refineable_spacetime_navier_stokes_elements.h

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// Header file for refineable space-time Navier Stokes elements
#ifndef OOMPH_REFINEABLE_SPACE_TIME_NAVIER_STOKES_ELEMENTS_HEADER
#define OOMPH_REFINEABLE_SPACE_TIME_NAVIER_STOKES_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// Oomph-lib headers
#include "generic/refineable_quad_element.h"
#include "generic/refineable_brick_element.h"
#include "generic/hp_refineable_elements.h"
#include "generic/error_estimator.h"
#include "spacetime_navier_stokes_elements.h"
 
namespace oomph
{
  //======================================================================
  /// 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 RefineableFpPressureAdvDiffRobinBCSpaceTimeElement
    : public virtual FpPressureAdvDiffRobinBCSpaceTimeElement<ELEMENT>
  {
  public:
    /// Constructor, which takes a "bulk" element and the value of the
    /// index and its limit
    RefineableFpPressureAdvDiffRobinBCSpaceTimeElement(
      FiniteElement* const& element_pt, const int& face_index)
      : FaceGeometry<ELEMENT>(),
        FaceElement(),
        FpPressureAdvDiffRobinBCSpaceTimeElement<ELEMENT>(
          element_pt, face_index, true)
    {
    }
 
    /// This function returns the residuals for the traction
    /// function. If construct_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& construct_jacobian_flag);
  };
 
 
  //======================================================================
  /// Get residuals and Jacobian of Robin boundary conditions in pressure
  /// advection diffusion problem in Fp preconditoner. Refineable version.
  //======================================================================
  template<class ELEMENT>
  void RefineableFpPressureAdvDiffRobinBCSpaceTimeElement<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 just yet!",
                        OOMPH_CURRENT_FUNCTION,
                        OOMPH_EXCEPTION_LOCATION);
 
    // Pointers to first hang info object
    HangInfo* hang_info_pt = 0;
 
    // Pointers to second hang info object
    HangInfo* hang_info2_pt = 0;
 
    // 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();
 
    // Which nodal value represents the pressure? (Negative if pressure
    // is not based on nodal interpolation).
    int p_index = bulk_el_pt->p_nodal_index_nst();
 
    // Local array of booleans that are true if the l-th pressure value is
    // hanging (avoid repeated virtual function calls)
    bool pressure_dof_is_hanging[n_pres];
 
    // If the pressure is stored at a node
    if (p_index >= 0)
    {
      // Read out whether the pressure is hanging
      for (unsigned l = 0; l < n_pres; ++l)
      {
        // Check the hang status of the pressure node
        pressure_dof_is_hanging[l] =
          bulk_el_pt->pressure_node_pt(l)->is_hanging(p_index);
      }
    }
    // Otherwise the pressure is not stored at a node and so cannot hang
    else
    {
      // Create an output stream
      std::ostringstream error_message_stream;
 
      // Create an error message
      error_message_stream << "Pressure advection diffusion does not work "
                           << "in this case!" << std::endl;
 
      // Throw an error
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
 
      // Loop over the pressure dofs
      for (unsigned l = 0; l < n_pres; ++l)
      {
        // DRAIG: This makes no sense...
        pressure_dof_is_hanging[l] = false;
      }
    } // if (p_index>=0)
 
    // Get the Reynolds number from the bulk element
    double re = bulk_el_pt->re();
 
    // Set up memory for pressure shape and test functions
    Shape psip(n_pres), testp(n_pres);
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Get the integral weight
      double w = this->integral_pt()->weight(ipt);
 
      // Assign values of local coordinate in FaceElement
      for (unsigned i = 0; i < my_dim; i++)
        s[i] = this->integral_pt()->knot(ipt, i);
 
      // Find corresponding coordinate in the the bulk element
      s_bulk = this->local_coordinate_in_bulk(s);
 
      /// Get outer unit normal
      this->outer_unit_normal(ipt, unit_normal);
 
      // Get velocity in bulk element
      bulk_el_pt->interpolated_u_nst(s_bulk, velocity);
 
      // Get normal component of veloc
      double flux = 0.0;
      for (unsigned i = 0; i < my_dim + 1; i++)
      {
        flux += velocity[i] * unit_normal[i];
      }
 
      // Modify bc: If outflow (flux>0) apply Neumann condition instead
      if (flux > 0.0) flux = 0.0;
 
      // Get pressure
      double interpolated_press = bulk_el_pt->interpolated_p_nst(s_bulk);
 
      // Call the pressure shape and test functions in bulk element
      bulk_el_pt->pshape_nst(s_bulk, psip, testp);
 
      // Find the Jacobian of the mapping within the FaceElement
      double J = this->J_eulerian(s);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
 
      // Number of master nodes and storage for the weight of the shape function
      unsigned n_master = 1;
      double hang_weight = 1.0;
 
      // Loop over the pressure shape functions
      for (unsigned l = 0; l < n_pres; l++)
      {
        // If the pressure dof is hanging
        if (pressure_dof_is_hanging[l])
        {
          // Pressure dof is hanging so it must be nodal-based
          // Get the hang info object
          hang_info_pt = bulk_el_pt->pressure_node_pt(l)->hanging_pt(p_index);
 
          // Get the number of master nodes from the pressure node
          n_master = hang_info_pt->nmaster();
        }
        // Otherwise the node is its own master
        else
        {
          n_master = 1;
        }
 
        // Loop over the master nodes
        for (unsigned m = 0; m < n_master; m++)
        {
          // Get the number of the unknown
          // If the pressure dof is hanging
          if (pressure_dof_is_hanging[l])
          {
            // Get the local equation from the master node
            local_eqn = bulk_el_pt->local_hang_eqn(
              hang_info_pt->master_node_pt(m), p_index);
            // Get the weight for the node
            hang_weight = hang_info_pt->master_weight(m);
          }
          else
          {
            local_eqn = bulk_el_pt->p_local_eqn(l);
            hang_weight = 1.0;
          }
 
          // If the equation is not pinned
          if (local_eqn >= 0)
          {
            residuals[local_eqn] -=
              re * flux * interpolated_press * testp[l] * W * hang_weight;
 
            // Jacobian too?
            if (flag)
            {
              // Number of master nodes and weights
              unsigned n_master2 = 1;
              double hang_weight2 = 1.0;
 
              // Loop over the pressure shape functions
              for (unsigned l2 = 0; l2 < n_pres; l2++)
              {
                // If the pressure dof is hanging
                if (pressure_dof_is_hanging[l2])
                {
                  hang_info2_pt =
                    bulk_el_pt->pressure_node_pt(l2)->hanging_pt(p_index);
                  // Pressure dof is hanging so it must be nodal-based
                  // Get the number of master nodes from the pressure node
                  n_master2 = hang_info2_pt->nmaster();
                }
                // Otherwise the node is its own master
                else
                {
                  n_master2 = 1;
                }
 
                // Loop over the master nodes
                for (unsigned m2 = 0; m2 < n_master2; m2++)
                {
                  // Get the number of the unknown
                  // If the pressure dof is hanging
                  if (pressure_dof_is_hanging[l2])
                  {
                    // Get the unknown from the master node
                    local_unknown = bulk_el_pt->local_hang_eqn(
                      hang_info2_pt->master_node_pt(m2), p_index);
                    // Get the weight from the hanging object
                    hang_weight2 = hang_info2_pt->master_weight(m2);
                  }
                  else
                  {
                    local_unknown = bulk_el_pt->p_local_eqn(l2);
                    hang_weight2 = 1.0;
                  }
 
                  // If the unknown is not pinned
                  if (local_unknown >= 0)
                  {
                    jacobian(local_eqn, local_unknown) -=
                      (re * flux * psip[l2] * testp[l] * W * hang_weight *
                       hang_weight2);
                  }
                } // for (unsigned m2=0;m2<n_master2;m2++)
              } // for (unsigned l2=0;l2<n_pres;l2++)
            } // if (flag)
          } // if (local_eqn>=0)
        } // for (unsigned m=0;m<n_master;m++)
      } // 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
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
  //======================================================================
  /// Refineable version of the Navier-Stokes equations
  //======================================================================
  template<unsigned DIM>
  class RefineableSpaceTimeNavierStokesEquations
    : public virtual SpaceTimeNavierStokesEquations<DIM>,
      public virtual RefineableElement,
      public virtual ElementWithZ2ErrorEstimator
  {
  protected:
    /// Unpin all pressure dofs in the element
    virtual void unpin_elemental_pressure_dofs() = 0;
 
    /// Pin unused nodal pressure dofs (empty by default, because
    /// by default pressure dofs are not associated with nodes)
    virtual void pin_elemental_redundant_nodal_pressure_dofs() {}
 
  public:
    /// Constructor
    RefineableSpaceTimeNavierStokesEquations()
      : SpaceTimeNavierStokesEquations<DIM>(),
        RefineableElement(),
        ElementWithZ2ErrorEstimator()
    {
    }
 
 
    ///  Loop over all elements in Vector (which typically contains
    /// all the elements in a fluid mesh) and pin the nodal pressure degrees
    /// of freedom that are not being used. Function uses
    /// the member function
    /// - \c RefineableSpaceTimeNavierStokesEquations::
    ///      pin_elemental_redundant_nodal_pressure_dofs()
    /// .
    /// which is empty by default and should be implemented for
    /// elements with nodal pressure degrees of freedom
    /// (e.g. for refineable Taylor-Hood.)
    static void pin_redundant_nodal_pressures(
      const Vector<GeneralisedElement*>& element_pt)
    {
      // How many elements are there?
      unsigned n_element = element_pt.size();
 
      // Loop over all elements
      for (unsigned e = 0; e < n_element; e++)
      {
        // Call the function that pins the unused nodal pressure data
        dynamic_cast<RefineableSpaceTimeNavierStokesEquations<DIM>*>(
          element_pt[e])
          ->pin_elemental_redundant_nodal_pressure_dofs();
      }
    } // End of pin_redundant_nodal_pressures
 
 
    /// Unpin all pressure dofs in elements listed in vector.
    static void unpin_all_pressure_dofs(
      const Vector<GeneralisedElement*>& element_pt)
    {
      // How many elements are there?
      unsigned n_element = element_pt.size();
 
      // Loop over all elements
      for (unsigned e = 0; e < n_element; e++)
      {
        // Unpin the pressure dofs in the e-th element
        dynamic_cast<RefineableSpaceTimeNavierStokesEquations<DIM>*>(
          element_pt[e])
          ->unpin_elemental_pressure_dofs();
      }
    } // End of unpin_all_pressure_dofs
 
 
    /// Pointer to n_p-th pressure node (Default: NULL,
    /// indicating that pressure is not based on nodal interpolation).
    virtual Node* pressure_node_pt(const unsigned& n_p)
    {
      // Return a null pointer
      return NULL;
    } // End of pressure_node_pt
 
 
    /// 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 'flux' terms for Z2 error estimation
    unsigned num_Z2_flux_terms()
    {
      // DIM diagonal strain rates, DIM(DIM-1)/2 off diagonal rates (plus
      // DIM velocity time-derivatives)
      return (DIM + (DIM * (DIM - 1)) / 2) + DIM;
    } // End of num_Z2_flux_terms
 
 
    /// Get 'flux' for Z2 error recovery: Upper triangular entries
    /// in strain rate tensor.
    void get_Z2_flux(const Vector<double>& s, Vector<double>& flux)
    {
#ifdef PARANOID
      // Calculate the number of entries there should be
      unsigned num_entries = (DIM + (DIM * (DIM - 1)) / 2) + DIM;
 
      // Check if the flux vector has the correct size
      if (flux.size() < num_entries)
      {
        std::ostringstream error_message;
        error_message << "The flux vector has the wrong number of entries, "
                      << flux.size() << ", whereas it should be at least "
                      << num_entries << std::endl;
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Allocate space for the strain-rate
      DenseMatrix<double> strainrate(DIM, DIM, 0.0);
 
      // Get strain rate matrix
      this->strain_rate(s, strainrate);
 
      // Pack into flux Vector
      unsigned icount = 0;
 
      // Loop over the diagonal entries
      for (unsigned i = 0; i < DIM; i++)
      {
        // Add the next strain rate entry to the flux vector
        flux[icount] = strainrate(i, i);
 
        // Increment the counter
        icount++;
      }
 
      // Loop over the rows of the matrix
      for (unsigned i = 0; i < DIM; i++)
      {
        // Loop over the lower triangular columns
        for (unsigned j = i + 1; j < DIM; j++)
        {
          // Add the next strain rate entry to the flux vector
          flux[icount] = strainrate(i, j);
 
          // Increment the counter
          icount++;
        }
      } // for (unsigned i=0;i<DIM;i++)
 
      // The number of nodes in the element
      unsigned n_node = this->nnode();
 
      // Set up memory for the shape and test functions
      Shape psif(n_node);
 
      // Set up memory for the derivatives of the shape and test functions
      DShape dpsifdx(n_node, DIM + 1);
 
      // Call the derivatives of the shape and test functions
      dshape_eulerian(s, psif, dpsifdx);
 
      // Loop over velocity components
      for (unsigned j = 0; j < DIM; j++)
      {
        // Find the index at which the variable is stored
        unsigned u_nodal_index = this->u_index_nst(j);
 
        // Loop over nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // Add the time-derivative contribution from the l-th node
          flux[icount] += this->nodal_value(l, u_nodal_index) * dpsifdx(l, DIM);
        }
 
        // Increment the counter
        icount++;
      } // for (unsigned j=0;j<DIM;j++)
    } // End of get_Z2_flux
 
 
    ///  Further build, pass the pointers down to the sons
    void further_build()
    {
      // Find the father element
      RefineableSpaceTimeNavierStokesEquations<DIM>* cast_father_element_pt =
        dynamic_cast<RefineableSpaceTimeNavierStokesEquations<DIM>*>(
          this->father_element_pt());
 
      // Set the viscosity ratio pointer
      this->Viscosity_Ratio_pt = cast_father_element_pt->viscosity_ratio_pt();
 
      // Set the density ratio pointer
      this->Density_Ratio_pt = cast_father_element_pt->density_ratio_pt();
 
      // Set pointer to global Reynolds number
      this->Re_pt = cast_father_element_pt->re_pt();
 
      // Set pointer to global Reynolds number x Strouhal number (=Womersley)
      this->St_pt = cast_father_element_pt->st_pt();
 
      // The Strouhal number (which is a proxy for the period here) is not
      // stored as external data
      this->Strouhal_is_stored_as_external_data =
        cast_father_element_pt->is_strouhal_stored_as_external_data();
 
      // If we're storing the Strouhal number as external data
      if (this->Strouhal_is_stored_as_external_data)
      {
        // The index of the external data (which contains the st)
        unsigned data_index = 0;
 
        // Get the external data pointer from the father and store it
        this->store_strouhal_as_external_data(
          cast_father_element_pt->external_data_pt(data_index));
      }
 
      // Set pointer to global Reynolds number x inverse Froude number
      this->ReInvFr_pt = cast_father_element_pt->re_invfr_pt();
 
      // Set pointer to global gravity Vector
      this->G_pt = cast_father_element_pt->g_pt();
 
      // Set pointer to body force function
      this->Body_force_fct_pt = cast_father_element_pt->body_force_fct_pt();
 
      // Set pointer to volumetric source function
      this->Source_fct_pt = cast_father_element_pt->source_fct_pt();
 
      // Set the ALE flag
      this->ALE_is_disabled = cast_father_element_pt->ALE_is_disabled;
    } // End of further_build
 
 
    /// 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.
    /// Overload the non-refineable version to take account of hanging node
    /// information
    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 in the element
      unsigned n_node = this->nnode();
      // Local shape function
      Shape psi(n_node);
      // Find values of shape function at the given local coordinate
      this->shape(s, psi);
 
      // Find the index at which the velocity component is stored
      const unsigned u_nodal_index = this->u_index_nst(i);
 
      // Storage for hang info pointer
      HangInfo* hang_info_pt = 0;
      // Storage for global equation
      int global_eqn = 0;
 
      // Find the number of dofs associated with interpolated u
      unsigned n_u_dof = 0;
      for (unsigned l = 0; l < n_node; l++)
      {
        unsigned n_master = 1;
 
        // Local bool (is the node hanging)
        bool is_node_hanging = this->node_pt(l)->is_hanging();
 
        // If the node is hanging, get the number of master nodes
        if (is_node_hanging)
        {
          hang_info_pt = this->node_pt(l)->hanging_pt();
          n_master = hang_info_pt->nmaster();
        }
        // Otherwise there is just one master node, the node itself
        else
        {
          n_master = 1;
        }
 
        // Loop over the master nodes
        for (unsigned m = 0; m < n_master; m++)
        {
          // Get the equation number
          if (is_node_hanging)
          {
            // Get the equation number from the master node
            global_eqn =
              hang_info_pt->master_node_pt(m)->eqn_number(u_nodal_index);
          }
          else
          {
            // Global equation number
            global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
          }
 
          // If it's positive add to the count
          if (global_eqn >= 0)
          {
            ++n_u_dof;
          }
        }
      }
 
      // Now resize the storage schemes
      du_ddata.resize(n_u_dof, 0.0);
      global_eqn_number.resize(n_u_dof, 0);
 
      // Loop over the nodes again and set the derivatives
      unsigned count = 0;
 
      // Loop over the nodes in the element
      for (unsigned l = 0; l < n_node; l++)
      {
        // Initialise the number of master nodes to one
        unsigned n_master = 1;
 
        // Initialise the hang weight to one
        double hang_weight = 1.0;
 
        // Local bool (is the node hanging)
        bool is_node_hanging = this->node_pt(l)->is_hanging();
 
        // If the node is hanging, get the number of master nodes
        if (is_node_hanging)
        {
          // Get the HangInfo pointer associated with the l-th node
          hang_info_pt = this->node_pt(l)->hanging_pt();
 
          // How many master nodes does this node have?
          n_master = hang_info_pt->nmaster();
        }
        // Otherwise there is just one master node, the node itself
        else
        {
          // Set n_master to one
          n_master = 1;
        }
 
        // Loop over the master nodes
        for (unsigned m = 0; m < n_master; m++)
        {
          // If the node is hanging get weight from master node
          if (is_node_hanging)
          {
            // Get the hang weight from the master node
            hang_weight = hang_info_pt->master_weight(m);
          }
          else
          {
            // Node contributes with full weight
            hang_weight = 1.0;
          }
 
          // Get the equation number
          if (is_node_hanging)
          {
            // Get the equation number from the master node
            global_eqn =
              hang_info_pt->master_node_pt(m)->eqn_number(u_nodal_index);
          }
          else
          {
            // Get the global equation number
            global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
          }
 
          // If it's a proper degree of freedom
          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] * hang_weight;
 
            // Increase the counter
            ++count;
          }
        } // for (unsigned m=0;m<n_master;m++)
      } // for (unsigned l=0;l<n_node;l++)
    } // End of dinterpolated_u_nst_ddata
 
  protected:
    /// Add the elements contribution to elemental residual vector
    /// and/or Jacobian matrix.
    ///           compute_jacobian_flag=0: compute residual vector only
    ///           compute_jacobian_flag=1: compute both
    void fill_in_generic_residual_contribution_nst(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      const unsigned& compute_jacobian_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.
    void fill_in_generic_pressure_advection_diffusion_contribution_nst(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& compute_jacobian_flag);
 
 
    /// 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);
  };
 
 
  //======================================================================
  /// Refineable version of Taylor Hood elements. These classes
  /// can be written in total generality.
  //======================================================================
  template<unsigned DIM>
  class RefineableQTaylorHoodSpaceTimeElement
    : public QTaylorHoodSpaceTimeElement<DIM>,
      public virtual RefineableSpaceTimeNavierStokesEquations<DIM>,
      public virtual RefineableQElement<DIM + 1>
  {
  public:
    /// Constructor
    RefineableQTaylorHoodSpaceTimeElement()
      : RefineableElement(),
        RefineableSpaceTimeNavierStokesEquations<DIM>(),
        RefineableQElement<DIM + 1>(),
        QTaylorHoodSpaceTimeElement<DIM>()
    {
    }
 
 
    /// Number of values required at local node n. In order to simplify
    /// matters, we allocate storage for pressure variables at all the nodes
    /// and then pin those that are not used.
    unsigned required_nvalue(const unsigned& n) const
    {
      // There are DIM velocity components and 1 pressure component
      return DIM + 1;
    } // End of required_nvalue
 
 
    /// Number of continuously interpolated values
    unsigned ncont_interpolated_values() const
    {
      // There are DIM velocities and 1 pressure component
      return DIM + 1;
    } // End of ncont_interpolated_values
 
 
    /// Rebuild from sons: empty
    void rebuild_from_sons(Mesh*& mesh_pt) {}
 
 
    /// Order of recovery shape functions for Z2 error estimation:
    /// Same order as shape functions.
    unsigned nrecovery_order()
    {
      // Using quadratic interpolation
      return 2;
    } // End of nrecovery_order
 
 
    /// Number of vertex nodes in the element
    unsigned nvertex_node() const
    {
      // Call the base class implementation of the function
      return QTaylorHoodSpaceTimeElement<DIM>::nvertex_node();
    } // End of nvertex_node
 
 
    /// Pointer to the j-th vertex node in the element
    Node* vertex_node_pt(const unsigned& j) const
    {
      // Call the base class implementation of the function
      return QTaylorHoodSpaceTimeElement<DIM>::vertex_node_pt(j);
    } // End of vertex_node_pt
 
 
    /// Get the function value u in Vector.
    /// Note: Given the generality of the interface (this function is usually
    /// called from black-box documentation or interpolation routines), the
    /// values Vector sets its own size in here.
    void get_interpolated_values(const Vector<double>& s,
                                 Vector<double>& values)
    {
      // Set size of Vector: u,v,p and initialise to zero
      values.resize(DIM + 1, 0.0);
 
      // Calculate velocities: values[0],...
      for (unsigned i = 0; i < DIM; i++)
      {
        // Calculate the i-th velocity component at local coordinates s
        values[i] = this->interpolated_u_nst(s, i);
      }
 
      // Calculate the pressure field at local coordinates s
      values[DIM] = this->interpolated_p_nst(s);
    } // End of get_interpolated_values
 
 
    /// Get the function value u in Vector.
    /// Note: Given the generality of the interface (this function
    /// is usually called from black-box documentation or interpolation
    /// routines), the values Vector sets its own size in here.
    void get_interpolated_values(const unsigned& t,
                                 const Vector<double>& s,
                                 Vector<double>& values)
    {
      // The only time this makes sense (if you can even say that...)
      if (t == 0)
      {
        // Call the other function
        get_interpolated_values(s, values);
      }
      else
      {
        // Create an output stream
        std::ostringstream error_message_stream;
 
        // Create an error message
        error_message_stream << "History values don't make sense in "
                             << "space-time elements!" << std::endl;
 
        // Throw an error message
        throw OomphLibError(error_message_stream.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
    } // End of get_interpolated_values
 
 
    /// Perform additional hanging node procedures for variables
    /// that are not interpolated by all nodes. The pressures are stored
    /// at the p_nodal_index_nst-th location in each node
    void further_setup_hanging_nodes()
    {
      // Set up the hang info for the pressure nodes
      this->setup_hang_for_value(this->p_nodal_index_nst());
    } // End of further_setup_hanging_nodes
 
 
    /// Pointer to n_p-th pressure node
    Node* pressure_node_pt(const unsigned& n_p)
    {
      // Return a pointer to the n_p-th pressure node
      return this->node_pt(this->Pconv[n_p]);
    } // End of pressure node_pt
 
 
    /// The velocities are isoparametric and so the "nodes" interpolating
    /// the velocities are the geometric nodes. The pressure "nodes" are a
    /// subset of the nodes, so when value_id==DIM, the n-th pressure
    /// node is returned.
    Node* interpolating_node_pt(const unsigned& n, const int& value_id)
 
    {
      // The only different nodes are the pressure nodes
      if (value_id == DIM)
      {
        // Return a pointer to the n-th pressure node
        return this->pressure_node_pt(n);
      }
      // The other variables are interpolated via the usual nodes
      else
      {
        // Return a pointer to the n-th regular node
        return this->node_pt(n);
      }
    } // End of interpolating_node_pt
 
 
    /// The pressure nodes are the corner nodes, so when n_value==DIM,
    /// the fraction is the same as the 1D node number, 0 or 1.
    double local_one_d_fraction_of_interpolating_node(const unsigned& n_1d,
                                                      const unsigned& i,
                                                      const int& value_id)
    {
      // If we're dealing with a pressure node
      if (value_id == DIM)
      {
        // The pressure nodes are just located on the boundaries at 0 or 1
        return double(n_1d);
      }
      // Otherwise we're dealing with a velocity node
      else
      {
        // The velocity nodes are the same as the geometric ones
        return this->local_one_d_fraction_of_node(n_1d, i);
      }
    } // End of local_one_d_fraction_of_interpolating_node
 
 
    /// The velocity nodes are the same as the geometric nodes. The
    /// pressure nodes must be calculated by using the same methods as
    /// the geometric nodes, but by recalling that there are only two pressure
    /// nodes per edge.
    Node* get_interpolating_node_at_local_coordinate(const Vector<double>& s,
                                                     const int& value_id)
    {
      // If we are calculating pressure nodes
      if (value_id == DIM)
      {
        // Storage for the index of the pressure node
        unsigned total_index = 0;
 
        // The number of nodes along each 1D edge is 2.
        unsigned nnode_1d = 2;
 
        // Storage for the index along each boundary
        Vector<int> index(DIM + 1);
 
        // Loop over the coordinate directions
        for (unsigned i = 0; i < DIM + 1; i++)
        {
          // If we are at the lower limit, the index is zero
          if (s[i] == -1.0)
          {
            // We're at the first node
            index[i] = 0;
          }
          // If we are at the upper limit, the index is the number of nodes
          // minus 1
          else if (s[i] == 1.0)
          {
            // We're on the last node
            index[i] = nnode_1d - 1;
          }
          // Otherwise, we have to calculate the index in general
          else
          {
            // For uniformly spaced nodes this should produce an integer when
            // s[i] is associated with a nodal location
            double float_index = 0.5 * (1.0 + s[i]) * (nnode_1d - 1);
 
            // Take the integer part of the float_index value
            index[i] = int(float_index);
 
            // What is the excess. This should be safe because the taking the
            // integer part rounds down
            double excess = float_index - index[i];
 
            // If the excess is bigger than our tolerance there is no node
            if ((excess > FiniteElement::Node_location_tolerance) &&
                ((1.0 - excess) > FiniteElement::Node_location_tolerance))
            {
              // As there is no node, return null
              return 0;
            }
          } // if (s[i]==-1.0)
 
          // Construct the general pressure index from the components.
          total_index +=
            index[i] * static_cast<unsigned>(pow(static_cast<float>(nnode_1d),
                                                 static_cast<int>(i)));
        } // for (unsigned i=0;i<DIM;i++)
 
        // If we've got here we have a node, so let's return a pointer to it
        return this->pressure_node_pt(total_index);
      }
      // Otherwise velocity nodes are the same as pressure nodes
      else
      {
        // Call the regular helper function to find the right node
        return this->get_node_at_local_coordinate(s);
      } // if (value_id==DIM)
    } // End of get_interpolating_node_at_local_coordinate
 
 
    /// The number of 1D pressure nodes is 2, the number of 1D velocity
    /// nodes is the same as the number of 1D geometric nodes.
    unsigned ninterpolating_node_1d(const int& value_id)
    {
      // If we're dealing with the pressure nodes
      if (value_id == DIM)
      {
        // Using linear interpolation for the pressure
        return 2;
      }
      // If we're dealing with the velocity nodes
      else
      {
        // Every node is a velocity interpolating node
        return this->nnode_1d();
      }
    } // End of ninterpolating_node_1d
 
 
    /// The number of pressure nodes is 2^DIM. The number of
    /// velocity nodes is the same as the number of geometric nodes.
    unsigned ninterpolating_node(const int& value_id)
    {
      // If we want the pressure basis functions
      if (value_id == DIM)
      {
        // There are 2^{DIM+1} pressure dofs (also interpolating in time)
        return static_cast<unsigned>(pow(2.0, static_cast<int>(DIM + 1)));
      }
      // If we want the velocity basis functions
      else
      {
        // Every node is a velocity interpolating node
        return this->nnode();
      }
    } // End if ninterpolating_node
 
 
    /// The basis interpolating the pressure is given by pshape().
    /// / The basis interpolating the velocity is shape().
    void interpolating_basis(const Vector<double>& s,
                             Shape& psi,
                             const int& value_id) const
    {
      // If we want the pressure basis functions
      if (value_id == DIM)
      {
        // Call the pressure shape function
        return this->pshape_nst(s, psi);
      }
      // If we want the velocity basis functions
      else
      {
        // Call the velocity shape function
        return this->shape(s, psi);
      }
    } // End of interpolating_basis
 
 
    /// Build FaceElements that apply the Robin boundary condition
    /// to the pressure advection diffusion problem required by
    /// Fp preconditioner
    void build_fp_press_adv_diff_robin_bc_element(const unsigned& face_index)
    {
      this->Pressure_advection_diffusion_robin_element_pt.push_back(
        new RefineableFpPressureAdvDiffRobinBCSpaceTimeElement<
          RefineableQTaylorHoodSpaceTimeElement<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.
    /// (Overloads non-refineable version and takes hanging nodes
    /// into account)
    void identify_load_data(
      std::set<std::pair<Data*, unsigned>>& paired_load_data)
    {
      // Allocate space for the velocity component indices
      unsigned u_index[DIM];
 
      // Loop over the velocity components
      for (unsigned i = 0; i < DIM; i++)
      {
        // Get the nodal indices at which the velocities are stored
        u_index[i] = this->u_index_nst(i);
      }
 
      // Get the number of nodes in the element
      unsigned n_node = this->nnode();
 
      // Loop over the nodes
      for (unsigned n = 0; n < n_node; n++)
      {
        // Pointer to current node
        Node* nod_pt = this->node_pt(n);
 
        // Check if it's hanging:
        if (nod_pt->is_hanging())
        {
          // It's hanging -- get number of master nodes
          unsigned nmaster = nod_pt->hanging_pt()->nmaster();
 
          // Loop over master nodes
          for (unsigned j = 0; j < nmaster; j++)
          {
            // Create a pointer to the j-th master node
            Node* master_nod_pt = nod_pt->hanging_pt()->master_node_pt(j);
 
            // Loop over the velocity components and add a pointer to their data
            // and indices to the vectors
            for (unsigned i = 0; i < DIM; i++)
            {
              // Create a pair and add it to the storage
              paired_load_data.insert(
                std::make_pair(master_nod_pt, u_index[i]));
            }
          } // for (unsigned j=0;j<nmaster;j++)
        }
        // If the node is not hanging
        else
        {
          // Loop over the velocity components and add pointer to their data
          // and indices to the vectors
          for (unsigned i = 0; i < DIM; i++)
          {
            // Create a pair and add it to the storage
            paired_load_data.insert(
              std::make_pair(this->node_pt(n), u_index[i]));
          }
        } // if (nod_pt->is_hanging())
      } // for (unsigned n=0;n<n_node;n++)
 
      // Get the nodal index at which the pressure is stored
      int p_index = this->p_nodal_index_nst();
 
      // Get the number of pressure degrees of freedom
      unsigned n_pres = this->npres_nst();
 
      // Loop over the pressure data
      for (unsigned l = 0; l < n_pres; l++)
      {
        // Get the pointer to the l-th pressure node
        Node* pres_node_pt = this->pressure_node_pt(l);
 
        // Check if the pressure dof is hanging
        if (pres_node_pt->is_hanging(p_index))
        {
          // Get the pointer to the hang info object (pressure is stored
          // as p_index-th nodal dof).
          HangInfo* hang_info_pt = pres_node_pt->hanging_pt(p_index);
 
          // Get number of pressure master nodes (pressure is stored
          unsigned nmaster = hang_info_pt->nmaster();
 
          // Loop over pressure master nodes
          for (unsigned m = 0; m < nmaster; m++)
          {
            // The p_index-th entry in each nodal data is the pressure, which
            // affects the traction
            paired_load_data.insert(
              std::make_pair(hang_info_pt->master_node_pt(m), p_index));
          }
        }
        // If the pressure dof is not hanging
        else
        {
          // The p_index-th entry in each nodal data is the pressure, which
          // affects the traction
          paired_load_data.insert(std::make_pair(pres_node_pt, p_index));
        } // if (pres_node_pt->is_hanging(p_index))
      } // for (unsigned l=0;l<n_pres;l++)
    } // End of identify_load_data
 
  private:
    /// Unpin all pressure dofs
    void unpin_elemental_pressure_dofs()
    {
      // Find the index at which the pressure is stored
      int p_index = this->p_nodal_index_nst();
 
      // Get the number of nodes in the element
      unsigned n_node = this->nnode();
 
      // Loop over nodes
      for (unsigned n = 0; n < n_node; n++)
      {
        // Unpin the p_index-th dof at node n
        this->node_pt(n)->unpin(p_index);
      }
    } // End of unpin_elemental_pressure_dofs
 
 
    /// Pin all nodal pressure dofs that are not required
    void pin_elemental_redundant_nodal_pressure_dofs()
    {
      // Find the pressure index
      int p_index = this->p_nodal_index_nst();
 
      // Loop over all nodes
      unsigned n_node = this->nnode();
 
      // Loop over all nodes and pin all the nodal pressures
      for (unsigned n = 0; n < n_node; n++)
      {
        // Pin the p_index-th dof at node n
        this->node_pt(n)->pin(p_index);
      }
 
      // Loop over all actual pressure nodes and unpin if they're not hanging
      unsigned n_pres = this->npres_nst();
 
      // Loop over all pressure nodes
      for (unsigned l = 0; l < n_pres; l++)
      {
        // Get a pointer to the l-th pressure node
        Node* nod_pt = this->pressure_node_pt(l);
 
        // If the node isn't hanging
        if (!nod_pt->is_hanging(p_index))
        {
          // Unpin the p_index-th dof at this node
          nod_pt->unpin(p_index);
        }
      } // for (unsigned l=0;l<n_pres;l++)
    } // End of pin_elemental_redundant_nodal_pressure_dofs
  };
 
 
  //=======================================================================
  /// Face geometry of the RefineableQTaylorHoodSpaceTimeElements is the
  /// same as the Face geometry of the QTaylorHoodSpaceTimeElements.
  //=======================================================================
  template<unsigned DIM>
  class FaceGeometry<RefineableQTaylorHoodSpaceTimeElement<DIM>>
    : public virtual FaceGeometry<QTaylorHoodSpaceTimeElement<DIM>>
  {
  public:
    /// Constructor; empty
    FaceGeometry() : FaceGeometry<QTaylorHoodSpaceTimeElement<DIM>>() {}
  };
 
 
  //=======================================================================
  /// Face geometry of the face geometry of the
  /// RefineableQTaylorHoodSpaceTimeElements is the same as the Face geometry
  /// of the Face geometry of QTaylorHoodSpaceTimeElements.
  //=======================================================================
  template<unsigned DIM>
  class FaceGeometry<FaceGeometry<RefineableQTaylorHoodSpaceTimeElement<DIM>>>
    : public virtual FaceGeometry<
        FaceGeometry<QTaylorHoodSpaceTimeElement<DIM>>>
  {
  public:
    /// Constructor; empty
    FaceGeometry()
      : FaceGeometry<FaceGeometry<QTaylorHoodSpaceTimeElement<DIM>>>()
    {
    }
  };
 
} // End of namespace oomph
 
#endif