(*Reaction_fct_pt)(C,R);
          const double fd_step = GeneralisedElement::Default_fd_jacobian_step;
          // Now loop over all the reagents
          for (unsigned p = 0; p < NREAGENT; p++)
          {
            // Store the old value
            double old_var = C_local[p];
            // Increment the value
            C_local[p] += fd_step;
            // Get the new values
            (*Reaction_fct_pt)(C_local, R_plus);
 
            // Reset the value
            C_local[p] = old_var;
            // Decrement the value
            C_local[p] -= fd_step;
            // Get the new values
            (*Reaction_fct_pt)(C_local, R_minus);
 
            // Assemble the column of the jacobian
            for (unsigned r = 0; r < NREAGENT; r++)
            {
              dRdC(r, p) = (R_plus[r] - R_minus[r]) / (2.0 * fd_step);
            }
 
            // Reset the value
            C_local[p] = old_var;
          }
        }
        // Otherwise get the terms from the function
        else
        {
          (*Reaction_deriv_fct_pt)(C, dRdC);
        }
      }
    }
 
    /// Get flux: \f$\mbox{flux}[DIM r + i] = \mbox{d}C_{r} / \mbox{d}x_i \f$
    void get_flux(const Vector<double>& s, Vector<double>& flux) const
    {
      // Find out how many nodes there are in the element
      const unsigned n_node = nnode();
 
      // 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 * NREAGENT; j++)
      {
        flux[j] = 0.0;
      }
 
      // Loop over the reagent terms
      for (unsigned r = 0; r < NREAGENT; r++)
      {
        unsigned c_nodal_index = c_index_adv_diff_react(r);
 
        // Loop over derivative directions
        for (unsigned j = 0; j < DIM; j++)
        {
          unsigned index = r * DIM + j;
          // Loop over the nodes
          for (unsigned l = 0; l < n_node; l++)
          {
            flux[index] += nodal_value(l, c_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_react(
        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_react(
        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_react(
        residuals, jacobian, mass_matrix, 2);
    }
 
 
    /// Return FE representation of function value c_i(s) at local coordinate s
    inline double interpolated_c_adv_diff_react(const Vector<double>& s,
                                                const unsigned& i) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      unsigned c_nodal_index = c_index_adv_diff_react(i);
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      double interpolated_c = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_c += nodal_value(l, c_nodal_index) * psi[l];
      }
 
      return (interpolated_c);
    }
 
 
    /// Self-test: Return 0 for OK
    unsigned self_test();
 
    /// Return the integrated reagent concentrations
    void integrate_reagents(Vector<double>& C) const;
 
  protected:
    // Static variable that holds the number of reagents
    static const unsigned N_reagent;
 
 
    /// 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_react(
      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_react(
      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_react(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
    /// Pointer to global diffusion coefficients
    Vector<double>* Diff_pt;
 
    /// Pointer to global timescales
    Vector<double>* Tau_pt;
 
    /// Pointer to source function:
    AdvectionDiffusionReactionSourceFctPt Source_fct_pt;
 
    /// Pointer to wind function:
    AdvectionDiffusionReactionWindFctPt Wind_fct_pt;
 
    /// Pointer to reaction function
    AdvectionDiffusionReactionReactionFctPt Reaction_fct_pt;
 
    /// Pointer to reaction derivatives
    AdvectionDiffusionReactionReactionDerivFctPt Reaction_deriv_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 dimensionless numbers
    static Vector<double> Default_dimensionless_number;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// QAdvectionDiffusionReactionElement elements are
  /// linear/quadrilateral/brick-shaped Advection Diffusion elements with
  /// isoparametric interpolation for the function.
  //======================================================================
  template<unsigned NREAGENT, unsigned DIM, unsigned NNODE_1D>
  class QAdvectionDiffusionReactionElement
    : public virtual QElement<DIM, NNODE_1D>,
      public virtual AdvectionDiffusionReactionEquations<NREAGENT, DIM>
  {
  public:
    /// Constructor: Call constructors for QElement and
    /// Advection Diffusion equations
    QAdvectionDiffusionReactionElement()
      : QElement<DIM, NNODE_1D>(),
        AdvectionDiffusionReactionEquations<NREAGENT, DIM>()
    {
    }
 
    /// Broken copy constructor
    QAdvectionDiffusionReactionElement(
      const QAdvectionDiffusionReactionElement<NREAGENT, DIM, NNODE_1D>&
        dummy) = delete;
 
    /// Broken assignment operator
    void operator=(
      const QAdvectionDiffusionReactionElement<NREAGENT, DIM, NNODE_1D>&) =
      delete;
 
    ///  Required  # of `values' (pinned or dofs)
    /// at node n
    inline unsigned required_nvalue(const unsigned& n) const
    {
      return NREAGENT;
    }
 
    /// Output function:
    ///  x,y,u   or    x,y,z,u
    void output(std::ostream& outfile)
    {
      AdvectionDiffusionReactionEquations<NREAGENT, 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)
    {
      AdvectionDiffusionReactionEquations<NREAGENT, DIM>::output(outfile,
                                                                 n_plot);
    }
 
 
    /// C-style output function:
    ///  x,y,u   or    x,y,z,u
    void output(FILE* file_pt)
    {
      AdvectionDiffusionReactionEquations<NREAGENT, 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)
    {
      AdvectionDiffusionReactionEquations<NREAGENT, 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)
    {
      AdvectionDiffusionReactionEquations<NREAGENT, 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)
    {
      AdvectionDiffusionReactionEquations<NREAGENT, 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_react(
      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_react(
      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 NREAGENT, unsigned DIM, unsigned NNODE_1D>
  double QAdvectionDiffusionReactionElement<NREAGENT, DIM, NNODE_1D>::
    dshape_and_dtest_eulerian_adv_diff_react(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 NREAGENT, unsigned DIM, unsigned NNODE_1D>
  double QAdvectionDiffusionReactionElement<NREAGENT, DIM, NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_adv_diff_react(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 QAdvectionDiffusionReactionElement 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 NREAGENT, unsigned DIM, unsigned NNODE_1D>
  class FaceGeometry<
    QAdvectionDiffusionReactionElement<NREAGENT, 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 QAdvectionDiffusionReaction elements: Point
  /// elements
  //=======================================================================
  template<unsigned NREAGENT, unsigned NNODE_1D>
  class FaceGeometry<QAdvectionDiffusionReactionElement<NREAGENT, 1, NNODE_1D>>
    : public virtual PointElement
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : PointElement() {}
  };
 
 
  //==========================================================
  /// AdvectionDiffusionReaction upgraded to become projectable
  //==========================================================
  template<class ADR_ELEMENT>
  class ProjectableAdvectionDiffusionReactionElement
    : public virtual ProjectableElement<ADR_ELEMENT>
  {
  public:
    /// 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)
    {
      // 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: Just one
    unsigned nfields_for_projection()
    {
      return this->N_reagent;
    }
 
    /// Number of history values to be stored for fld-th field
    /// (includes current value!)
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
      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)
    {
      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_adv_diff_react(
        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)
    {
      // Find the index at which the variable is stored
      unsigned c_nodal_index = this->c_index_adv_diff_react(fld);
 
      // Local shape function
      unsigned n_node = this->nnode();
      Shape psi(n_node);
 
      // Find values of shape function
      this->shape(s, psi);
 
      // Initialise value of c
      double interpolated_c = 0.0;
 
      // Sum over the local nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_c += this->nodal_value(t, l, c_nodal_index) * psi[l];
      }
      return interpolated_c;
    }
 
 
    /// Return number of values in field fld: One per node
    unsigned nvalue_of_field(const unsigned& fld)
    {
      return this->nnode();
    }
 
 
    /// Return local equation number of value j in field fld.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
      const unsigned c_nodal_index = this->c_index_adv_diff_react(fld);
      return this->nodal_local_eqn(j, c_nodal_index);
    }
  };
 
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<ProjectableAdvectionDiffusionReactionElement<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<ProjectableAdvectionDiffusionReactionElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
 
} // namespace oomph
 
#endif