spacetime_navier_stokes_elements.h

Go to the documentation of this file.

// LIC// ====================================================================
// LIC// This file forms part of oomph-lib, the object-oriented,
// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2024 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
// LIC// modify it under the terms of the GNU Lesser General Public
// LIC// License as published by the Free Software Foundation; either
// LIC// version 2.1 of the License, or (at your option) any later version.
// LIC//
// LIC// This library is distributed in the hope that it will be useful,
// LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
// LIC// You should have received a copy of the GNU Lesser General Public
// LIC// License along with this library; if not, write to the Free Software
// LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
// LIC// 02110-1301  USA.
// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header file for SpaceTimeNavierStokes elements
#ifndef OOMPH_SPACETIME_NAVIER_STOKES_ELEMENTS_HEADER
#define OOMPH_SPACETIME_NAVIER_STOKES_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// Oomph-lib headers
#include "generic.h"
 
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 FpPressureAdvDiffRobinBCSpaceTimeElementBase
    : public virtual FaceElement
  {
  public:
    /// Constructor
    FpPressureAdvDiffRobinBCSpaceTimeElementBase() {}
 
    /// Empty virtual destructor
    virtual ~FpPressureAdvDiffRobinBCSpaceTimeElementBase() {}
 
    /// This function returns the residuals for the traction
    /// function. If compute_jacobian_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,
      const unsigned& compute_jacobian_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 FpPressureAdvDiffRobinBCSpaceTimeElement
    : public virtual FaceGeometry<ELEMENT>,
      public virtual FaceElement,
      public virtual FpPressureAdvDiffRobinBCSpaceTimeElementBase
  {
  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.
    FpPressureAdvDiffRobinBCSpaceTimeElement(
      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 a three-dimensional element
        if (element_pt->dim() == 3)
        {
          // Upcast the element to a RefineableElement
          RefineableElement* ref_el_pt =
            dynamic_cast<RefineableElement*>(element_pt);
 
          // Is it actually refineable?
          if (ref_el_pt != 0)
          {
            // If it is refineable then check if it has any hanging nodes
            if (this->has_hanging_nodes())
            {
              // Create an output stream
              std::ostringstream error_message_stream;
 
              // Create an error message
              error_message_stream
                << "This flux element will not work correctly "
                << "if nodes are hanging!" << std::endl;
 
              // Throw an error message
              throw OomphLibError(error_message_stream.str(),
                                  OOMPH_CURRENT_FUNCTION,
                                  OOMPH_EXCEPTION_LOCATION);
            }
          } // if (ref_el_pt!=0)
        } // if (element_pt->dim()==3)
      } // if (!called_from_refineable_constructor)
#endif
    } // End of FpPressureAdvDiffRobinBCSpaceTimeElement
 
 
    /// Empty destructor
    ~FpPressureAdvDiffRobinBCSpaceTimeElement() {}
 
 
    /// 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,
      const unsigned& flag);
 
 
    /// This function returns just the residuals
    inline void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Create an output stream
      std::ostringstream error_message;
 
      // Create an error message
      error_message << "fill_in_contribution_to_residuals() must not be "
                    << "called directly.\nsince it uses the local equation "
                    << "numbering of the bulk element\nwhich calls the "
                    << "relevant helper function directly." << std::endl;
 
      // Throw an error
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    } // End of fill_in_contribution_to_residuals
 
 
    /// This function returns the residuals and the jacobian
    inline void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                                 DenseMatrix<double>& jacobian)
    {
      // Create an output stream
      std::ostringstream error_message;
 
      // Create an error message
      error_message << "fill_in_contribution_to_jacobian() must not be "
                    << "called directly.\nsince it uses the local equation "
                    << "numbering of the bulk element\nwhich calls the "
                    << "relevant helper function directly." << std::endl;
 
      // Throw an error
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    } // End of fill_in_contribution_to_jacobian
 
 
    /// Overload the output function
    void output(std::ostream& outfile)
    {
      // Call the output function from the FiniteElement base class
      FiniteElement::output(outfile);
    } // End of output
 
 
    /// Output function: x,y,[z],u,v,[w],p in tecplot format
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      // Call the output function from the FiniteElement base class
      FiniteElement::output(outfile, nplot);
    } // End of output
  };
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
  //============================================================================
  /// Get residuals and Jacobian of Robin boundary conditions in pressure
  /// advection diffusion problem in Fp preconditoner
  /// NOTE: Care has to be taken here as fluxes are spatially dependent and
  /// not temporally dependent but the template elements are space-time
  /// elements!
  //============================================================================
  template<class ELEMENT>
  void FpPressureAdvDiffRobinBCSpaceTimeElement<ELEMENT>::
    fill_in_generic_residual_contribution_fp_press_adv_diff_robin_bc(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag)
  {
    // Throw a warning
    throw OomphLibError("You shouldn't be using this yet!",
                        OOMPH_CURRENT_FUNCTION,
                        OOMPH_EXCEPTION_LOCATION);
 
    // Get the dimension of the element
    // DRAIG: Should be 2 as bulk (space-time) element is 3D...
    unsigned my_dim = this->dim();
 
    // Storage for local coordinates in FaceElement
    Vector<double> s(my_dim, 0.0);
 
    // Storage for local coordinates in the associated bulk element
    Vector<double> s_bulk(my_dim + 1, 0.0);
 
    // Storage for outer unit normal
    // DRAIG: Need to be careful here...
    Vector<double> unit_normal(my_dim + 1, 0.0);
 
    // Storage for velocity in bulk element
    // DRAIG: Need to be careful here...
    Vector<double> velocity(my_dim, 0.0);
 
    // Set the value of n_intpt
    unsigned n_intpt = this->integral_pt()->nweight();
 
    // Integer to store local equation number
    int local_eqn = 0;
 
    // Integer to store local unknown number
    int local_unknown = 0;
 
    // Get upcast 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 functions
    Shape psip(n_pres);
 
    // Set up memory for pressure test functions
    Shape 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++)
      {
        // Get the i-th local coordinate at this knot point
        s[i] = this->integral_pt()->knot(ipt, i);
      }
 
      // Find corresponding coordinate in the bulk element
      s_bulk = this->local_coordinate_in_bulk(s);
 
      // Get outer unit normal
      // DRAIG: Make sure the normal is calculated properly; i.e.
      // the normal needs to be calculated in the spatial direction
      // and NOT in the temporal direction!!!
      this->outer_unit_normal(ipt, unit_normal);
 
      // Get velocity in bulk element
      bulk_el_pt->interpolated_u_nst(s_bulk, velocity);
 
      // Get normal component of velocity
      double flux = 0.0;
 
      // Loop over the velocity components
      for (unsigned i = 0; i < my_dim; i++)
      {
        // Update the flux quantity
        flux += velocity[i] * unit_normal[i];
      }
 
      // Modify BC: If outflow (flux>0) apply Neumann condition instead
      if (flux > 0.0)
      {
        // Apply no flux condition
        flux = 0.0;
      }
 
      // Get the pressure at these local coordinates
      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++)
      {
        // Get the local equation number associated with this pressure dof
        local_eqn = bulk_el_pt->p_local_eqn(l);
 
        // If it's not a boundary condition
        if (local_eqn >= 0)
        {
          // Update the residuals
          residuals[local_eqn] -= re * flux * interpolated_press * testp[l] * W;
 
          // If the Jacobian needs to be computed too
          if (flag)
          {
            // Loop over the shape functions in bulk
            for (unsigned l2 = 0; l2 < n_pres; l2++)
            {
              // Get the equation number associated with this pressure dof
              local_unknown = bulk_el_pt->p_local_eqn(l2);
 
              // If it's not a boundary condition
              if (local_unknown >= 0)
              {
                // Update the appropriate Jacobian entry
                jacobian(local_eqn, local_unknown) -=
                  re * flux * psip[l2] * testp[l] * W;
              }
            } // for (unsigned l2=0;l2<n_pres;l2++)
          } // if (flag)
        } // if (local_eqn>=0)
      } // for (unsigned l=0;l<n_pres;l++)
    } // for (unsigned ipt=0;ipt<n_intpt;ipt++)
  } // End of fill_in_generic_residual_contribution_fp_press_adv_diff_robin_bc
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
  //======================================================================
  /// Template-free base class for Navier-Stokes equations to avoid
  /// casting problems
  //======================================================================
  class TemplateFreeSpaceTimeNavierStokesEquationsBase
    : public virtual NavierStokesElementWithDiagonalMassMatrices,
      public virtual FiniteElement
  {
  public:
    /// Constructor (empty)
    TemplateFreeSpaceTimeNavierStokesEquationsBase(){};
 
 
    /// Virtual destructor (empty)
    virtual ~TemplateFreeSpaceTimeNavierStokesEquationsBase(){};
 
 
    /// 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.
  ///
  /// --------------------
  /// SPACE-TIME ELEMENTS:
  /// --------------------
  /// The space-time extension is written ONLY for the 2D Navier-Stokes
  /// equations. The result is a 3D problem (x,y,t) to be solved on a
  /// 3D mesh. The template parameter DIM now corresponds to the
  /// dimension of the space-time problem (i.e. DIM=3 for the 2D flow).
  //======================================================================
  template<unsigned DIM>
  class SpaceTimeNavierStokesEquations
    : public virtual FSIFluidElement,
      public virtual TemplateFreeSpaceTimeNavierStokesEquationsBase
  {
  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;
 
    /// Boolean to indicate whether or not the Strouhal value is
    /// stored as external data (if it's also an unknown of the problem)
    bool ReynoldsStrouhal_is_stored_as_external_data;
 
    /// 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<FpPressureAdvDiffRobinBCSpaceTimeElementBase*>
      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;
 
 
    /// 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;
 
 
    /// 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 a function has not been provided
      if (Body_force_fct_pt == 0)
      {
        // Set the body forces to zero
        result.initialise(0.0);
      }
      // If the function pointer is non-zero
      else
      {
        // Call the function
        (*Body_force_fct_pt)(time, x, result);
      } // if (Body_force_fct_pt!=0)
    } // End of get_body_force_nst
 
 
    /// 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)
    {
      // Reference value
      Vector<double> body_force(DIM, 0.0);
 
      // Get the body force vector
      get_body_force_nst(time, ipt, s, x, body_force);
 
      // Get the finite-difference step size
      double eps_fd = GeneralisedElement::Default_fd_jacobian_step;
 
      // The body force computed at the perturbed coordinates
      Vector<double> body_force_pls(DIM, 0.0);
 
      // Copy the (Eulerian) coordinate vector x
      Vector<double> x_pls(x);
 
      // Loop over the coordinate directions
      for (unsigned i = 0; i < DIM; i++)
      {
        // Update the i-th entry
        x_pls[i] += eps_fd;
 
        // Get the body force at the current time
        get_body_force_nst(time, ipt, s, x_pls, body_force_pls);
 
        // Loop over the coordinate directions
        for (unsigned j = 0; j < DIM; j++)
        {
          // Finite-difference the body force derivative
          d_body_force_dx(j, i) = (body_force_pls[j] - body_force[j]) / eps_fd;
        }
 
        // Reset the i-th entry
        x_pls[i] = x[i];
      }
    } // End of get_body_force_gradient_nst
 
 
    /// Calculate the source fct at a 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 null
      if (Source_fct_pt == 0)
      {
        // Set the source function value to zero
        return 0;
      }
      // Otherwise call the function pointer
      else
      {
        // Return the appropriate value
        return (*Source_fct_pt)(time, x);
      }
    } // End of get_source_nst
 
 
    /// 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)
    {
      // Reference value
      double source = get_source_nst(time, ipt, x);
 
      // Get the finite-difference step size
      double eps_fd = GeneralisedElement::Default_fd_jacobian_step;
 
      // The source function value computed at the perturbed coordinates
      double source_pls = 0.0;
 
      // Copy the (Eulerian) coordinate vector, x
      Vector<double> x_pls(x);
 
      // Loop over the coordinate directions
      for (unsigned i = 0; i < DIM; i++)
      {
        // Update the i-th entry
        x_pls[i] += eps_fd;
 
        // Get the body force at the current time
        source_pls = get_source_nst(time, ipt, x_pls);
 
        // Loop over the coordinate directions
        for (unsigned j = 0; j < DIM; j++)
        {
          // Finite-difference the source function gradient
          gradient[i] = (source_pls - source) / eps_fd;
        }
 
        // Reset the i-th entry
        x_pls[i] = x[i];
      }
    } // End of get_source_gradient_nst
 
 
    /// 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,
      const 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,
      const 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,
      const 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.
    SpaceTimeNavierStokesEquations()
      : 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:
 
      // Set the Reynolds number to the value zero
      Re_pt = &Default_Physical_Constant_Value;
 
      // Set the Reynolds x Strouhal (= Womersley) number to the value zero
      ReSt_pt = &Default_Physical_Constant_Value;
 
      // The Strouhal number (which is a proxy for the period here) is not
      // stored as external data
      ReynoldsStrouhal_is_stored_as_external_data = false;
 
      // Set the Reynolds / Froude value to zero
      ReInvFr_pt = &Default_Physical_Constant_Value;
 
      // Set the gravity vector to be the zero vector
      G_pt = &Default_Gravity_vector;
 
      // Set the Physical ratios:
 
      // Set the viscosity ratio to the value one
      Viscosity_Ratio_pt = &Default_Physical_Ratio_Value;
 
      // Set the density ratio to the value one
      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;
 
 
    /// Function that tells us whether the period is stored as external data
    void store_strouhal_as_external_data(Data* strouhal_data_pt)
    {
#ifdef PARANOID
      // Sanity check; make sure it's not a null pointer
      if (strouhal_data_pt == 0)
      {
        // Create an output stream
        std::ostringstream error_message_stream;
 
        // Create an error message
        error_message_stream
          << "User supplied Strouhal number as external data\n"
          << "but the pointer provided is a null pointer!" << std::endl;
 
        // Throw an error
        throw OomphLibError(error_message_stream.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Set the Strouhal value pointer (the Strouhal number is stored as the
      // only piece of internal data in the phase condition element)
      this->add_external_data(strouhal_data_pt);
 
      // Indicate that the Strouhal number is store as external data
      ReynoldsStrouhal_is_stored_as_external_data = true;
    } // End of store_strouhal_as_external_data
 
 
    // Access functions for the physical constants:
 
    /// Are we storing the Strouhal number as external data?
    bool is_reynolds_strouhal_stored_as_external_data() const
    {
      // Return the flags value
      return ReynoldsStrouhal_is_stored_as_external_data;
    } // End of is_reynolds_strouhal_stored_as_external_data
 
 
    /// Reynolds number
    const double& re() const
    {
      // Use the Reynolds number pointer to get the Reynolds value
      return *Re_pt;
    } // End of re
 
 
    /// Pointer to Reynolds number
    double*& re_pt()
    {
      // Return the Reynolds number pointer
      return Re_pt;
    } // End of re_pt
 
 
    /// ReSt parameter (const. version)
    const double& re_st() const
    {
      // If the st is stored as external data
      if (ReynoldsStrouhal_is_stored_as_external_data)
      {
        // The index of the external data (which contains the st)
        unsigned data_index = 0;
 
        // The index of the value at which the ReynoldsStrouhal number is stored
        unsigned re_st_index = 0;
 
        // Return the value of the st in the external data
        return *(this->external_data_pt(data_index)->value_pt(re_st_index));
      }
      // Otherwise the st is just stored as a pointer
      else
      {
        // Return the value of St
        return *ReSt_pt;
      }
    } // End of re_st
 
 
    /// Pointer to Strouhal parameter (const. version)
    double* re_st_pt() const
    {
      // If the strouhal is stored as external data
      if (ReynoldsStrouhal_is_stored_as_external_data)
      {
        // The index of the external data (which contains the strouhal)
        unsigned data_index = 0;
 
        // The index of the value at which the ReynoldsStrouhal number is stored
        unsigned re_st_index = 0;
 
        // Return the value of the st in the external data
        return this->external_data_pt(data_index)->value_pt(re_st_index);
      }
      // Otherwise the strouhal is just stored as a pointer
      else
      {
        // Return the value of Strouhal
        return ReSt_pt;
      }
    } // End of re_st_pt
 
 
    /// Pointer to ReSt number (can only assign to private member data)
    double*& re_st_pt()
    {
      // Return the ReSt number pointer
      return ReSt_pt;
    } // End of st_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 the appropriate equation number
      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;
 
    /// 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
    {
#ifdef PARANOID
      // Since we're using space-time elements, this only makes sense if t=0
      if (t != 0)
      {
        // Throw an error
        throw OomphLibError("Space-time elements cannot have history values!",
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Return the appropriate nodal value (noting that t=0 here)
      return nodal_value(t, n, u_index_nst(i));
    } // End of u_nst
 
    /// 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
    {
#ifdef PARANOID
      if (i > DIM - 1)
      {
        // Create an output stream
        std::ostringstream error_message_stream;
 
        // Create an error message
        error_message_stream << "Input index " << i << " does not correspond "
                             << "to a velocity component when DIM=" << DIM
                             << "!" << std::endl;
 
        // Throw an error
        throw OomphLibError(error_message_stream.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Return the appropriate entry
      return i;
    } // End of u_index_nst
 
 
    /// Return the number of velocity components (used in
    /// FluidInterfaceElements)
    inline unsigned n_u_nst() const
    {
      // Return the number of equations to solve
      return DIM;
    } // End of n_u_nst
 
 
    /// i-th component of du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    /// NOTE: This is essentially a wrapper for du_dt_nst()
    /// so it can be called externally.
    double get_du_dt(const unsigned& n, const unsigned& i) const
    {
      // Return the value calculated by du_dt_vdp
      return du_dt_nst(n, i);
    } // End of get_du_dt
 
 
    /// 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
    {
      // Storage for the local coordinates
      Vector<double> s(DIM + 1, 0.0);
 
      // Get the local coordinate at the n-th node
      local_coordinate_of_node(n, s);
 
      // Return the interpolated du_i/dt value
      return interpolated_du_dt_nst(s, i);
    } // End of du_dt_nst
 
 
    /// Return FE representation of function value du_i/dt(s) at local
    /// coordinate s
    inline double interpolated_du_dt_nst(const Vector<double>& s,
                                         const unsigned& i) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Allocate space for the derivatives of the shape functions
      DShape dpsidx(n_node, DIM + 1);
 
      // Compute the geometric shape functions and also first derivatives
      // w.r.t. global coordinates at local coordinate s
      dshape_eulerian(s, psi, dpsidx);
 
      // Initialise value of du_i/dt
      double interpolated_dudt = 0.0;
 
      // Find the index at which the variable is stored
      unsigned u_nodal_index = u_index_nst(i);
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        // Update the interpolated du_i/dt value
        interpolated_dudt += nodal_value(l, u_nodal_index) * dpsidx(l, DIM);
      }
 
      // Return the interpolated du_i/dt value
      return interpolated_dudt;
    } // End of interpolated_du_dt_nst
 
 
    /// Disable ALE, i.e. assert the mesh is not moving -- you do this
    /// at your own risk!
    void disable_ALE()
    {
      ALE_is_disabled = true;
    } // End of disable_ALE
 
 
    /// (Re-)enable ALE, i.e. take possible mesh motion into account
    /// when evaluating the time-derivative. Note: By default, ALE is
    /// enabled, at the expense of possibly creating unnecessary work
    /// in problems where the mesh is, in fact, stationary.
    void enable_ALE()
    {
      ALE_is_disabled = false;
    } // End of enable_ALE
 
 
    /// 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 vorticity vector at local coordinate s
    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
      // expecting 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
    {
      // The number of velocity components plus the pressure field
      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 + 1, 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);
 
        // Velocities
        if (i < DIM)
        {
          // Output the i-th velocity component
          file_out << interpolated_u_nst(s, i) << std::endl;
        }
        // Pressure
        else if (i == DIM)
        {
          // Output the pressure at this point
          file_out << interpolated_p_nst(s) << std::endl;
        }
        // Never get here
        else
        {
#ifdef PARANOID
          // 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
        }
      } // for (unsigned iplot=0;iplot<num_plot_points;iplot++)
    } // End of scalar_value_paraview
 
 
    /// Write values of the i-th scalar field at the plot points. Needs
    /// to be implemented for each new specific element type.
    void scalar_value_fct_paraview(
      std::ofstream& file_out,
      const unsigned& i,
      const unsigned& nplot,
      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 time value
      double interpolated_t = 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 i = 0; i < DIM; i++)
        {
          // Assign the i-th spatial coordinate
          spatial_coordinates[i] = interpolated_x(s, i);
        }
 
        // Get the time value
        interpolated_t = interpolated_x(s, DIM);
 
        // ------ DRAIG: REMOVE ----------------------------------------
        // Exact solution vector (here it's simply a scalar)
        Vector<double> exact_soln(DIM + 1, 0.0);
        // DRAIG: Needs to be changed to a Vector of size DIM
        // ------ DRAIG: REMOVE ----------------------------------------
 
        // Get the exact solution at this point
        (*exact_soln_pt)(interpolated_t, spatial_coordinates, exact_soln);
 
        // Output the interpolated solution value
        file_out << exact_soln[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 the appropriate string for the i-th velocity component
        return "Velocity " + StringConversion::to_string(i);
      }
      // Pressure
      else if (i == DIM)
      {
        // Return the name for the pressure
        return "Pressure";
      }
      // Never get here
      else
      {
        // Create an output stream
        std::stringstream error_stream;
 
        // Create the error message
        error_stream << "These Navier Stokes elements only store " << DIM + 1
                     << " fields,\nbut i is currently " << i << std::endl;
 
        // Throw the error message
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
 
        // Dummy return so the compiler doesn't complain
        return " ";
      }
    } // End of scalar_name_paraview
 
 
    /// Output function: x,y,t,u,v,p in tecplot format. The
    /// default number of plot points is five
    void output(std::ostream& outfile)
    {
      // Set the number of plot points in each direction
      unsigned n_plot = 5;
 
      // Forward the call to the appropriate output function
      output(outfile, n_plot);
    } // End of output
 
 
    /// Output function: x,y,[z],u,v,[w],p in tecplot format. Here,
    /// we use n_plot plot points in each coordinate direction
    void output(std::ostream& outfile, const unsigned& n_plot);
 
 
    /// C-style output function: x,y,[z],u,v,[w],p in tecplot format. The
    /// default number of plot points is five
    void output(FILE* file_pt)
    {
      // Set the number of plot points in each direction
      unsigned n_plot = 5;
 
      // Forward the call to the appropriate output function
      output(file_pt, n_plot);
    } // End of output
 
 
    /// C-style output function: x,y,[z],u,v,[w],p in tecplot format. Use
    /// n_plot points in each coordinate direction
    void output(FILE* file_pt, const unsigned& n_plot);
 
 
    /// Full output function:
    /// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation in tecplot format. The
    /// default number of plot points is five
    void full_output(std::ostream& outfile)
    {
      // Set the number of plot points in each direction
      unsigned n_plot = 5;
 
      // Forward the call to the appropriate output function
      full_output(outfile, n_plot);
    } // End of full_output
 
 
    /// Full output function:
    /// x,y,t,u,v,p,du/dt,dv/dt,dissipation in tecplot format. Use
    /// n_plot plot points in each coordinate direction
    void full_output(std::ostream& outfile, const unsigned& n_plot);
 
 
    /// Output function: x,y,t,u,v in tecplot format. Use
    /// n_plot 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,t,omega
    /// 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);
 
    /// 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);
 
 
    /// 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,
                       Vector<double>& error,
                       Vector<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)
    {
      // Do we want to compute the Jacobian? ANSWER: No!
      unsigned compute_jacobian_flag = 0;
 
      // Call the generic residuals function using a dummy matrix argument
      fill_in_generic_residual_contribution_nst(
        residuals,
        GeneralisedElement::Dummy_matrix,
        GeneralisedElement::Dummy_matrix,
        compute_jacobian_flag);
    } // End of fill_in_contribution_to_residuals
 
 
    /// 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)
    {
      // Do we want to compute the Jacobian? ANSWER: Yes!
      unsigned compute_jacobian_flag = 1;
 
      // Call the generic routine with the flag set to 1
      fill_in_generic_residual_contribution_nst(
        residuals,
        jacobian,
        GeneralisedElement::Dummy_matrix,
        compute_jacobian_flag);
    } // End of fill_in_contribution_to_residuals
 
 
    /// 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)
    {
      // Do we want to compute the Jacobian AND mass matrix? ANSWER: Yes!
      unsigned compute_matrices_flag = 2;
 
      // Call the generic routine with the flag set to 2
      fill_in_generic_residual_contribution_nst(
        residuals, jacobian, mass_matrix, compute_matrices_flag);
    } // End of fill_in_contribution_to_jacobian_and_mass_matrix
 
 
    /// Compute the element's residual Vector (differentiated w.r.t. a
    /// parameter)
    void fill_in_contribution_to_dresiduals_dparameter(
      double* const& parameter_pt, Vector<double>& dres_dparam)
    {
      // Do we want to compute the Jacobian? ANSWER: No!
      unsigned compute_jacobian_flag = 0;
 
      // Call the generic residuals function using a dummy matrix argument
      fill_in_generic_dresidual_contribution_nst(
        parameter_pt,
        dres_dparam,
        GeneralisedElement::Dummy_matrix,
        GeneralisedElement::Dummy_matrix,
        compute_jacobian_flag);
    } // End of fill_in_contribution_to_dresiduals_dparameter
 
 
    /// 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)
    {
      // Do we want to compute the Jacobian? ANSWER: Yes!
      unsigned compute_jacobian_flag = 1;
 
      // 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,
        compute_jacobian_flag);
    } // End of fill_in_contribution_to_djacobian_dparameter
 
 
    /// 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)
    {
      // Do we want to compute the Jacobian AND mass matrix? ANSWER: Yes!
      unsigned compute_matrices_flag = 2;
 
      // Call the generic routine with the appropriate flag
      fill_in_generic_dresidual_contribution_nst(parameter_pt,
                                                 dres_dparam,
                                                 djac_dparam,
                                                 dmass_matrix_dparam,
                                                 compute_matrices_flag);
    } // End of fill_in_contribution_to_djacobian_and_dmass_matrix_dparameter
 
 
    /// 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)
    {
      // Do we want to compute the Jacobian? ANSWER: No!
      unsigned compute_jacobian_flag = 0;
 
      // Call the generic routine with the appropriate flag
      fill_in_generic_pressure_advection_diffusion_contribution_nst(
        residuals, GeneralisedElement::Dummy_matrix, compute_jacobian_flag);
    } // End of fill_in_pressure_advection_diffusion_residuals
 
 
    /// 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)
    {
      // Do we want to compute the Jacobian? ANSWER: Yes!
      unsigned compute_jacobian_flag = 1;
 
      // Call the generic routine with the appropriate flag
      fill_in_generic_pressure_advection_diffusion_contribution_nst(
        residuals, jacobian, compute_jacobian_flag);
    } // End of fill_in_pressure_advection_diffusion_jacobian
 
 
    /// 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);
              }
            }
          }
        }
      }
    } // End of pin_all_non_pressure_dofs
 
 
    /// 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 + 1);
        Vector<double> s(DIM);
        unsigned n = face_el_pt->integral_pt()->nweight();
        for (unsigned ipt = 0; ipt < n; ipt++)
        {
          for (unsigned i = 0; i < DIM; 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 + 1; i++)
          {
            outfile << x[i] << " ";
          }
          for (unsigned i = 0; i < DIM; i++)
          {
            outfile << unit_normal[i] << " ";
          }
          outfile << std::endl;
        }
      }
    } // End of output_pressure_advection_diffusion_robin_elements
 
 
    /// 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();
    } // End of delete_pressure_advection_diffusion_robin_elements
 
 
    /// 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>& velocity) const
    {
      // Find the number of nodes
      unsigned n_node = nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Loop over the velocity components
      for (unsigned i = 0; i < DIM; i++)
      {
        // Index at which the nodal value is stored
        unsigned u_nodal_index = u_index_nst(i);
 
        // Initialise the i-th value of the velocity vector
        velocity[i] = 0.0;
 
        // Loop over the local nodes and sum
        for (unsigned l = 0; l < n_node; l++)
        {
          // Update the i-th velocity component approximation
          velocity[i] += nodal_value(l, u_nodal_index) * psi[l];
        }
      } // for (unsigned i=0;i<DIM;i++)
    } // End of interpolated_u_nst
 
 
    /// Return FE interpolated velocity u[i] at local coordinate s
    double interpolated_u_nst(const Vector<double>& s, const unsigned& i) const
    {
      // Find the 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++)
      {
        // Update the i-th velocity component approximation
        interpolated_u += nodal_value(l, u_nodal_index) * psi[l];
      }
 
      // Return the calculated approximation to the i-th velocity component
      return interpolated_u;
    } // End of interpolated_u_nst
 
 
    /// Return FE interpolated velocity u[i] at local coordinate s
    /// time level, t. Purposely broken for space-time elements.
    double interpolated_u_nst(const unsigned& t,
                              const Vector<double>& s,
                              const unsigned& i) const
    {
      // Create an output stream
      std::ostringstream error_message_stream;
 
      // Create an error message
      error_message_stream << "This interface doesn't make sense in "
                           << "space-time elements!" << std::endl;
 
      // Throw an error
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    } // End of interpolated_u_nst
 
 
    /// 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 the 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);
 
      // Find the number of dofs associated with interpolated u
      unsigned n_u_dof = 0;
 
      // Loop over the nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the global equation number of the dof
        int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
 
        // If it's positive add to the count
        if (global_eqn >= 0)
        {
          // Increment the counter
          n_u_dof++;
        }
      } // End of dinterpolated_u_nst_ddata
 
      // 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;
        }
      }
    } // End of dinterpolated_u_nst_ddata
 
 
    /// 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);
    }
 
 
    /// Output solution in data vector at local cordinates s:
    /// x,y,z,u,v,p
    void point_output_data(const Vector<double>& s, Vector<double>& data)
    {
      // Resize data for values
      data.resize(2 * DIM + 2);
 
      // Write values in the vector
      for (unsigned i = 0; i < DIM + 1; i++)
      {
        // Output the i-th (Eulerian) coordinate at these local coordinates
        data[i] = interpolated_x(s, i);
      }
 
      // Write values in the vector
      for (unsigned i = 0; i < DIM; i++)
      {
        // Output the i-th velocity component at these local coordinates
        data[i + (DIM + 1)] = this->interpolated_u_nst(s, i);
      }
 
      // Output the pressure field value at these local coordinates
      data[2 * DIM + 1] = this->interpolated_p_nst(s);
    } // End of point_output_data
  };
 
  /// ///////////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////////
 
  //=======================================================================
  /// 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 QTaylorHoodSpaceTimeElement
    : public virtual QElement<DIM + 1, 3>,
      public virtual SpaceTimeNavierStokesEquations<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;
 
 
    /// 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;
 
  public:
    /// Constructor, no internal data points
    QTaylorHoodSpaceTimeElement()
      : QElement<DIM + 1, 3>(), SpaceTimeNavierStokesEquations<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 the appropriate entry from Initial_Nvalue
      return Initial_Nvalue[n];
    } // End of required_nvalue
 
 
    /// 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
    {
      // The pressure is stored straight after the velocity components
      return static_cast<int>(DIM);
    } // End of p_nodal_index_nst
 
 
    /// Return the local equation numbers for the pressure values.
    inline int p_local_eqn(const unsigned& n) const
    {
      // Get the local equation number associated with the n-th pressure dof
      return this->nodal_local_eqn(Pconv[n], p_nodal_index_nst());
    } // End of p_local_eqn
 
 
    /// Access function for the pressure values at local pressure
    /// node n_p (const version)
    double p_nst(const unsigned& n_p) const
    {
      // Get the nodal value associated with the n_p-th pressure dof
      return this->nodal_value(Pconv[n_p], this->p_nodal_index_nst());
    } // End of p_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 the pressure value of the n_p-th pressure dof at time-level t
      return this->nodal_value(t, Pconv[n_p], this->p_nodal_index_nst());
    } // End of p_nst
 
 
    /// Return number of pressure values
    unsigned npres_nst() const
    {
      // There are 2^{DIM+1} pressure dofs where DIM is the spatial dimension
      // (rememebering that these are space-time elements)
      return static_cast<unsigned>(pow(2.0, static_cast<int>(DIM + 1)));
    } // End of npres_nst
 
 
    /// 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)
    {
      // Pin the pressure dof
      this->node_pt(Pconv[p_dof])->pin(this->p_nodal_index_nst());
 
      // Now set its value
      this->node_pt(Pconv[p_dof])
        ->set_value(this->p_nodal_index_nst(), p_value);
    } // End of fix_pressure
 
 
    /// 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)
    {
      // Create a new Robic BC element and add it to the storage
      this->Pressure_advection_diffusion_robin_element_pt.push_back(
        new FpPressureAdvDiffRobinBCSpaceTimeElement<
          QTaylorHoodSpaceTimeElement<DIM>>(this, face_index));
    } // End of build_fp_press_adv_diff_robin_bc_element
 
 
    /// 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)
    {
      // Call the base class implementation
      SpaceTimeNavierStokesEquations<DIM>::output(outfile);
    } // End of output
 
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      // Call the base class implementation
      SpaceTimeNavierStokesEquations<DIM>::output(outfile, nplot);
    } // End of output
 
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt)
    {
      // Call the base class implementation
      SpaceTimeNavierStokesEquations<DIM>::output(file_pt);
    } // End of output
 
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt, const unsigned& nplot)
    {
      // Call the base class implementation
      SpaceTimeNavierStokesEquations<DIM>::output(file_pt, nplot);
    } // End of output
 
    /*
    /// Returns the number of "DOF types" that degrees of freedom
    /// in this element are sub-divided into: Velocity and pressure.
    unsigned ndof_types() const
    {
     // There are DIM velocity components and 1 pressure component
     return DIM+1;
    } // End of ndof_types
 
 
    /// 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 QTaylorHoodSpaceTimeElement<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;
 
    // The test function derivatives are equal to the shape function derivatives
    dtestdx = dpsidx;
 
    // Return the Jacobian
    return J;
  } // End of dshape_and_dtest_eulerian_nst
 
  //==========================================================================
  /// 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 QTaylorHoodSpaceTimeElement<
    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;
 
    // The test function derivatives are equal to the shape function derivatives
    dtestdx = dpsidx;
 
    // Return the Jacobian
    return J;
  } // End of dshape_and_dtest_eulerian_at_knot_nst
 
  //==========================================================================
  /// 2D (in space): Pressure shape and test functions and derivs w.r.t. to
  /// Eulerian coordinates. Return Jacobian of mapping between local and
  /// global coordinates.
  //==========================================================================
  template<>
  inline double QTaylorHoodSpaceTimeElement<2>::dpshape_and_dptest_eulerian_nst(
    const Vector<double>& s,
    Shape& ppsi,
    DShape& dppsidx,
    Shape& ptest,
    DShape& dptestdx) const
  {
    // Local storage for the shape function (x-direction)
    double psi1[2];
 
    // Local storage for the shape function (y-direction)
    double psi2[2];
 
    // Local storage for the shape function (z-direction)
    double psi3[2];
 
    // Local storage for the shape function derivatives (x-direction)
    double dpsi1[2];
 
    // Local storage for the test function derivatives (y-direction)
    double dpsi2[2];
 
    // Local storage for the test function derivatives (z-direction)
    double dpsi3[2];
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[0], psi1);
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[1], psi2);
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[2], psi3);
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::dshape<2>(s[0], dpsi1);
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::dshape<2>(s[1], dpsi2);
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::dshape<2>(s[2], dpsi3);
 
    //--------------------------------------------------------------------
    // Now let's loop over the nodal points in the element with s1 being
    // the "x" coordinate, s2 being the "y" coordinate and s3 being the
    // "z" coordinate:
    //--------------------------------------------------------------------
    // Loop over the points in the z-direction
    for (unsigned i = 0; i < 2; i++)
    {
      // Loop over the points in the y-direction
      for (unsigned j = 0; j < 2; j++)
      {
        // Loop over the points in the x-direction
        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];
 
          // Multiply the appropriate shape and shape function derivatives
          // together
          dppsidx(4 * i + 2 * j + k, 0) = psi3[i] * psi2[j] * dpsi1[k];
 
          // Multiply the appropriate shape and shape function derivatives
          // together
          dppsidx(4 * i + 2 * j + k, 1) = psi3[i] * dpsi2[j] * psi1[k];
 
          // Multiply the appropriate shape and shape function derivatives
          // together
          dppsidx(4 * i + 2 * j + k, 2) = dpsi3[i] * psi2[j] * psi1[k];
        }
      } // for (unsigned j=0;j<2;j++)
    } // for (unsigned i=0;i<2;i++)
 
    // Allocate space for the shape functions
    Shape psi(27);
 
    // Allocate space for the local shape function derivatives
    DShape dpsi(27, 3);
 
    // Get the values of the shape functions and their local derivatives
    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 derivatives w.r.t. to
    // the Eulerian coordinates
    transform_derivatives(inverse_jacobian, dppsidx);
 
    // The test functions are equal to the shape functions
    ptest = ppsi;
 
    // The test function derivatives are equal to the shape function derivatives
    dptestdx = dppsidx;
 
    // Return the determinant of the jacobian
    return det;
  } // End of dpshape_and_dptest_eulerian_nst
 
  //==========================================================================
  /// 2D (in space):
  /// 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 QTaylorHoodSpaceTimeElement<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
    for (unsigned i = 0; i < 27; i++)
    {
      // The test functions are the same as the shape functions
      test[i] = psi[i];
 
      // Loop over the spatial derivatives
      for (unsigned k = 0; k < 3; k++)
      {
        // Set the test function derivatives to the shape function derivatives
        dtestdx(i, k) = dpsidx(i, k);
 
        // Loop over the dimensions
        for (unsigned p = 0; p < 3; p++)
        {
          // Loop over test functions
          for (unsigned q = 0; q < 27; q++)
          {
            // Set the test function derivatives to the shape function
            // derivatives
            d_dtestdx_dX(p, q, i, k) = d_dpsidx_dX(p, q, i, k);
          }
        } // for (unsigned p=0;p<3;p++)
      } // for (unsigned k=0;k<3;k++)
    } // for (unsigned i=0;i<27;i++)
 
    // Return the jacobian
    return J;
  } // End of dshape_and_dtest_eulerian_at_knot_nst
 
  //==========================================================================
  /// 2D (in space):
  /// Pressure shape functions
  //==========================================================================
  template<>
  inline void QTaylorHoodSpaceTimeElement<2>::pshape_nst(
    const Vector<double>& s, Shape& psi) const
  {
    // Local storage for the shape function value (in the x-direction)
    double psi1[2];
 
    // Local storage for the shape function value (in the y-direction)
    double psi2[2];
 
    // Local storage for the shape function value (in the z-direction)
    double psi3[2];
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[0], psi1);
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[1], psi2);
 
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[2], psi3);
 
    //--------------------------------------------------------------------
    // Now let's loop over the nodal points in the element with s1 being
    // the "x" coordinate, s2 being the "y" coordinate and s3 being the
    // "z" coordinate:
    //--------------------------------------------------------------------
    // Loop over the points in the z-direction
    for (unsigned i = 0; i < 2; i++)
    {
      // Loop over the points in the y-direction
      for (unsigned j = 0; j < 2; j++)
      {
        // Loop over the points in the x-direction
        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];
        }
      } // for (unsigned j=0;j<2;j++)
    } // for (unsigned i=0;i<2;i++)
  } // End of pshape_nst
 
 
  //==========================================================================
  /// Pressure shape and test functions
  //==========================================================================
  template<unsigned DIM>
  inline void QTaylorHoodSpaceTimeElement<DIM>::pshape_nst(
    const Vector<double>& s, Shape& psi, Shape& test) const
  {
    // Call the pressure shape functions
    this->pshape_nst(s, psi);
 
    // Set the test functions equal to the shape functions
    test = psi;
  } // End of pshape_nst
 
 
  //=======================================================================
  /// Face geometry of the 2D space-time TaylorHood elements
  //=======================================================================
  template<>
  class FaceGeometry<QTaylorHoodSpaceTimeElement<2>>
    : public virtual QElement<2, 3>
  {
  public:
    /// Constructor; empty
    FaceGeometry() : QElement<2, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of the face geometry of the 2D Taylor Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<FaceGeometry<QTaylorHoodSpaceTimeElement<2>>>
    : public virtual QElement<1, 3>
  {
  public:
    /// Constructor; empty
    FaceGeometry() : QElement<1, 3>() {}
  };
 
  /// /////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////
 
  //==========================================================
  /// Taylor Hood upgraded to become projectable
  //==========================================================
  template<class TAYLOR_HOOD_ELEMENT>
  class ProjectableTaylorHoodSpaceTimeElement
    : public virtual ProjectableElement<TAYLOR_HOOD_ELEMENT>
  {
  public:
    /// Constructor [this was only required explicitly
    /// from gcc 4.5.2 onwards...]
    ProjectableTaylorHoodSpaceTimeElement() {}
 
    /// 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;
 
      // If we're dealing with the velocity dofs
      if (fld < this->dim() - 1)
      {
        // How many nodes in the element?
        unsigned nnod = this->nnode();
 
        // Loop over all nodes
        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));
        }
      }
      // If we're dealing with the pressure dof
      else
      {
        // How many pressure dofs are there?
        // DRAIG: Shouldn't there be more?
        unsigned Pconv_size = this->dim();
 
        // Loop over all vertex nodes
        for (unsigned j = 0; j < Pconv_size; j++)
        {
          // Get the vertex index associated with the j-th pressure dof
          unsigned vertex_index = this->Pconv[j];
 
          // Add the data value associated with the pressure components
          data_values.push_back(
            std::make_pair(this->node_pt(vertex_index), fld));
        }
      } // if (fld<this->dim())
 
      // Return the vector
      return data_values;
    } // End of data_values_of_field
 
 
    /// Number of fields to be projected: dim+1, corresponding to
    /// velocity components and pressure
    unsigned nfields_for_projection()
    {
      // There are dim velocity dofs and 1 pressure dof
      return this->dim();
    } // End of nfields_for_projection
 
 
    /// 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: The count includes the current value!
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
      // If we're dealing with the pressure dof
      if (fld == this->dim())
      {
        // Pressure doesn't have history values
        return this->node_pt(0)->ntstorage();
      }
      // If we're dealing with the velocity dofs
      else
      {
        // The velocity dofs have ntstorage() history values
        return this->node_pt(0)->ntstorage();
      }
    } // End of nhistory_values_for_projection
 
 
    /// Number of positional history values
    /// (Note: count includes current value!)
    unsigned nhistory_values_for_coordinate_projection()
    {
      // Return the number of positional history values
      return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
    } // End of nhistory_values_for_coordinate_projection
 
 
    /// Return the Jacobian of the mapping and the shape functions
    /// of field fld at local coordinate s
    double jacobian_and_shape_of_field(const unsigned& fld,
                                       const Vector<double>& s,
                                       Shape& psi)
    {
      // How many dimensions in this element?
      unsigned n_dim = this->dim();
 
      // How many nodes in this element?
      unsigned n_node = this->nnode();
 
      // If we're on the pressure dof
      if (fld == n_dim)
      {
        // Call the pressure interpolation function
        this->pshape_nst(s, psi);
 
        // Allocate space for the pressure shape function
        Shape psif(n_node);
 
        // Allocate space for the pressure test function
        Shape testf(n_node);
 
        // Allocate space for the pressure shape function derivatives
        DShape dpsifdx(n_node, n_dim);
 
        // Allocate space for the pressure test function derivatives
        DShape dtestfdx(n_node, n_dim);
 
        // Calculate the Jacobian of the mapping
        double J = this->dshape_and_dtest_eulerian_nst(
          s, psif, dpsifdx, testf, dtestfdx);
 
        // Return the Jacobian
        return J;
      }
      // If we're on the velocity components
      else
      {
        // Allocate space for the test functions
        Shape testf(n_node);
 
        // Allocate space for the shape function derivatives
        DShape dpsifdx(n_node, n_dim);
 
        // Allocate space for the test function derivatives
        DShape dtestfdx(n_node, n_dim);
 
        // Calculate the Jacobian of the mapping
        double J =
          this->dshape_and_dtest_eulerian_nst(s, psi, dpsifdx, testf, dtestfdx);
 
        // Return the Jacobian
        return J;
      }
    } // End of jacobian_and_shape_of_field
 
 
    /// Return interpolated field fld at local coordinate s, at time
    /// level t (t=0: present; t>0: history values)
    double get_field(const unsigned& t,
                     const unsigned& fld,
                     const Vector<double>& s)
    {
      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<ProjectableTaylorHoodSpaceTimeElement<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<ProjectableTaylorHoodSpaceTimeElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
} // End of namespace oomph
 
#endif