navier_stokes_elements.h

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// LIC// ====================================================================
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// LIC// multi-physics finite-element library, available
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// LIC//
// LIC// Copyright (C) 2006-2024 Matthias Heil and Andrew Hazel
// LIC//
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// LIC//====================================================================
// Header file for Navier Stokes elements
 
#ifndef OOMPH_NAVIER_STOKES_ELEMENTS_HEADER
#define OOMPH_NAVIER_STOKES_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/fsi.h"
#include "../generic/projection.h"
 
#include <algorithm>
#include <iterator>
 
namespace oomph
{
  //======================================================================
  /// Helper class for elements that impose Robin boundary conditions
  /// on pressure advection diffusion problem required by Fp preconditioner
  /// (class used to get around some templating issues)
  //======================================================================
  class FpPressureAdvDiffRobinBCElementBase : public virtual FaceElement
  {
  public:
    /// Constructor
    FpPressureAdvDiffRobinBCElementBase() {}
 
    /// Empty virtual destructor
    virtual ~FpPressureAdvDiffRobinBCElementBase() {}
 
    /// This function returns the residuals for the
    /// traction function. flag=1 (or 0): do (or don't) compute the
    /// Jacobian as well.
    virtual void fill_in_generic_residual_contribution_fp_press_adv_diff_robin_bc(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      unsigned flag) = 0;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// A class for elements that allow the imposition of Robin boundary
  /// conditions for the pressure advection diffusion problem in the
  /// Fp preconditioner.
  /// The geometrical information can be read from the FaceGeometry<ELEMENT>
  /// class and and thus, we can be generic enough without the need to have
  /// a separate equations class
  //======================================================================
  template<class ELEMENT>
  class FpPressureAdvDiffRobinBCElement
    : public virtual FaceGeometry<ELEMENT>,
      public virtual FaceElement,
      public virtual FpPressureAdvDiffRobinBCElementBase
  {
  public:
    /// Constructor, which takes a "bulk" element and the value of the index
    /// and its limit. Optional boolean flag indicates if it's called
    // refineable constructor.
    FpPressureAdvDiffRobinBCElement(
      FiniteElement* const& element_pt,
      const int& face_index,
      const bool& called_from_refineable_constructor = false)
      : FaceGeometry<ELEMENT>(), FaceElement()
    {
      // Attach the geometrical information to the element. N.B. This function
      // also assigns nbulk_value from the required_nvalue of the bulk element
      element_pt->build_face_element(face_index, this);
 
#ifdef PARANOID
      {
        // Check that the element is not a refineable 3d element
        if (!called_from_refineable_constructor)
        {
          // If it's three-d
          if (element_pt->dim() == 3)
          {
            // Is it refineable
            RefineableElement* ref_el_pt =
              dynamic_cast<RefineableElement*>(element_pt);
            if (ref_el_pt != 0)
            {
              if (this->has_hanging_nodes())
              {
                throw OomphLibError("This flux element will not work correctly "
                                    "if nodes are hanging\n",
                                    OOMPH_CURRENT_FUNCTION,
                                    OOMPH_EXCEPTION_LOCATION);
              }
            }
          }
        }
      }
#endif
    }
 
    /// Empty destructor
    ~FpPressureAdvDiffRobinBCElement() {}
 
    /// This function returns the residuals for the
    /// traction function. flag=1 (or 0): do (or don't) compute the
    /// Jacobian as well.
    virtual void fill_in_generic_residual_contribution_fp_press_adv_diff_robin_bc(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag);
 
 
    /// This function returns just the residuals
    inline void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      std::ostringstream error_message;
      error_message
        << "fill_in_contribution_to_residuals() must not be called directly.\n"
        << "since it uses the local equation numbering of the bulk element\n"
        << "which calls the relevant helper function directly.\n";
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// This function returns the residuals and the jacobian
    inline void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                                 DenseMatrix<double>& jacobian)
    {
      std::ostringstream error_message;
      error_message
        << "fill_in_contribution_to_jacobian() must not be called directly.\n"
        << "since it uses the local equation numbering of the bulk element\n"
        << "which calls the relevant helper function directly.\n";
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Overload the output function
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    /// Output function: x,y,[z],u,v,[w],p in tecplot format
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      FiniteElement::output(outfile, nplot);
    }
  };
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //============================================================================
  /// Get residuals and Jacobian of Robin boundary conditions in pressure
  /// advection diffusion problem in Fp preconditoner
  //============================================================================
  template<class ELEMENT>
  void FpPressureAdvDiffRobinBCElement<ELEMENT>::
    fill_in_generic_residual_contribution_fp_press_adv_diff_robin_bc(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag)
  {
    // Storage for local coordinates in FaceElement and associted bulk element
    unsigned my_dim = this->dim();
    Vector<double> s(my_dim);
    Vector<double> s_bulk(my_dim + 1);
 
    // Storage for outer unit normal
    Vector<double> unit_normal(my_dim + 1);
 
    // Storage for veloc in bulk element
    Vector<double> veloc(my_dim + 1);
 
    // Set the value of n_intpt
    unsigned n_intpt = this->integral_pt()->nweight();
 
    // Integers to store local equation numbers
    int local_eqn = 0;
    int local_unknown = 0;
 
    // Get cast bulk element
    ELEMENT* bulk_el_pt = dynamic_cast<ELEMENT*>(this->bulk_element_pt());
 
    // Find out how many pressure dofs there are in the bulk element
    unsigned n_pres = bulk_el_pt->npres_nst();
 
    // Get the Reynolds number from the bulk element
    double re = bulk_el_pt->re();
 
    // Set up memory for pressure shape and test functions
    Shape psip(n_pres), testp(n_pres);
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Get the integral weight
      double w = this->integral_pt()->weight(ipt);
 
      // Assign values of local coordinate in FaceElement
      for (unsigned i = 0; i < my_dim; i++)
        s[i] = this->integral_pt()->knot(ipt, i);
 
      // Find corresponding coordinate in the the bulk element
      s_bulk = this->local_coordinate_in_bulk(s);
 
      /// Get outer unit normal
      this->outer_unit_normal(ipt, unit_normal);
 
      // Get velocity in bulk element
      bulk_el_pt->interpolated_u_nst(s_bulk, veloc);
 
      // Get normal component of veloc
      double flux = 0.0;
      for (unsigned i = 0; i < my_dim + 1; i++)
      {
        flux += veloc[i] * unit_normal[i];
      }
 
      // Modify bc: If outflow (flux>0) apply Neumann condition instead
      if (flux > 0.0) flux = 0.0;
 
      // Get pressure
      double interpolated_press = bulk_el_pt->interpolated_p_nst(s_bulk);
 
      // Call the pressure shape and test functions in bulk element
      bulk_el_pt->pshape_nst(s_bulk, psip, testp);
 
      // Find the Jacobian of the mapping within the FaceElement
      double J = this->J_eulerian(s);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Loop over the pressure shape functions in bulk
      //(wasteful but they'll be zero on the boundary)
      for (unsigned l = 0; l < n_pres; l++)
      {
        local_eqn = bulk_el_pt->p_local_eqn(l);
 
        // If not a boundary conditions
        if (local_eqn >= 0)
        {
          residuals[local_eqn] -= re * flux * interpolated_press * testp[l] * W;
 
          // Jacobian too?
          if (flag)
          {
            // Loop over the shape functions in bulk
            for (unsigned l2 = 0; l2 < n_pres; l2++)
            {
              local_unknown = bulk_el_pt->p_local_eqn(l2);
 
              // If not a boundary conditions
              if (local_unknown >= 0)
              {
                jacobian(local_eqn, local_unknown) -=
                  re * flux * psip[l2] * testp[l] * W;
              }
            }
          } /*End of Jacobian calculation*/
        } // End of if not boundary condition
      } // End of loop over l
    }
  }
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// Template-free base class for Navier-Stokes equations to avoid
  /// casting problems
  //======================================================================
  class TemplateFreeNavierStokesEquationsBase
    : public virtual NavierStokesElementWithDiagonalMassMatrices,
      public virtual FiniteElement
  {
  public:
    /// Constructor (empty)
    TemplateFreeNavierStokesEquationsBase(){};
 
    /// Virtual destructor (empty)
    virtual ~TemplateFreeNavierStokesEquationsBase(){};
 
    /// Compute the residuals for the associated pressure advection
    /// diffusion problem. Used by the Fp preconditioner.
    virtual void fill_in_pressure_advection_diffusion_residuals(
      Vector<double>& residuals) = 0;
 
    /// Compute the residuals and Jacobian for the associated
    /// pressure advection diffusion problem. Used by the Fp preconditioner.
    virtual void fill_in_pressure_advection_diffusion_jacobian(
      Vector<double>& residuals, DenseMatrix<double>& jacobian) = 0;
 
    /// Return the index at which the pressure is stored if it is
    /// stored at the nodes. If not stored at the nodes this will return
    /// a negative number.
    virtual int p_nodal_index_nst() const = 0;
 
    /// Access function for the local equation number information for
    /// the pressure.
    /// p_local_eqn[n] = local equation number or < 0 if pinned
    virtual int p_local_eqn(const unsigned& n) const = 0;
 
    /// Global eqn number of pressure dof that's pinned in pressure
    /// adv diff problem
    virtual int& pinned_fp_pressure_eqn() = 0;
 
 
    /// Pin all non-pressure dofs and backup eqn numbers of all Data
    virtual void pin_all_non_pressure_dofs(
      std::map<Data*, std::vector<int>>& eqn_number_backup) = 0;
 
    /// Build FaceElements that apply the Robin boundary condition
    /// to the pressure advection diffusion problem required by
    /// Fp preconditioner
    virtual void build_fp_press_adv_diff_robin_bc_element(
      const unsigned& face_index) = 0;
 
    /// Delete the FaceElements that apply the Robin boundary condition
    /// to the pressure advection diffusion problem required by
    /// Fp preconditioner
    virtual void delete_pressure_advection_diffusion_robin_elements() = 0;
 
 
    /// Compute the diagonal of the velocity/pressure mass matrices.
    /// If which one=0, both are computed, otherwise only the pressure
    /// (which_one=1) or the velocity mass matrix (which_one=2 -- the
    /// LSC version of the preconditioner only needs that one)
    virtual void get_pressure_and_velocity_mass_matrix_diagonal(
      Vector<double>& press_mass_diag,
      Vector<double>& veloc_mass_diag,
      const unsigned& which_one = 0) = 0;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// A class for elements that solve the cartesian Navier--Stokes equations,
  /// templated by the dimension DIM.
  /// This contains the generic maths -- any concrete implementation must
  /// be derived from this.
  ///
  /// We're solving:
  ///
  ///  \f$ { Re \left( St \frac{\partial u_i}{\partial t} + (u_j - u_j^{M}) \frac{\partial u_i}{\partial x_j} \right) = - \frac{\partial p}{\partial x_i} - R_\rho B_i(x_j) - \frac{Re}{Fr} G_i + \frac{\partial }{\partial x_j} \left[ R_\mu \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \right] } \f$
  ///
  ///  and
  ///
  ///  \f$ { \frac{\partial u_i}{\partial x_i} = Q } \f$
  ///
  /// We also provide all functions required to use this element
  /// in FSI problems, by deriving it from the FSIFluidElement base
  /// class.
  //======================================================================
  template<unsigned DIM>
  class NavierStokesEquations
    : public virtual FSIFluidElement,
      public virtual TemplateFreeNavierStokesEquationsBase
  {
  public:
    /// Function pointer to body force function fct(t,x,f(x))
    /// x is a Vector!
    typedef void (*NavierStokesBodyForceFctPt)(const double& time,
                                               const Vector<double>& x,
                                               Vector<double>& body_force);
 
    /// Function pointer to source function fct(t,x)
    /// x is a Vector!
    typedef double (*NavierStokesSourceFctPt)(const double& time,
                                              const Vector<double>& x);
 
 
    /// Function pointer to source function fct(x) for the
    /// pressure advection diffusion equation (only used during
    /// validation!). x is a Vector!
    typedef double (*NavierStokesPressureAdvDiffSourceFctPt)(
      const Vector<double>& x);
 
  private:
    /// Static "magic" number that indicates that the pressure is
    /// not stored at a node
    static int Pressure_not_stored_at_node;
 
    /// Static default value for the physical constants (all initialised to
    /// zero)
    static double Default_Physical_Constant_Value;
 
    /// Static default value for the physical ratios (all are initialised to
    /// one)
    static double Default_Physical_Ratio_Value;
 
    /// Static default value for the gravity vector
    static Vector<double> Default_Gravity_vector;
 
  protected:
    // Physical constants
 
    /// Pointer to the viscosity ratio (relative to the
    /// viscosity used in the definition of the Reynolds number)
    double* Viscosity_Ratio_pt;
 
    /// Pointer to the density ratio (relative to the density used in the
    /// definition of the Reynolds number)
    double* Density_Ratio_pt;
 
    // Pointers to global physical constants
 
    /// Pointer to global Reynolds number
    double* Re_pt;
 
    /// Pointer to global Reynolds number x Strouhal number (=Womersley)
    double* ReSt_pt;
 
    /// Pointer to global Reynolds number x inverse Froude number
    /// (= Bond number / Capillary number)
    double* ReInvFr_pt;
 
    /// Pointer to global gravity Vector
    Vector<double>* G_pt;
 
    /// Pointer to body force function
    NavierStokesBodyForceFctPt Body_force_fct_pt;
 
    /// Pointer to volumetric source function
    NavierStokesSourceFctPt Source_fct_pt;
 
    /// Pointer to source function pressure advection diffusion equation
    /// (only to be used during validation)
    NavierStokesPressureAdvDiffSourceFctPt Press_adv_diff_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;
 
    /// Storage for FaceElements that apply Robin BC for pressure adv
    /// diff equation used in Fp preconditioner.
    Vector<FpPressureAdvDiffRobinBCElementBase*>
      Pressure_advection_diffusion_robin_element_pt;
 
    /// Global eqn number of pressure dof that's pinned in
    /// pressure advection diffusion problem (defaults to -1)
    int Pinned_fp_pressure_eqn;
 
    /// Compute the shape functions and derivatives
    /// w.r.t. global coords at local coordinate s.
    /// Return Jacobian of mapping between local and global coordinates.
    virtual double dshape_and_dtest_eulerian_nst(const Vector<double>& s,
                                                 Shape& psi,
                                                 DShape& dpsidx,
                                                 Shape& test,
                                                 DShape& dtestdx) const = 0;
 
    /// Compute the shape functions and derivatives
    /// w.r.t. global coords at ipt-th integration point
    /// Return Jacobian of mapping between local and global coordinates.
    virtual double dshape_and_dtest_eulerian_at_knot_nst(
      const unsigned& ipt,
      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 (J). Also compute
    /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
    virtual double dshape_and_dtest_eulerian_at_knot_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const = 0;
 
    /// Calculate the body force at a given time and local and/or
    /// Eulerian position. This function is virtual so that it can be
    /// overloaded in multi-physics elements where the body force might
    /// depend on another variable.
    virtual void get_body_force_nst(const double& time,
                                    const unsigned& ipt,
                                    const Vector<double>& s,
                                    const Vector<double>& x,
                                    Vector<double>& result)
    {
      // If the function pointer is zero return zero
      if (Body_force_fct_pt == 0)
      {
        // Loop over dimensions and set body forces to zero
        for (unsigned i = 0; i < DIM; i++)
        {
          result[i] = 0.0;
        }
      }
      // Otherwise call the function
      else
      {
        (*Body_force_fct_pt)(time, x, result);
      }
    }
 
    /// Get gradient of body force 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. Computed via function pointer (if set) or by finite
    /// differencing (default)
    inline virtual void get_body_force_gradient_nst(
      const double& time,
      const unsigned& ipt,
      const Vector<double>& s,
      const Vector<double>& x,
      DenseMatrix<double>& d_body_force_dx)
    {
      // hierher: Implement function pointer version
      /*    //If no gradient function has been set, FD it */
      /*    if(Body_force_fct_gradient_pt==0) */
      {
        // Reference value
        Vector<double> body_force(DIM, 0.0);
        get_body_force_nst(time, ipt, s, x, body_force);
 
        // FD it
        double eps_fd = GeneralisedElement::Default_fd_jacobian_step;
        Vector<double> body_force_pls(DIM, 0.0);
        Vector<double> x_pls(x);
        for (unsigned i = 0; i < DIM; i++)
        {
          x_pls[i] += eps_fd;
          get_body_force_nst(time, ipt, s, x_pls, body_force_pls);
          for (unsigned j = 0; j < DIM; j++)
          {
            d_body_force_dx(j, i) =
              (body_force_pls[j] - body_force[j]) / eps_fd;
          }
          x_pls[i] = x[i];
        }
      }
      /*    else */
      /*     { */
      /*      // Get gradient */
      /*      (*Source_fct_gradient_pt)(time,x,gradient); */
      /*     } */
    }
 
 
    /// Calculate the source fct at given time and
    /// Eulerian position
    virtual double get_source_nst(const double& time,
                                  const unsigned& ipt,
                                  const Vector<double>& x)
    {
      // If the function pointer is zero return zero
      if (Source_fct_pt == 0)
      {
        return 0;
      }
      // Otherwise call the function
      else
      {
        return (*Source_fct_pt)(time, x);
      }
    }
 
 
    /// Get gradient of 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. Computed via function pointer
    /// (if set) or by finite differencing (default)
    inline virtual void get_source_gradient_nst(const double& time,
                                                const unsigned& ipt,
                                                const Vector<double>& x,
                                                Vector<double>& gradient)
    {
      // hierher: Implement function pointer version
      /*    //If no gradient function has been set, FD it */
      /*    if(Source_fct_gradient_pt==0) */
      {
        // Reference value
        double source = get_source_nst(time, ipt, x);
 
        // FD it
        double eps_fd = GeneralisedElement::Default_fd_jacobian_step;
        double source_pls = 0.0;
        Vector<double> x_pls(x);
        for (unsigned i = 0; i < DIM; i++)
        {
          x_pls[i] += eps_fd;
          source_pls = get_source_nst(time, ipt, x_pls);
          gradient[i] = (source_pls - source) / eps_fd;
          x_pls[i] = x[i];
        }
      }
      /*    else */
      /*     { */
      /*      // Get gradient */
      /*      (*Source_fct_gradient_pt)(time,x,gradient); */
      /*     } */
    }
 
 
    /// Compute the residuals for the Navier--Stokes equations.
    /// Flag=1 (or 0): do (or don't) compute the Jacobian as well.
    /// Flag=2: Fill in mass matrix too.
    virtual void fill_in_generic_residual_contribution_nst(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
 
    /// Compute the residuals for the associated pressure advection
    /// diffusion problem. Used by the Fp preconditioner.
    /// flag=1(or 0): do (or don't) compute the Jacobian as well.
    virtual void fill_in_generic_pressure_advection_diffusion_contribution_nst(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag);
 
    /// Compute the derivatives of the
    /// residuals for the Navier--Stokes equations with respect to a parameter
    /// Flag=1 (or 0): do (or don't) compute the Jacobian as well.
    /// Flag=2: Fill in mass matrix too.
    virtual void fill_in_generic_dresidual_contribution_nst(
      double* const& parameter_pt,
      Vector<double>& dres_dparam,
      DenseMatrix<double>& djac_dparam,
      DenseMatrix<double>& dmass_matrix_dparam,
      unsigned flag);
 
    /// Compute the hessian tensor vector products required to
    /// perform continuation of bifurcations analytically
    void fill_in_contribution_to_hessian_vector_products(
      Vector<double> const& Y,
      DenseMatrix<double> const& C,
      DenseMatrix<double>& product);
 
 
  public:
    /// Constructor: NULL the body force and source function
    /// and make sure the ALE terms are included by default.
    NavierStokesEquations()
      : Body_force_fct_pt(0),
        Source_fct_pt(0),
        Press_adv_diff_source_fct_pt(0),
        ALE_is_disabled(false),
        Pinned_fp_pressure_eqn(-1)
    {
      // Set all the Physical parameter pointers to the default value zero
      Re_pt = &Default_Physical_Constant_Value;
      ReSt_pt = &Default_Physical_Constant_Value;
      ReInvFr_pt = &Default_Physical_Constant_Value;
      G_pt = &Default_Gravity_vector;
      // Set the Physical ratios to the default value of 1
      Viscosity_Ratio_pt = &Default_Physical_Ratio_Value;
      Density_Ratio_pt = &Default_Physical_Ratio_Value;
    }
 
    /// Vector to decide whether the stress-divergence form is used or not
    // N.B. This needs to be public so that the intel compiler gets things
    // correct somehow the access function messes things up when going to
    // refineable navier--stokes
    static Vector<double> Gamma;
 
    // Access functions for the physical constants
 
    /// Reynolds number
    const double& re() const
    {
      return *Re_pt;
    }
 
    /// Product of Reynolds and Strouhal number (=Womersley number)
    const double& re_st() const
    {
      return *ReSt_pt;
    }
 
    /// Pointer to Reynolds number
    double*& re_pt()
    {
      return Re_pt;
    }
 
    /// Pointer to product of Reynolds and Strouhal number (=Womersley number)
    double*& re_st_pt()
    {
      return ReSt_pt;
    }
 
    /// Viscosity ratio for element: Element's viscosity relative to the
    /// viscosity used in the definition of the Reynolds number
    const double& viscosity_ratio() const
    {
      return *Viscosity_Ratio_pt;
    }
 
    /// Pointer to Viscosity Ratio
    double*& viscosity_ratio_pt()
    {
      return Viscosity_Ratio_pt;
    }
 
    /// Density ratio for element: Element's density relative to the
    ///  viscosity used in the definition of the Reynolds number
    const double& density_ratio() const
    {
      return *Density_Ratio_pt;
    }
 
    /// Pointer to Density ratio
    double*& density_ratio_pt()
    {
      return Density_Ratio_pt;
    }
 
    /// Global inverse Froude number
    const double& re_invfr() const
    {
      return *ReInvFr_pt;
    }
 
    /// Pointer to global inverse Froude number
    double*& re_invfr_pt()
    {
      return ReInvFr_pt;
    }
 
    /// Vector of gravitational components
    const Vector<double>& g() const
    {
      return *G_pt;
    }
 
    /// Pointer to Vector of gravitational components
    Vector<double>*& g_pt()
    {
      return G_pt;
    }
 
    /// Access function for the body-force pointer
    NavierStokesBodyForceFctPt& body_force_fct_pt()
    {
      return Body_force_fct_pt;
    }
 
    /// Access function for the body-force pointer. Const version
    NavierStokesBodyForceFctPt body_force_fct_pt() const
    {
      return Body_force_fct_pt;
    }
 
    /// Access function for the source-function pointer
    NavierStokesSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
    /// Access function for the source-function pointer. Const version
    NavierStokesSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
    /// Access function for the source-function pointer for pressure
    /// advection diffusion  (used for validation only).
    NavierStokesPressureAdvDiffSourceFctPt& source_fct_for_pressure_adv_diff()
    {
      return Press_adv_diff_source_fct_pt;
    }
 
    /// Access function for the source-function pointer for pressure
    /// advection diffusion  (used for validation only). Const version.
    NavierStokesPressureAdvDiffSourceFctPt source_fct_for_pressure_adv_diff()
      const
    {
      return Press_adv_diff_source_fct_pt;
    }
 
    /// Global eqn number of pressure dof that's pinned in pressure
    /// adv diff problem
    int& pinned_fp_pressure_eqn()
    {
      return Pinned_fp_pressure_eqn;
    }
 
    /// Function to return number of pressure degrees of freedom
    virtual unsigned npres_nst() const = 0;
 
    /// Compute the pressure shape functions at local coordinate s
    virtual void pshape_nst(const Vector<double>& s, Shape& psi) const = 0;
 
    /// Compute the pressure shape and test functions
    /// at local coordinate s
    virtual void pshape_nst(const Vector<double>& s,
                            Shape& psi,
                            Shape& test) const = 0;
 
    /// Compute the pressure shape and test functions and derivatives
    /// w.r.t. global coords at local coordinate s.
    /// Return Jacobian of mapping between local and global coordinates.
    virtual double dpshape_and_dptest_eulerian_nst(const Vector<double>& s,
                                                   Shape& ppsi,
                                                   DShape& dppsidx,
                                                   Shape& ptest,
                                                   DShape& dptestdx) const = 0;
 
    /// Velocity i at local node n. Uses suitably interpolated value
    /// for hanging nodes. The use of u_index_nst() permits the use of this
    /// element as the basis for multi-physics elements. The default
    /// is to assume that the i-th velocity component is stored at the
    /// i-th location of the node
    double u_nst(const unsigned& n, const unsigned& i) const
    {
      return nodal_value(n, u_index_nst(i));
    }
 
    /// Velocity i at local node n at timestep t (t=0: present;
    /// t>0: previous). Uses suitably interpolated value for hanging nodes.
    double u_nst(const unsigned& t, const unsigned& n, const unsigned& i) const
    {
      return nodal_value(t, n, u_index_nst(i));
    }
 
    /// Return the index at which the i-th unknown velocity component
    /// is stored. The default value, i, is appropriate for single-physics
    /// problems.
    /// In derived multi-physics elements, this function should be overloaded
    /// to reflect the chosen storage scheme. Note that these equations require
    /// that the unknowns are always stored at the same indices at each node.
    virtual inline unsigned u_index_nst(const unsigned& i) const
    {
      return i;
    }
 
    /// Return the number of velocity components
    /// Used in FluidInterfaceElements
    inline unsigned n_u_nst() const
    {
      return DIM;
    }
 
    /// i-th component of du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double du_dt_nst(const unsigned& n, const unsigned& i) 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 dof is stored
        const unsigned u_nodal_index = this->u_index_nst(i);
 
        // Number of timsteps (past & present)
        const unsigned n_time = time_stepper_pt->ntstorage();
        // Loop over the timesteps
        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;
    }
 
    /// Pressure at local pressure "node" n_p
    /// Uses suitably interpolated value for hanging nodes.
    virtual double p_nst(const unsigned& n_p) const = 0;
 
    /// Pressure at local pressure "node" n_p at time level t
    virtual double p_nst(const unsigned& t, const unsigned& n_p) const = 0;
 
    /// Pin p_dof-th pressure dof and set it to value specified by p_value.
    virtual void fix_pressure(const unsigned& p_dof, const double& p_value) = 0;
 
    /// Return the index at which the pressure is stored if it is
    /// stored at the nodes. If not stored at the nodes this will return
    /// a negative number.
    virtual int p_nodal_index_nst() const
    {
      return Pressure_not_stored_at_node;
    }
 
    /// Integral of pressure over element
    double pressure_integral() const;
 
    /// Return integral of dissipation over element
    double dissipation() const;
 
    /// Return dissipation at local coordinate s
    double dissipation(const Vector<double>& s) const;
 
    /// Compute the vorticity vector at local coordinate s
    void get_vorticity(const Vector<double>& s,
                       Vector<double>& vorticity) const;
 
    /// Compute the scalar vorticity at local coordinate s (2D)
    void get_vorticity(const Vector<double>& s, double& vorticity) const;
 
    /// Get integral of kinetic energy over element
    double kin_energy() const;
 
    /// Get integral of time derivative of kinetic energy over element
    double d_kin_energy_dt() const;
 
    /// Strain-rate tensor: 1/2 (du_i/dx_j + du_j/dx_i)
    void strain_rate(const Vector<double>& s,
                     DenseMatrix<double>& strain_rate) const;
 
    /// Compute traction (on the viscous scale) exerted onto
    /// the fluid at local coordinate s. N has to be outer unit normal
    /// to the fluid.
    void get_traction(const Vector<double>& s,
                      const Vector<double>& N,
                      Vector<double>& traction);
 
    /// Compute traction (on the viscous scale) exerted onto
    /// the fluid at local coordinate s, decomposed into pressure and
    /// normal and tangential viscous components.
    /// N has to be outer unit normal to the fluid.
    void get_traction(const Vector<double>& s,
                      const Vector<double>& N,
                      Vector<double>& traction_p,
                      Vector<double>& traction_visc_n,
                      Vector<double>& traction_visc_t);
 
    /// This implements a pure virtual function defined
    /// in the FSIFluidElement class. The function computes
    /// the traction (on the viscous scale), at the element's local
    /// coordinate s, that the fluid element exerts onto an adjacent
    /// solid element. The number of arguments is imposed by
    /// the interface defined in the FSIFluidElement -- only the
    /// unit normal N (pointing into the fluid!) is actually used
    /// in the computation.
    void get_load(const Vector<double>& s,
                  const Vector<double>& N,
                  Vector<double>& load)
    {
      // Note: get_traction() computes the traction onto the fluid
      // if N is the outer unit normal onto the fluid; here we're
      // exepcting N to point into the fluid so we're getting the
      // traction onto the adjacent wall instead!
      get_traction(s, N, load);
    }
 
    /// Compute the diagonal of the velocity/pressure mass matrices.
    /// If which one=0, both are computed, otherwise only the pressure
    /// (which_one=1) or the velocity mass matrix (which_one=2 -- the
    /// LSC version of the preconditioner only needs that one)
    void get_pressure_and_velocity_mass_matrix_diagonal(
      Vector<double>& press_mass_diag,
      Vector<double>& veloc_mass_diag,
      const unsigned& which_one = 0);
 
    /// 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);
 
        // Velocities
        if (i < DIM)
        {
          file_out << interpolated_u_nst(s, i) << std::endl;
        }
 
        // Pressure
        else if (i == DIM)
        {
          file_out << interpolated_p_nst(s) << std::endl;
        }
 
        // Never get here
        else
        {
#ifdef PARANOID
          std::stringstream error_stream;
          error_stream << "These Navier Stokes elements only store " << DIM + 1
                       << " fields, "
                       << "but i is currently  " << i << std::endl;
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
#endif
        }
      }
    }
 
 
    /// 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 > DIM)
      {
        // Create an output stream
        std::stringstream error_stream;
 
        // Create the error message
        error_stream << "These Navier Stokes elements only store " << DIM + 1
                     << " fields, but i is currently " << i << std::endl;
 
        // Throw the error message
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Vector of local coordinates
      Vector<double> s(DIM + 1, 0.0);
 
      // Storage for the spatial coordinates
      Vector<double> spatial_coordinates(DIM, 0.0);
 
      // How many plot points do we have in total?
      unsigned num_plot_points = nplot_points_paraview(nplot);
 
      // Loop over plot points
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get the local coordinates of the iplot-th plot point
        get_s_plot(iplot, nplot, s);
 
        // Loop over the spatial coordinates
        for (unsigned j = 0; j < DIM; j++)
        {
          // Assign the i-th spatial coordinate
          spatial_coordinates[j] = interpolated_x(s, j);
        }
 
        // Exact solution vector (here it's simply a scalar)
        Vector<double> exact_soln(DIM + 1, 0.0);
 
        // Get the exact solution at this point
        (*exact_soln_pt)(time, spatial_coordinates, exact_soln);
 
        // Output the interpolated solution value
        file_out << exact_soln[i] << std::endl;
      } // for (unsigned iplot=0;iplot<num_plot_points;iplot++)
    } // 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. Can (should!) be
    /// overloaded with more meaningful names in specific elements.
    std::string scalar_name_paraview(const unsigned& i) const
    {
      // Velocities
      if (i < DIM)
      {
        return "Velocity " + StringConversion::to_string(i);
      }
      // Preussre
      else if (i == DIM)
      {
        return "Pressure";
      }
      // Never get here
      else
      {
        std::stringstream error_stream;
        error_stream << "These Navier Stokes elements only store " << DIM + 1
                     << "  fields,\n"
                     << "but i is currently  " << i << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
        // Dummy return
        return " ";
      }
    }
 
    /// Output function: x,y,[z],u,v,[w],p
    /// in tecplot format. Default number of plot points
    void output(std::ostream& outfile)
    {
      unsigned nplot = 5;
      output(outfile, nplot);
    }
 
    /// Output function: x,y,[z],u,v,[w],p
    /// in tecplot format. nplot points in each coordinate direction
    void output(std::ostream& outfile, const unsigned& nplot);
 
    /// C-style output function: x,y,[z],u,v,[w],p
    /// in tecplot format. Default number of plot points
    void output(FILE* file_pt)
    {
      unsigned nplot = 5;
      output(file_pt, nplot);
    }
 
    /// C-style output function: x,y,[z],u,v,[w],p
    /// in tecplot format. nplot points in each coordinate direction
    void output(FILE* file_pt, const unsigned& nplot);
 
    /// Full output function:
    /// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation
    /// in tecplot format. Default number of plot points
    void full_output(std::ostream& outfile)
    {
      unsigned nplot = 5;
      full_output(outfile, nplot);
    }
 
    /// Full output function:
    /// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation
    /// in tecplot format. nplot points in each coordinate direction
    void full_output(std::ostream& outfile, const unsigned& nplot);
 
 
    /// Output function: x,y,[z],u,v,[w] in tecplot format.
    /// nplot points in each coordinate direction at timestep t
    /// (t=0: present; t>0: previous timestep)
    void output_veloc(std::ostream& outfile,
                      const unsigned& nplot,
                      const unsigned& t);
 
 
    /// Output function: x,y,[z], [omega_x,omega_y,[and/or omega_z]]
    /// in tecplot format. nplot points in each coordinate direction
    void output_vorticity(std::ostream& outfile, const unsigned& nplot);
 
    /// Output exact solution specified via function pointer
    /// at a given number of plot points. Function prints as
    /// many components as are returned in solution Vector
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output exact solution specified via function pointer
    /// at a given time and at a given number of plot points.
    /// Function prints as many components as are returned in solution Vector.
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt);
 
    /// Compute norm of solution: square of the L2 norm of the velocities
    void compute_norm(double& norm);
 
    /// Compute the vector norm of the FEM solution
    void compute_norm(Vector<double>& norm);
 
    /// Validate against exact solution at given time
    /// Solution is provided via function pointer.
    /// Plot at a given number of plot points and compute L2 error
    /// and L2 norm of velocity solution over element
    void compute_error(std::ostream& outfile,
                       FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
                       const double& time,
                       double& error,
                       double& norm);
 
    /// Validate against exact solution.
    /// Solution is provided via function pointer.
    /// Plot at a given number of plot points and compute L2 error
    /// and L2 norm of velocity solution over element
    void compute_error(std::ostream& outfile,
                       FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
    /// Validate against exact solution. Solution is provided via
    /// function pointer. Compute L2 error and L2 norm of velocity solution
    /// over element.
    void compute_error(FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
                       const double& time,
                       double& error,
                       double& norm);
 
    /// Validate against exact solution. Solution is provided via
    /// function pointer. Compute L2 error and L2 norm of velocity solution
    /// over element.
    void compute_error(FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
    /// Compute the element's residual Vector
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Call the generic residuals function with flag set to 0
      // and using a dummy matrix argument
      fill_in_generic_residual_contribution_nst(
        residuals,
        GeneralisedElement::Dummy_matrix,
        GeneralisedElement::Dummy_matrix,
        0);
    }
 
    /// Compute the element's residual Vector and the jacobian matrix
    /// Virtual function can be overloaded by hanging-node version
    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_nst(
        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_nst(
        residuals, jacobian, mass_matrix, 2);
    }
 
    /// Compute the element's residual Vector
    void fill_in_contribution_to_dresiduals_dparameter(
      double* const& parameter_pt, Vector<double>& dres_dparam)
    {
      // Call the generic residuals function with flag set to 0
      // and using a dummy matrix argument
      fill_in_generic_dresidual_contribution_nst(
        parameter_pt,
        dres_dparam,
        GeneralisedElement::Dummy_matrix,
        GeneralisedElement::Dummy_matrix,
        0);
    }
 
    /// Compute the element's residual Vector and the jacobian matrix
    /// Virtual function can be overloaded by hanging-node version
    void fill_in_contribution_to_djacobian_dparameter(
      double* const& parameter_pt,
      Vector<double>& dres_dparam,
      DenseMatrix<double>& djac_dparam)
    {
      // Call the generic routine with the flag set to 1
      fill_in_generic_dresidual_contribution_nst(
        parameter_pt,
        dres_dparam,
        djac_dparam,
        GeneralisedElement::Dummy_matrix,
        1);
    }
 
    /// Add the element's contribution to its residuals vector,
    /// jacobian matrix and mass matrix
    void fill_in_contribution_to_djacobian_and_dmass_matrix_dparameter(
      double* const& parameter_pt,
      Vector<double>& dres_dparam,
      DenseMatrix<double>& djac_dparam,
      DenseMatrix<double>& dmass_matrix_dparam)
    {
      // Call the generic routine with the flag set to 2
      fill_in_generic_dresidual_contribution_nst(
        parameter_pt, dres_dparam, djac_dparam, dmass_matrix_dparam, 2);
    }
 
 
    /// Compute the residuals for the associated pressure advection
    /// diffusion problem. Used by the Fp preconditioner.
    void fill_in_pressure_advection_diffusion_residuals(
      Vector<double>& residuals)
    {
      fill_in_generic_pressure_advection_diffusion_contribution_nst(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
    /// Compute the residuals and Jacobian for the associated
    /// pressure advection diffusion problem. Used by the Fp preconditioner.
    void fill_in_pressure_advection_diffusion_jacobian(
      Vector<double>& residuals, DenseMatrix<double>& jacobian)
    {
      fill_in_generic_pressure_advection_diffusion_contribution_nst(
        residuals, jacobian, 1);
    }
 
 
    /// Pin all non-pressure dofs and backup eqn numbers
    void pin_all_non_pressure_dofs(
      std::map<Data*, std::vector<int>>& eqn_number_backup)
    {
      // Loop over internal data and pin the values (having established that
      // pressure dofs aren't amongst those)
      unsigned nint = this->ninternal_data();
      for (unsigned j = 0; j < nint; j++)
      {
        Data* data_pt = this->internal_data_pt(j);
        if (eqn_number_backup[data_pt].size() == 0)
        {
          unsigned nvalue = data_pt->nvalue();
          eqn_number_backup[data_pt].resize(nvalue);
          for (unsigned i = 0; i < nvalue; i++)
          {
            // Backup
            eqn_number_backup[data_pt][i] = data_pt->eqn_number(i);
 
            // Pin everything
            data_pt->pin(i);
          }
        }
      }
 
      // Now deal with nodal values
      unsigned nnod = this->nnode();
      for (unsigned j = 0; j < nnod; j++)
      {
        Node* nod_pt = this->node_pt(j);
        if (eqn_number_backup[nod_pt].size() == 0)
        {
          unsigned nvalue = nod_pt->nvalue();
          eqn_number_backup[nod_pt].resize(nvalue);
          for (unsigned i = 0; i < nvalue; i++)
          {
            // Pin everything apart from the nodal pressure
            // value
            if (int(i) != this->p_nodal_index_nst())
            {
              // Backup
              eqn_number_backup[nod_pt][i] = nod_pt->eqn_number(i);
 
              // Pin
              nod_pt->pin(i);
            }
            // Else it's a pressure value
            else
            {
              // Exclude non-nodal pressure based elements
              if (this->p_nodal_index_nst() >= 0)
              {
                // Backup
                eqn_number_backup[nod_pt][i] = nod_pt->eqn_number(i);
              }
            }
          }
 
 
          // If it's a solid node deal with its positional data too
          SolidNode* solid_nod_pt = dynamic_cast<SolidNode*>(nod_pt);
          if (solid_nod_pt != 0)
          {
            Data* solid_posn_data_pt = solid_nod_pt->variable_position_pt();
            if (eqn_number_backup[solid_posn_data_pt].size() == 0)
            {
              unsigned nvalue = solid_posn_data_pt->nvalue();
              eqn_number_backup[solid_posn_data_pt].resize(nvalue);
              for (unsigned i = 0; i < nvalue; i++)
              {
                // Backup
                eqn_number_backup[solid_posn_data_pt][i] =
                  solid_posn_data_pt->eqn_number(i);
 
                // Pin
                solid_posn_data_pt->pin(i);
              }
            }
          }
        }
      }
    }
 
 
    /// Build FaceElements that apply the Robin boundary condition
    /// to the pressure advection diffusion problem required by
    /// Fp preconditioner
    virtual void build_fp_press_adv_diff_robin_bc_element(
      const unsigned& face_index) = 0;
 
    /// Output the FaceElements that apply the Robin boundary condition
    /// to the pressure advection diffusion problem required by
    /// Fp preconditioner
    void output_pressure_advection_diffusion_robin_elements(
      std::ostream& outfile)
    {
      unsigned nel = Pressure_advection_diffusion_robin_element_pt.size();
      for (unsigned e = 0; e < nel; e++)
      {
        FaceElement* face_el_pt =
          Pressure_advection_diffusion_robin_element_pt[e];
        outfile << "ZONE" << std::endl;
        Vector<double> unit_normal(DIM);
        Vector<double> x(DIM);
        Vector<double> s(DIM - 1);
        unsigned n = face_el_pt->integral_pt()->nweight();
        for (unsigned ipt = 0; ipt < n; ipt++)
        {
          for (unsigned i = 0; i < DIM - 1; i++)
          {
            s[i] = face_el_pt->integral_pt()->knot(ipt, i);
          }
          face_el_pt->interpolated_x(s, x);
          face_el_pt->outer_unit_normal(ipt, unit_normal);
          for (unsigned i = 0; i < DIM; i++)
          {
            outfile << x[i] << " ";
          }
          for (unsigned i = 0; i < DIM; i++)
          {
            outfile << unit_normal[i] << " ";
          }
          outfile << std::endl;
        }
      }
    }
 
    /// Delete the FaceElements that apply the Robin boundary condition
    /// to the pressure advection diffusion problem required by
    /// Fp preconditioner
    void delete_pressure_advection_diffusion_robin_elements()
    {
      unsigned nel = Pressure_advection_diffusion_robin_element_pt.size();
      for (unsigned e = 0; e < nel; e++)
      {
        delete Pressure_advection_diffusion_robin_element_pt[e];
      }
      Pressure_advection_diffusion_robin_element_pt.clear();
    }
 
    /// Compute derivatives of elemental residual vector with respect
    /// to nodal coordinates. Overwrites default implementation in
    /// FiniteElement base class.
    /// dresidual_dnodal_coordinates(l,i,j) = d res(l) / dX_{ij}
    virtual void get_dresidual_dnodal_coordinates(
      RankThreeTensor<double>& dresidual_dnodal_coordinates);
 
 
    /// Compute vector of FE interpolated velocity u at local coordinate s
    void interpolated_u_nst(const Vector<double>& s,
                            Vector<double>& veloc) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
      // Local shape function
      Shape psi(n_node);
      // Find values of shape function
      shape(s, psi);
 
      for (unsigned i = 0; i < DIM; i++)
      {
        // Index at which the nodal value is stored
        unsigned u_nodal_index = u_index_nst(i);
        // Initialise value of u
        veloc[i] = 0.0;
        // Loop over the local nodes and sum
        for (unsigned l = 0; l < n_node; l++)
        {
          veloc[i] += nodal_value(l, u_nodal_index) * psi[l];
        }
      }
    }
 
    /// Return FE interpolated velocity u[i] at local coordinate s
    double interpolated_u_nst(const Vector<double>& s, const unsigned& i) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
      // Local shape function
      Shape psi(n_node);
      // Find values of shape function
      shape(s, psi);
 
      // Get nodal index at which i-th velocity is stored
      unsigned u_nodal_index = u_index_nst(i);
 
      // 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 FE interpolated velocity u[i] at local coordinate s
    /// at time level t (t=0: present; t>0: history)
    double interpolated_u_nst(const unsigned& t,
                              const Vector<double>& s,
                              const unsigned& i) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Get nodal index at which i-th velocity is stored
      unsigned u_nodal_index = u_index_nst(i);
 
      // 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(t, l, u_nodal_index) * psi[l];
      }
 
      return (interpolated_u);
    }
 
    /// Compute the derivatives of the i-th component of
    /// velocity at point s with respect
    /// to all data that can affect its value. In addition, return the global
    /// equation numbers corresponding to the data. The function is virtual
    /// so that it can be overloaded in the refineable version
    virtual void dinterpolated_u_nst_ddata(const Vector<double>& s,
                                           const unsigned& i,
                                           Vector<double>& du_ddata,
                                           Vector<unsigned>& global_eqn_number)
    {
      // Find number of nodes
      unsigned n_node = nnode();
      // Local shape function
      Shape psi(n_node);
      // Find values of shape function
      shape(s, psi);
 
      // Find the index at which the velocity component is stored
      const unsigned u_nodal_index = u_index_nst(i);
 
      // 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 th nodes again and set the derivatives
      unsigned count = 0;
      // Loop over the local nodes and sum
      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 (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;
        }
      }
    }
 
 
    /// Return FE interpolated pressure at local coordinate s
    virtual double interpolated_p_nst(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_pres = npres_nst();
      // Local shape function
      Shape psi(n_pres);
      // Find values of shape function
      pshape_nst(s, psi);
 
      // Initialise value of p
      double interpolated_p = 0.0;
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_pres; l++)
      {
        interpolated_p += p_nst(l) * psi[l];
      }
 
      return (interpolated_p);
    }
 
 
    /// Return FE interpolated pressure at local coordinate s at time level t
    double interpolated_p_nst(const unsigned& t, const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_pres = npres_nst();
      // Local shape function
      Shape psi(n_pres);
      // Find values of shape function
      pshape_nst(s, psi);
 
      // Initialise value of p
      double interpolated_p = 0.0;
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_pres; l++)
      {
        interpolated_p += p_nst(t, l) * psi[l];
      }
 
      return (interpolated_p);
    }
 
 
    /// Return FE interpolated derivatives of velocity component u[i]
    /// w.r.t spatial global coordinate direction x[j] at local coordinate s
    double interpolated_dudx_nst(const Vector<double>& s,
                                 const unsigned& i,
                                 const unsigned& j) const
    {
      // Determine number of nodes
      const unsigned n_node = nnode();
 
      // Allocate storage for local shape function and its derivatives
      // with respect to space
      Shape psif(n_node);
      DShape dpsifdx(n_node, DIM);
 
      // Find values of shape function (ignore jacobian)
      (void)this->dshape_eulerian(s, psif, dpsifdx);
 
      // Get the index at which the velocity is stored
      const unsigned u_nodal_index = u_index_nst(i);
 
      // Initialise value of dudx
      double interpolated_dudx = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_dudx += nodal_value(l, u_nodal_index) * dpsifdx(l, j);
      }
 
      return (interpolated_dudx);
    }
 
 
    /// Output solution in data vector at local cordinates s:
    /// x,y [,z], u,v,[w], p
    void point_output_data(const Vector<double>& s, Vector<double>& data)
    {
      // Dimension
      unsigned dim = s.size();
 
      // Resize data for values
      data.resize(2 * dim + 1);
 
      // Write values in the vector
      for (unsigned i = 0; i < dim; i++)
      {
        data[i] = interpolated_x(s, i);
        data[i + dim] = this->interpolated_u_nst(s, i);
      }
      data[2 * dim] = this->interpolated_p_nst(s);
    }
  };
 
  /// ///////////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////////
 
 
  //==========================================================================
  /// Crouzeix_Raviart elements are Navier--Stokes elements with quadratic
  /// interpolation for velocities and positions, but a discontinuous linear
  /// pressure interpolation. They can be used within oomph-lib's
  /// block preconditioning framework.
  //==========================================================================
  template<unsigned DIM>
  class QCrouzeixRaviartElement : public virtual QElement<DIM, 3>,
                                  public virtual NavierStokesEquations<DIM>
  {
  private:
    /// Static array of ints to hold required number of variables at nodes
    static const unsigned Initial_Nvalue[];
 
  protected:
    /// Internal index that indicates at which internal data the pressure
    /// is stored
    unsigned P_nst_internal_index;
 
 
    /// Velocity shape and test functions and their derivs
    /// w.r.t. to global coords  at local coordinate s (taken from geometry)
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_nst(const Vector<double>& s,
                                                Shape& psi,
                                                DShape& dpsidx,
                                                Shape& test,
                                                DShape& dtestdx) const;
 
    /// Velocity shape and test functions and their derivs
    /// w.r.t. to global coords at ipt-th integation point (taken from geometry)
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_nst(const unsigned& ipt,
                                                        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 (J). Also compute
    /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const;
 
 
  public:
    /// Constructor, there are DIM+1 internal values (for the pressure)
    QCrouzeixRaviartElement() : QElement<DIM, 3>(), NavierStokesEquations<DIM>()
    {
      // Allocate and add one Internal data object that stored DIM+1 pressure
      // values;
      P_nst_internal_index = this->add_internal_data(new Data(DIM + 1));
    }
 
    /// Number of values (pinned or dofs) required at local node n.
    virtual unsigned required_nvalue(const unsigned& n) const;
 
 
    /// Pressure shape functions at local coordinate s
    inline void pshape_nst(const Vector<double>& s, Shape& psi) const;
 
    /// Pressure shape and test functions at local coordinte s
    inline void pshape_nst(const Vector<double>& s,
                           Shape& psi,
                           Shape& test) const;
 
    /// Return the i-th pressure value
    /// (Discontinous pressure interpolation -- no need to cater for hanging
    /// nodes).
    double p_nst(const unsigned& i) const
    {
      return this->internal_data_pt(P_nst_internal_index)->value(i);
    }
 
    /// Return the i-th pressure value
    /// (Discontinous pressure interpolation -- no need to cater for hanging
    /// nodes).
    double p_nst(const unsigned& t, const unsigned& i) const
    {
      return this->internal_data_pt(P_nst_internal_index)->value(t, i);
    }
 
    /// Return number of pressure values
    unsigned npres_nst() const
    {
      return DIM + 1;
    }
 
    /// Pressure shape and test functions and their derivs
    /// w.r.t. to global coords  at local coordinate s (taken from geometry)
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dpshape_and_dptest_eulerian_nst(const Vector<double>& s,
                                                  Shape& ppsi,
                                                  DShape& dppsidx,
                                                  Shape& ptest,
                                                  DShape& dptestdx) const;
 
    /// Return the local equation numbers for the pressure values.
    inline int p_local_eqn(const unsigned& n) const
    {
      return this->internal_local_eqn(P_nst_internal_index, n);
    }
 
    /// Pin p_dof-th pressure dof and set it to value specified by p_value.
    void fix_pressure(const unsigned& p_dof, const double& p_value)
    {
      this->internal_data_pt(P_nst_internal_index)->pin(p_dof);
      this->internal_data_pt(P_nst_internal_index)->set_value(p_dof, p_value);
    }
 
 
    /// Build FaceElements that apply the Robin boundary condition
    /// to the pressure advection diffusion problem required by
    /// Fp preconditioner
    void build_fp_press_adv_diff_robin_bc_element(const unsigned& face_index)
    {
      this->Pressure_advection_diffusion_robin_element_pt.push_back(
        new FpPressureAdvDiffRobinBCElement<QCrouzeixRaviartElement<DIM>>(
          this, face_index));
    }
 
    ///  Add to the set \c paired_load_data pairs containing
    /// - the pointer to a Data object
    /// and
    /// - the index of the value in that Data object
    /// .
    /// for all values (pressures, velocities) that affect the
    /// load computed in the \c get_load(...) function.
    void identify_load_data(
      std::set<std::pair<Data*, unsigned>>& paired_load_data);
 
    ///  Add to the set \c paired_pressure_data pairs
    /// containing
    /// - the pointer to a Data object
    /// and
    /// - the index of the value in that Data object
    /// .
    /// for all pressure values that affect the
    /// load computed in the \c get_load(...) function.
    void identify_pressure_data(
      std::set<std::pair<Data*, unsigned>>& paired_pressure_data);
 
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile)
    {
      NavierStokesEquations<DIM>::output(outfile);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      NavierStokesEquations<DIM>::output(outfile, nplot);
    }
 
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt)
    {
      NavierStokesEquations<DIM>::output(file_pt);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt, const unsigned& nplot)
    {
      NavierStokesEquations<DIM>::output(file_pt, nplot);
    }
 
 
    /// Full output function:
    /// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation
    /// in tecplot format. Default number of plot points
    void full_output(std::ostream& outfile)
    {
      NavierStokesEquations<DIM>::full_output(outfile);
    }
 
    /// Full output function:
    /// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation
    /// in tecplot format. nplot points in each coordinate direction
    void full_output(std::ostream& outfile, const unsigned& nplot)
    {
      NavierStokesEquations<DIM>::full_output(outfile, nplot);
    }
 
 
    /// The number of "DOF types" that degrees of freedom in this element
    /// are sub-divided into: Velocity and pressure.
    unsigned ndof_types() const
    {
      return DIM + 1;
    }
 
    /// Create a list of pairs for all unknowns in this element,
    /// so that the first entry in each pair contains the global equation
    /// number of the unknown, while the second one contains the number
    /// of the "DOF type" that this unknown is associated with.
    /// (Function can obviously only be called if the equation numbering
    /// scheme has been set up.) Velocity=0; Pressure=1
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const;
  };
 
  // Inline functions
 
  //=======================================================================
  /// Derivatives of the shape functions and test functions w.r.t. to global
  /// (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //=======================================================================
  template<unsigned DIM>
  inline double QCrouzeixRaviartElement<DIM>::dshape_and_dtest_eulerian_nst(
    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);
    // The test functions are equal to the shape functions
    test = psi;
    dtestdx = dpsidx;
    // Return the jacobian
    return J;
  }
 
  //=======================================================================
  /// Derivatives of the shape functions and test functions w.r.t. to global
  /// (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //=======================================================================
  template<unsigned DIM>
  inline double QCrouzeixRaviartElement<
    DIM>::dshape_and_dtest_eulerian_at_knot_nst(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);
    // The test functions are equal to the shape functions
    test = psi;
    dtestdx = dpsidx;
    // Return the jacobian
    return J;
  }
 
 
  //=======================================================================
  /// 2D
  /// Define the shape functions (psi) and test functions (test) and
  /// their derivatives w.r.t. global coordinates (dpsidx and dtestdx)
  /// and return Jacobian of mapping (J). Additionally compute the
  /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
  ///
  /// Galerkin: Test functions = shape functions
  //=======================================================================
  template<>
  inline double QCrouzeixRaviartElement<2>::
    dshape_and_dtest_eulerian_at_knot_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian_at_knot(
      ipt, psi, dpsidx, djacobian_dX, d_dpsidx_dX);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 9; i++)
    {
      test[i] = psi[i];
 
      for (unsigned k = 0; k < 2; k++)
      {
        dtestdx(i, k) = dpsidx(i, k);
 
        for (unsigned p = 0; p < 2; p++)
        {
          for (unsigned q = 0; q < 9; q++)
          {
            d_dtestdx_dX(p, q, i, k) = d_dpsidx_dX(p, q, i, k);
          }
        }
      }
    }
 
    // Return the jacobian
    return J;
  }
 
 
  //=======================================================================
  /// 3D
  /// Define the shape functions (psi) and test functions (test) and
  /// their derivatives w.r.t. global coordinates (dpsidx and dtestdx)
  /// and return Jacobian of mapping (J). Additionally compute the
  /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
  ///
  /// Galerkin: Test functions = shape functions
  //=======================================================================
  template<>
  inline double QCrouzeixRaviartElement<3>::
    dshape_and_dtest_eulerian_at_knot_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian_at_knot(
      ipt, psi, dpsidx, djacobian_dX, d_dpsidx_dX);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 27; i++)
    {
      test[i] = psi[i];
 
      for (unsigned k = 0; k < 3; k++)
      {
        dtestdx(i, k) = dpsidx(i, k);
 
        for (unsigned p = 0; p < 3; p++)
        {
          for (unsigned q = 0; q < 27; q++)
          {
            d_dtestdx_dX(p, q, i, k) = d_dpsidx_dX(p, q, i, k);
          }
        }
      }
    }
 
    // Return the jacobian
    return J;
  }
 
 
  //=======================================================================
  /// 2D :
  /// Pressure shape functions
  //=======================================================================
  template<>
  inline void QCrouzeixRaviartElement<2>::pshape_nst(const Vector<double>& s,
                                                     Shape& psi) const
  {
    psi[0] = 1.0;
    psi[1] = s[0];
    psi[2] = s[1];
  }
 
 
  //==========================================================================
  /// 2D :
  /// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
  /// Return Jacobian of mapping between local and global coordinates.
  //==========================================================================
  template<>
  inline double QCrouzeixRaviartElement<2>::dpshape_and_dptest_eulerian_nst(
    const Vector<double>& s,
    Shape& ppsi,
    DShape& dppsidx,
    Shape& ptest,
    DShape& dptestdx) const
  {
    // Initalise with shape fcts and derivs. w.r.t. to local coordinates
    ppsi[0] = 1.0;
    ppsi[1] = s[0];
    ppsi[2] = s[1];
 
    dppsidx(0, 0) = 0.0;
    dppsidx(1, 0) = 1.0;
    dppsidx(2, 0) = 0.0;
 
    dppsidx(0, 1) = 0.0;
    dppsidx(1, 1) = 0.0;
    dppsidx(2, 1) = 1.0;
 
 
    // Get the values of the shape functions and their local derivatives
    Shape psi(9);
    DShape dpsi(9, 2);
    dshape_local(s, psi, dpsi);
 
    // Allocate memory for the inverse 2x2 jacobian
    DenseMatrix<double> inverse_jacobian(2);
 
    // Now calculate the inverse jacobian
    const double det = local_to_eulerian_mapping(dpsi, inverse_jacobian);
 
    // Now set the values of the derivatives to be derivs w.r.t. to the
    // Eulerian coordinates
    transform_derivatives(inverse_jacobian, dppsidx);
 
    // The test functions are equal to the shape functions
    ptest = ppsi;
    dptestdx = dppsidx;
 
    // Return the determinant of the jacobian
    return det;
  }
 
 
  //=======================================================================
  /// Ppressure shape and test functions
  //=======================================================================
  template<unsigned DIM>
  inline void QCrouzeixRaviartElement<DIM>::pshape_nst(const Vector<double>& s,
                                                       Shape& psi,
                                                       Shape& test) const
  {
    // Call the pressure shape functions
    this->pshape_nst(s, psi);
    // Test functions are equal to shape functions
    test = psi;
  }
 
 
  //=======================================================================
  /// 3D :
  /// Pressure shape functions
  //=======================================================================
  template<>
  inline void QCrouzeixRaviartElement<3>::pshape_nst(const Vector<double>& s,
                                                     Shape& psi) const
  {
    psi[0] = 1.0;
    psi[1] = s[0];
    psi[2] = s[1];
    psi[3] = s[2];
  }
 
 
  //==========================================================================
  /// 3D :
  /// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
  /// Return Jacobian of mapping between local and global coordinates.
  //==========================================================================
  template<>
  inline double QCrouzeixRaviartElement<3>::dpshape_and_dptest_eulerian_nst(
    const Vector<double>& s,
    Shape& ppsi,
    DShape& dppsidx,
    Shape& ptest,
    DShape& dptestdx) const
  {
    // Initalise with shape fcts and derivs. w.r.t. to local coordinates
    ppsi[0] = 1.0;
    ppsi[1] = s[0];
    ppsi[2] = s[1];
    ppsi[3] = s[2];
 
    dppsidx(0, 0) = 0.0;
    dppsidx(1, 0) = 1.0;
    dppsidx(2, 0) = 0.0;
    dppsidx(3, 0) = 0.0;
 
    dppsidx(0, 1) = 0.0;
    dppsidx(1, 1) = 0.0;
    dppsidx(2, 1) = 1.0;
    dppsidx(3, 1) = 0.0;
 
    dppsidx(0, 2) = 0.0;
    dppsidx(1, 2) = 0.0;
    dppsidx(2, 2) = 0.0;
    dppsidx(3, 2) = 1.0;
 
 
    // Get the values of the shape functions and their local derivatives
    Shape psi(27);
    DShape dpsi(27, 3);
    dshape_local(s, psi, dpsi);
 
    // Allocate memory for the inverse 3x3 jacobian
    DenseMatrix<double> inverse_jacobian(3);
 
    // Now calculate the inverse jacobian
    const double det = local_to_eulerian_mapping(dpsi, inverse_jacobian);
 
    // Now set the values of the derivatives to be derivs w.r.t. to the
    // Eulerian coordinates
    transform_derivatives(inverse_jacobian, dppsidx);
 
    // The test functions are equal to the shape functions
    ptest = ppsi;
    dptestdx = dppsidx;
 
    // Return the determinant of the jacobian
    return det;
  }
 
 
  //=======================================================================
  /// Face geometry of the 2D Crouzeix_Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<QCrouzeixRaviartElement<2>> : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of the 3D Crouzeix_Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<QCrouzeixRaviartElement<3>> : public virtual QElement<2, 3>
  {
  public:
    FaceGeometry() : QElement<2, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of the FaceGeometry of the 2D Crouzeix_Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<FaceGeometry<QCrouzeixRaviartElement<2>>>
    : public virtual PointElement
  {
  public:
    FaceGeometry() : PointElement() {}
  };
 
 
  //=======================================================================
  /// Face geometry of the FaceGeometry of the 3D Crouzeix_Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<FaceGeometry<QCrouzeixRaviartElement<3>>>
    : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
 
  /// /////////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Taylor--Hood elements are Navier--Stokes elements
  /// with quadratic interpolation for velocities and positions and
  /// continuous linear pressure interpolation. They can be used
  /// within oomph-lib's block-preconditioning framework.
  //=======================================================================
  template<unsigned DIM>
  class QTaylorHoodElement : public virtual QElement<DIM, 3>,
                             public virtual NavierStokesEquations<DIM>
  {
  private:
    /// Static array of ints to hold number of variables at node
    static const unsigned Initial_Nvalue[];
 
  protected:
    /// Static array of ints to hold conversion from pressure
    /// node numbers to actual node numbers
    static const unsigned Pconv[];
 
    /// Velocity shape and test functions and their derivs
    /// w.r.t. to global coords  at local coordinate s (taken from geometry)
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_nst(const Vector<double>& s,
                                                Shape& psi,
                                                DShape& dpsidx,
                                                Shape& test,
                                                DShape& dtestdx) const;
 
    /// Velocity shape and test functions and their derivs
    /// w.r.t. to global coords  at local coordinate s (taken from geometry)
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_nst(const unsigned& ipt,
                                                        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 (J). Also compute
    /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const;
 
  public:
    /// Constructor, no internal data points
    QTaylorHoodElement() : QElement<DIM, 3>(), NavierStokesEquations<DIM>() {}
 
    /// Number of values (pinned or dofs) required at node n. Can
    /// be overwritten for hanging node version
    inline virtual unsigned required_nvalue(const unsigned& n) const
    {
      return Initial_Nvalue[n];
    }
 
 
    /// Pressure shape functions at local coordinate s
    inline void pshape_nst(const Vector<double>& s, Shape& psi) const;
 
    /// Pressure shape and test functions at local coordinte s
    inline void pshape_nst(const Vector<double>& s,
                           Shape& psi,
                           Shape& test) const;
 
    /// Set the value at which the pressure is stored in the nodes
    virtual int p_nodal_index_nst() const
    {
      return static_cast<int>(DIM);
    }
 
    /// Return the local equation numbers for the pressure values.
    inline int p_local_eqn(const unsigned& n) const
    {
      return this->nodal_local_eqn(Pconv[n], p_nodal_index_nst());
    }
 
    /// Access function for the pressure values at local pressure
    /// node n_p (const version)
    double p_nst(const unsigned& n_p) const
    {
      return this->nodal_value(Pconv[n_p], this->p_nodal_index_nst());
    }
 
    /// Access function for the pressure values at local pressure
    /// node n_p (const version)
    double p_nst(const unsigned& t, const unsigned& n_p) const
    {
      return this->nodal_value(t, Pconv[n_p], this->p_nodal_index_nst());
    }
 
    /// Pressure shape and test functions and their derivs
    /// w.r.t. to global coords  at local coordinate s (taken from geometry).
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dpshape_and_dptest_eulerian_nst(const Vector<double>& s,
                                                  Shape& ppsi,
                                                  DShape& dppsidx,
                                                  Shape& ptest,
                                                  DShape& dptestdx) const;
 
    /// Return number of pressure values
    unsigned npres_nst() const
    {
      return static_cast<unsigned>(pow(2.0, static_cast<int>(DIM)));
    }
 
 
    /// Deduce whether or not the provided node is a pressure node
    bool is_pressure_node(const unsigned& n) const
    {
      // The number of pressure nodes
      unsigned n_p = npres_nst();
 
      // See if the value n is in the array Pconv
      return std::find(this->Pconv, this->Pconv + n_p, n) != this->Pconv + n_p;
    } // End of is_pressure_node
 
 
    /// Pin p_dof-th pressure dof and set it to value specified by p_value.
    void fix_pressure(const unsigned& p_dof, const double& p_value)
    {
      this->node_pt(Pconv[p_dof])->pin(this->p_nodal_index_nst());
      this->node_pt(Pconv[p_dof])
        ->set_value(this->p_nodal_index_nst(), p_value);
    }
 
 
    /// Build FaceElements that apply the Robin boundary condition
    /// to the pressure advection diffusion problem required by
    /// Fp preconditioner
    void build_fp_press_adv_diff_robin_bc_element(const unsigned& face_index)
    {
      this->Pressure_advection_diffusion_robin_element_pt.push_back(
        new FpPressureAdvDiffRobinBCElement<QTaylorHoodElement<DIM>>(
          this, face_index));
    }
 
 
    ///  Add to the set \c paired_load_data pairs containing
    /// - the pointer to a Data object
    /// and
    /// - the index of the value in that Data object
    /// .
    /// for all values (pressures, velocities) that affect the
    /// load computed in the \c get_load(...) function.
    void identify_load_data(
      std::set<std::pair<Data*, unsigned>>& paired_load_data);
 
 
    ///  Add to the set \c paired_pressure_data pairs
    /// containing
    /// - the pointer to a Data object
    /// and
    /// - the index of the value in that Data object
    /// .
    /// for all pressure values that affect the
    /// load computed in the \c get_load(...) function.
    void identify_pressure_data(
      std::set<std::pair<Data*, unsigned>>& paired_pressure_data);
 
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile)
    {
      NavierStokesEquations<DIM>::output(outfile);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      NavierStokesEquations<DIM>::output(outfile, nplot);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt)
    {
      NavierStokesEquations<DIM>::output(file_pt);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt, const unsigned& nplot)
    {
      NavierStokesEquations<DIM>::output(file_pt, nplot);
    }
 
 
    /// Returns the number of "DOF types" that degrees of freedom
    /// in this element are sub-divided into: Velocity and pressure.
    unsigned ndof_types() const
    {
      return DIM + 1;
    }
 
    /// Create a list of pairs for all unknowns in this element,
    /// so that the first entry in each pair contains the global equation
    /// number of the unknown, while the second one contains the number
    /// of the "DOF type" that this unknown is associated with.
    /// (Function can obviously only be called if the equation numbering
    /// scheme has been set up.) Velocity=0; Pressure=1
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const;
  };
 
  // Inline functions
 
  //==========================================================================
  /// Derivatives of the shape functions and test functions w.r.t to
  /// global (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //==========================================================================
  template<unsigned DIM>
  inline double QTaylorHoodElement<DIM>::dshape_and_dtest_eulerian_nst(
    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);
 
    // The test functions are equal to the shape functions
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
 
  //==========================================================================
  /// Derivatives of the shape functions and test functions w.r.t to
  /// global (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //==========================================================================
  template<unsigned DIM>
  inline double QTaylorHoodElement<DIM>::dshape_and_dtest_eulerian_at_knot_nst(
    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);
 
    // The test functions are equal to the shape functions
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
 
  //==========================================================================
  /// 2D :
  /// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
  /// Return Jacobian of mapping between local and global coordinates.
  //==========================================================================
  template<>
  inline double QTaylorHoodElement<2>::dpshape_and_dptest_eulerian_nst(
    const Vector<double>& s,
    Shape& ppsi,
    DShape& dppsidx,
    Shape& ptest,
    DShape& dptestdx) const
  {
    // Local storage
    double psi1[2], psi2[2];
    double dpsi1[2], dpsi2[2];
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[0], psi1);
    OneDimLagrange::shape<2>(s[1], psi2);
    OneDimLagrange::dshape<2>(s[0], dpsi1);
    OneDimLagrange::dshape<2>(s[1], dpsi2);
 
    // Now let's loop over the nodal points in the element
    // s1 is the "x" coordinate, s2 the "y"
    for (unsigned i = 0; i < 2; i++)
    {
      for (unsigned j = 0; j < 2; j++)
      {
        /*Multiply the two 1D functions together to get the 2D function*/
        ppsi[2 * i + j] = psi2[i] * psi1[j];
        dppsidx(2 * i + j, 0) = psi2[i] * dpsi1[j];
        dppsidx(2 * i + j, 1) = dpsi2[i] * psi1[j];
      }
    }
 
 
    // Get the values of the shape functions and their local derivatives
    Shape psi(9);
    DShape dpsi(9, 2);
    dshape_local(s, psi, dpsi);
 
    // Allocate memory for the inverse 2x2 jacobian
    DenseMatrix<double> inverse_jacobian(2);
 
    // Now calculate the inverse jacobian
    const double det = local_to_eulerian_mapping(dpsi, inverse_jacobian);
 
    // Now set the values of the derivatives to be derivs w.r.t. to the
    // Eulerian coordinates
    transform_derivatives(inverse_jacobian, dppsidx);
 
    // The test functions are equal to the shape functions
    ptest = ppsi;
    dptestdx = dppsidx;
 
    // Return the determinant of the jacobian
    return det;
  }
 
 
  //==========================================================================
  /// 2D :
  /// Define the shape functions (psi) and test functions (test) and
  /// their derivatives w.r.t. global coordinates (dpsidx and dtestdx)
  /// and return Jacobian of mapping (J). Additionally compute the
  /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
  ///
  /// Galerkin: Test functions = shape functions
  //==========================================================================
  template<>
  inline double QTaylorHoodElement<2>::dshape_and_dtest_eulerian_at_knot_nst(
    const unsigned& ipt,
    Shape& psi,
    DShape& dpsidx,
    RankFourTensor<double>& d_dpsidx_dX,
    Shape& test,
    DShape& dtestdx,
    RankFourTensor<double>& d_dtestdx_dX,
    DenseMatrix<double>& djacobian_dX) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian_at_knot(
      ipt, psi, dpsidx, djacobian_dX, d_dpsidx_dX);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 9; i++)
    {
      test[i] = psi[i];
 
      for (unsigned k = 0; k < 2; k++)
      {
        dtestdx(i, k) = dpsidx(i, k);
 
        for (unsigned p = 0; p < 2; p++)
        {
          for (unsigned q = 0; q < 9; q++)
          {
            d_dtestdx_dX(p, q, i, k) = d_dpsidx_dX(p, q, i, k);
          }
        }
      }
    }
 
    // Return the jacobian
    return J;
  }
 
 
  //==========================================================================
  /// 3D :
  /// Define the shape functions (psi) and test functions (test) and
  /// their derivatives w.r.t. global coordinates (dpsidx and dtestdx)
  /// and return Jacobian of mapping (J). Additionally compute the
  /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
  ///
  /// Galerkin: Test functions = shape functions
  //==========================================================================
  template<>
  inline double QTaylorHoodElement<3>::dshape_and_dtest_eulerian_at_knot_nst(
    const unsigned& ipt,
    Shape& psi,
    DShape& dpsidx,
    RankFourTensor<double>& d_dpsidx_dX,
    Shape& test,
    DShape& dtestdx,
    RankFourTensor<double>& d_dtestdx_dX,
    DenseMatrix<double>& djacobian_dX) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian_at_knot(
      ipt, psi, dpsidx, djacobian_dX, d_dpsidx_dX);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 27; i++)
    {
      test[i] = psi[i];
 
      for (unsigned k = 0; k < 3; k++)
      {
        dtestdx(i, k) = dpsidx(i, k);
 
        for (unsigned p = 0; p < 3; p++)
        {
          for (unsigned q = 0; q < 27; q++)
          {
            d_dtestdx_dX(p, q, i, k) = d_dpsidx_dX(p, q, i, k);
          }
        }
      }
    }
 
    // Return the jacobian
    return J;
  }
 
 
  //==========================================================================
  /// 2D :
  /// Pressure shape functions
  //==========================================================================
  template<>
  inline void QTaylorHoodElement<2>::pshape_nst(const Vector<double>& s,
                                                Shape& psi) const
  {
    // Local storage
    double psi1[2], psi2[2];
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[0], psi1);
    OneDimLagrange::shape<2>(s[1], psi2);
 
    // Now let's loop over the nodal points in the element
    // s1 is the "x" coordinate, s2 the "y"
    for (unsigned i = 0; i < 2; i++)
    {
      for (unsigned j = 0; j < 2; j++)
      {
        /*Multiply the two 1D functions together to get the 2D function*/
        psi[2 * i + j] = psi2[i] * psi1[j];
      }
    }
  }
 
 
  //==========================================================================
  /// 3D :
  /// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
  /// Return Jacobian of mapping between local and global coordinates.
  //==========================================================================
  template<>
  inline double QTaylorHoodElement<3>::dpshape_and_dptest_eulerian_nst(
    const Vector<double>& s,
    Shape& ppsi,
    DShape& dppsidx,
    Shape& ptest,
    DShape& dptestdx) const
  {
    // Local storage
    double psi1[2], psi2[2], psi3[2];
    double dpsi1[2], dpsi2[2], dpsi3[2];
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[0], psi1);
    OneDimLagrange::shape<2>(s[1], psi2);
    OneDimLagrange::shape<2>(s[2], psi3);
    OneDimLagrange::dshape<2>(s[0], dpsi1);
    OneDimLagrange::dshape<2>(s[1], dpsi2);
    OneDimLagrange::dshape<2>(s[2], dpsi3);
 
    // Now let's loop over the nodal points in the element
    // s0 is the "x" coordinate, s1 the "y", s2 is the "z"
    for (unsigned i = 0; i < 2; i++)
    {
      for (unsigned j = 0; j < 2; j++)
      {
        for (unsigned k = 0; k < 2; k++)
        {
          /*Multiply the three 1D functions together to get the 3D function*/
          ppsi[4 * i + 2 * j + k] = psi3[i] * psi2[j] * psi1[k];
          dppsidx(4 * i + 2 * j + k, 0) = psi3[i] * psi2[j] * dpsi1[k];
          dppsidx(4 * i + 2 * j + k, 1) = psi3[i] * dpsi2[j] * psi1[k];
          dppsidx(4 * i + 2 * j + k, 2) = dpsi3[i] * psi2[j] * psi1[k];
        }
      }
    }
 
 
    // Get the values of the shape functions and their local derivatives
    Shape psi(27);
    DShape dpsi(27, 3);
    dshape_local(s, psi, dpsi);
 
    // Allocate memory for the inverse 3x3 jacobian
    DenseMatrix<double> inverse_jacobian(3);
 
    // Now calculate the inverse jacobian
    const double det = local_to_eulerian_mapping(dpsi, inverse_jacobian);
 
    // Now set the values of the derivatives to be derivs w.r.t. to the
    // Eulerian coordinates
    transform_derivatives(inverse_jacobian, dppsidx);
 
    // The test functions are equal to the shape functions
    ptest = ppsi;
    dptestdx = dppsidx;
 
    // Return the determinant of the jacobian
    return det;
  }
 
  //==========================================================================
  /// 3D :
  /// Pressure shape functions
  //==========================================================================
  template<>
  inline void QTaylorHoodElement<3>::pshape_nst(const Vector<double>& s,
                                                Shape& psi) const
  {
    // Local storage
    double psi1[2], psi2[2], psi3[2];
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[0], psi1);
    OneDimLagrange::shape<2>(s[1], psi2);
    OneDimLagrange::shape<2>(s[2], psi3);
 
    // Now let's loop over the nodal points in the element
    // s0 is the "x" coordinate, s1 the "y", s2 is the "z"
    for (unsigned i = 0; i < 2; i++)
    {
      for (unsigned j = 0; j < 2; j++)
      {
        for (unsigned k = 0; k < 2; k++)
        {
          /*Multiply the three 1D functions together to get the 3D function*/
          psi[4 * i + 2 * j + k] = psi3[i] * psi2[j] * psi1[k];
        }
      }
    }
  }
 
 
  //==========================================================================
  /// Pressure shape and test functions
  //==========================================================================
  template<unsigned DIM>
  inline void QTaylorHoodElement<DIM>::pshape_nst(const Vector<double>& s,
                                                  Shape& psi,
                                                  Shape& test) const
  {
    // Call the pressure shape functions
    this->pshape_nst(s, psi);
    // Test functions are shape functions
    test = psi;
  }
 
 
  //=======================================================================
  /// Face geometry of the 2D Taylor_Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<QTaylorHoodElement<2>> : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of the 3D Taylor_Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<QTaylorHoodElement<3>> : public virtual QElement<2, 3>
  {
  public:
    FaceGeometry() : QElement<2, 3>() {}
  };
 
 
  //=======================================================================
  /// Face geometry of the FaceGeometry of the 2D Taylor Hoodelements
  //=======================================================================
  template<>
  class FaceGeometry<FaceGeometry<QTaylorHoodElement<2>>>
    : public virtual PointElement
  {
  public:
    FaceGeometry() : PointElement() {}
  };
 
 
  //=======================================================================
  /// Face geometry of the FaceGeometry of the 3D Taylor_Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<FaceGeometry<QTaylorHoodElement<3>>>
    : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
 
  /// /////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////
 
 
  //==========================================================
  /// Taylor Hood upgraded to become projectable
  //==========================================================
  template<class TAYLOR_HOOD_ELEMENT>
  class ProjectableTaylorHoodElement
    : public virtual ProjectableElement<TAYLOR_HOOD_ELEMENT>
  {
  public:
    /// Constructor [this was only required explicitly
    /// from gcc 4.5.2 onwards...]
    ProjectableTaylorHoodElement() {}
 
 
    /// 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.
    /// In the underlying Taylor Hood elements the fld-th velocities are stored
    /// at the fld-th value of the nodes; the pressures (the dim-th
    /// field) are the dim-th values at the vertex nodes etc.
    Vector<std::pair<Data*, unsigned>> data_values_of_field(const unsigned& fld)
    {
      // Create the vector
      Vector<std::pair<Data*, unsigned>> data_values;
 
      // Velocities dofs
      if (fld < this->dim())
      {
        // Loop over all nodes
        unsigned nnod = this->nnode();
        for (unsigned j = 0; j < nnod; j++)
        {
          // Add the data value associated with the velocity components
          data_values.push_back(std::make_pair(this->node_pt(j), fld));
        }
      }
      // Pressure
      else
      {
        // Loop over all vertex nodes
        unsigned Pconv_size = this->dim() + 1;
        for (unsigned j = 0; j < Pconv_size; j++)
        {
          // Add the data value associated with the pressure components
          unsigned vertex_index = this->Pconv[j];
          data_values.push_back(
            std::make_pair(this->node_pt(vertex_index), fld));
        }
      }
 
      // Return the vector
      return data_values;
    }
 
    /// Number of fields to be projected: dim+1, corresponding to
    /// velocity components and  pressure
    unsigned nfields_for_projection()
    {
      return this->dim() + 1;
    }
 
    /// Number of history values to be stored for fld-th field. Whatever
    /// the timestepper has set up for the velocity components and
    /// none for the pressure field.
    /// (Note: count includes current value!)
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
      if (fld == this->dim())
      {
        // pressure doesn't have history values
        return this->node_pt(0)->ntstorage(); // 1;
      }
      else
      {
        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();
 
      if (fld == n_dim)
      {
        // We are dealing with the pressure
        this->pshape_nst(s, psi);
 
        Shape psif(n_node), testf(n_node);
        DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
        // Domain Shape
        double J = this->dshape_and_dtest_eulerian_nst(
          s, psif, dpsifdx, testf, dtestfdx);
        return J;
      }
      else
      {
        Shape testf(n_node);
        DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
        // Domain Shape
        double J =
          this->dshape_and_dtest_eulerian_nst(s, psi, dpsifdx, testf, dtestfdx);
        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)
    {
      unsigned n_dim = this->dim();
      unsigned n_node = this->nnode();
 
      // If fld=n_dim, we deal with the pressure
      if (fld == n_dim)
      {
        return this->interpolated_p_nst(t, s);
      }
      // Velocity
      else
      {
        // Find the index at which the variable is stored
        unsigned u_nodal_index = this->u_index_nst(fld);
 
        // 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;
 
        // Sum over the local nodes at that time
        for (unsigned l = 0; l < n_node; l++)
        {
          interpolated_u += this->nodal_value(t, l, u_nodal_index) * psi[l];
        }
        return interpolated_u;
      }
    }
 
 
    /// Return number of values in field fld
    unsigned nvalue_of_field(const unsigned& fld)
    {
      if (fld == this->dim())
      {
        return this->npres_nst();
      }
      else
      {
        return this->nnode();
      }
    }
 
 
    /// Return local equation number of value j in field fld.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
      if (fld == this->dim())
      {
        return this->p_local_eqn(j);
      }
      else
      {
        const unsigned u_nodal_index = this->u_index_nst(fld);
        return this->nodal_local_eqn(j, u_nodal_index);
      }
    }
  };
 
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<ProjectableTaylorHoodElement<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<ProjectableTaylorHoodElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
 
  //==========================================================
  /// Crouzeix Raviart upgraded to become projectable
  //==========================================================
  template<class CROUZEIX_RAVIART_ELEMENT>
  class ProjectableCrouzeixRaviartElement
    : public virtual ProjectableElement<CROUZEIX_RAVIART_ELEMENT>
  {
  public:
    /// Constructor [this was only required explicitly
    /// from gcc 4.5.2 onwards...]
    ProjectableCrouzeixRaviartElement() {}
 
    /// 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.
    /// In the underlying Crouzeix Raviart elements the
    /// fld-th velocities are stored
    /// at the fld-th value of the nodes; the pressures are stored internally
    Vector<std::pair<Data*, unsigned>> data_values_of_field(const unsigned& fld)
    {
      // Create the vector
      Vector<std::pair<Data*, unsigned>> data_values;
 
      // Velocities dofs
      if (fld < this->dim())
      {
        // Loop over all nodes
        const unsigned n_node = this->nnode();
        for (unsigned n = 0; n < n_node; n++)
        {
          // Add the data value associated with the velocity components
          data_values.push_back(std::make_pair(this->node_pt(n), fld));
        }
      }
      // Pressure
      else
      {
        // Need to push back the internal data
        const unsigned n_press = this->npres_nst();
        // Loop over all pressure values
        for (unsigned j = 0; j < n_press; j++)
        {
          data_values.push_back(std::make_pair(
            this->internal_data_pt(this->P_nst_internal_index), j));
        }
      }
 
      // Return the vector
      return data_values;
    }
 
    /// Number of fields to be projected: dim+1, corresponding to
    /// velocity components and  pressure
    unsigned nfields_for_projection()
    {
      return this->dim() + 1;
    }
 
    /// Number of history values to be stored for fld-th field. Whatever
    /// the timestepper has set up for the velocity components and
    /// none for the pressure field.
    /// (Note: count includes current value!)
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
      if (fld == this->dim())
      {
        // pressure doesn't have history values
        return 1;
      }
      else
      {
        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();
 
      if (fld == n_dim)
      {
        // We are dealing with the pressure
        this->pshape_nst(s, psi);
 
        Shape psif(n_node), testf(n_node);
        DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
        // Domain Shape
        double J = this->dshape_and_dtest_eulerian_nst(
          s, psif, dpsifdx, testf, dtestfdx);
        return J;
      }
      else
      {
        Shape testf(n_node);
        DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
        // Domain Shape
        double J =
          this->dshape_and_dtest_eulerian_nst(s, psi, dpsifdx, testf, dtestfdx);
        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)
    {
      unsigned n_dim = this->dim();
 
      // If fld=n_dim, we deal with the pressure
      if (fld == n_dim)
      {
        return this->interpolated_p_nst(s);
      }
      // Velocity
      else
      {
        return this->interpolated_u_nst(t, s, fld);
      }
    }
 
 
    /// Return number of values in field fld
    unsigned nvalue_of_field(const unsigned& fld)
    {
      if (fld == this->dim())
      {
        return this->npres_nst();
      }
      else
      {
        return this->nnode();
      }
    }
 
 
    /// Return local equation number of value j in field fld.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
      if (fld == this->dim())
      {
        return this->p_local_eqn(j);
      }
      else
      {
        const unsigned u_nodal_index = this->u_index_nst(fld);
        return this->nodal_local_eqn(j, u_nodal_index);
      }
    }
  };
 
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<ProjectableCrouzeixRaviartElement<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<ProjectableCrouzeixRaviartElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
 
} // namespace oomph
 
#endif