helmholtz_geometric_multigrid.h

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// 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) 2006-2022 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.
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// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
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// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
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// LIC//====================================================================
// Include guards
#ifndef OOMPH_HELMHOLTZ_GEOMETRIC_MULTIGRID_HEADER
#define OOMPH_HELMHOLTZ_GEOMETRIC_MULTIGRID_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <config.h>
#endif
 
// Oomph-lib headers
#include "generic/problem.h"
#include "generic/matrices.h"
#include "generic/preconditioner.h"
 
// Include the complex smoother
#include "complex_smoother.h"
 
// Namespace extension
namespace oomph
{
  //======================================================================
  //======================================================================
  class HelmholtzMGProblem : public virtual Problem
  {
  public:
    HelmholtzMGProblem() {}
 
    virtual ~HelmholtzMGProblem() {}
 
    virtual HelmholtzMGProblem* make_new_problem() = 0;
 
    virtual TreeBasedRefineableMeshBase* mg_bulk_mesh_pt() = 0;
 
  }; // End of HelmholtzMGProblem class
 
 
 
 
  //======================================================================
  // MG solver class
  //======================================================================
  template<unsigned DIM>
  class HelmholtzMGPreconditioner : public BlockPreconditioner<CRDoubleMatrix>
  {
  public:
    typedef HelmholtzSmoother* (*PreSmootherFactoryFctPt)();
 
    typedef HelmholtzSmoother* (*PostSmootherFactoryFctPt)();
 
    void set_pre_smoother_factory_function(
      PreSmootherFactoryFctPt pre_smoother_fn)
    {
      // Assign the function pointer
      Pre_smoother_factory_function_pt = pre_smoother_fn;
    }
 
    void set_post_smoother_factory_function(
      PostSmootherFactoryFctPt post_smoother_fn)
    {
      // Assign the function pointer
      Post_smoother_factory_function_pt = post_smoother_fn;
    }
 
    HelmholtzMGPreconditioner(HelmholtzMGProblem* mg_problem_pt)
      : BlockPreconditioner<CRDoubleMatrix>(),
        Pre_smoother_factory_function_pt(0),
        Post_smoother_factory_function_pt(0),
        Mg_problem_pt(mg_problem_pt),
        Tolerance(1.0e-09),
        Npre_smooth(2),
        Npost_smooth(2),
        Nvcycle(1),
        Doc_time(true),
        Suppress_v_cycle_output(false),
        Suppress_all_output(false),
        Has_been_setup(false),
        Has_been_solved(false),
        Stream_pt(0),
        Alpha_shift(0.0)
    {
      // Set the pointer to the finest level as the first
      // entry in Mg_hierarchy_pt
      Mg_hierarchy_pt.push_back(Mg_problem_pt);
    } // End of HelmholtzMGPreconditioner
 
    ~HelmholtzMGPreconditioner()
    {
      // Run the function written to clean up the memory
      clean_up_memory();
    } // End of ~HelmholtzMGPreconditioner
 
    void clean_up_memory()
    {
      // We only need to destroy data if the solver has been set up and
      // the data hasn't already been cleared
      if (Has_been_setup)
      {
        // Loop over all of the levels in the hierarchy
        for (unsigned i = 0; i < Nlevel - 1; i++)
        {
          // Delete the pre-smoother associated with this level
          delete Pre_smoothers_storage_pt[i];
 
          // Make it a null pointer
          Pre_smoothers_storage_pt[i] = 0;
 
          // Delete the post-smoother associated with this level
          delete Post_smoothers_storage_pt[i];
 
          // Make it a null pointer
          Post_smoothers_storage_pt[i] = 0;
 
          // Loop over the real and imaginary parts of the system matrix
          // associated with the i-th level
          for (unsigned j = 0; j < 2; j++)
          {
            // Delete the real/imaginary part of the system matrix
            delete Mg_matrices_storage_pt[i][j];
 
            // Make it a null pointer
            Mg_matrices_storage_pt[i][j] = 0;
          }
        }
 
        // Delete the expanded matrix associated with the problem on the
        // coarsest level
        delete Coarsest_matrix_mg_pt;
 
        // Make it a null pointer
        Coarsest_matrix_mg_pt = 0;
 
        // Loop over all but the coarsest of the levels in the hierarchy
        for (unsigned i = 0; i < Nlevel - 1; i++)
        {
          // Delete the interpolation matrix associated with the i-th level
          delete Interpolation_matrices_storage_pt[i];
 
          // Make it a null pointer
          Interpolation_matrices_storage_pt[i] = 0;
 
          // Delete the restriction matrix associated with the i-th level
          delete Restriction_matrices_storage_pt[i];
 
          // Make it a null pointer
          Restriction_matrices_storage_pt[i] = 0;
        }
 
        // Everything has been deleted now so we need to indicate that the
        // solver is not set up
        Has_been_setup = false;
      }
    } // End of clean_up_memory
 
    double& tolerance()
    {
      // Return the variable, Tolerance
      return Tolerance;
    } // End of tolerance
 
    double& alpha_shift()
    {
      // Return the alpha shift value
      return Alpha_shift;
    } // End of alpha_shift
 
    void disable_doc_time()
    {
      // Set the value of Doc_time to false
      Doc_time = false;
    } // End of disable_doc_time
 
    void disable_v_cycle_output()
    {
      // Set the value of Doc_time to false
      Doc_time = false;
 
      // Enable the suppression of output from the V-cycle
      Suppress_v_cycle_output = true;
    } // End of disable_v_cycle_output
 
    void disable_output()
    {
      // Set the value of Doc_time to false
      Doc_time = false;
 
      // Enable the suppression of output from the V-cycle
      Suppress_v_cycle_output = true;
 
      // Enable the suppression of everything
      Suppress_all_output = true;
 
      // Store the output stream pointer
      Stream_pt = oomph_info.stream_pt();
 
      // Now set the oomph_info stream pointer to the null stream to
      // disable all possible output
      oomph_info.stream_pt() = &oomph_nullstream;
    } // End of disable_output
 
    void enable_doc_time()
    {
      // Set the value of Doc_time to true
      Doc_time = true;
    } // End of enable_doc_time
 
    void enable_v_cycle_output()
    {
      // Enable time documentation
      Doc_time = true;
 
      // Enable output from the MG solver
      Suppress_v_cycle_output = false;
    } // End of enable_v_cycle_output
 
    void enable_output()
    {
      // Enable time documentation
      Doc_time = true;
 
      // Enable output from everything during the full setup of the solver
      Suppress_all_output = false;
 
      // Enable output from the MG solver
      Suppress_v_cycle_output = false;
    } // End of enable_output
 
    void disable_smoother_and_superlu_doc_time()
    {
      // Loop over all levels of the hierarchy
      for (unsigned i = 0; i < Nlevel - 1; i++)
      {
        // Disable time documentation on each level (for each pre-smoother)
        Pre_smoothers_storage_pt[i]->disable_doc_time();
 
        // Disable time documentation on each level (for each post-smoother)
        Post_smoothers_storage_pt[i]->disable_doc_time();
      }
 
      // We only need a direct solve on the coarsest level so this is the
      // only place we need to silence SuperLU
      Coarsest_matrix_mg_pt->linear_solver_pt()->disable_doc_time();
    } // End of disable_smoother_and_superlu_doc_time
 
    unsigned& npost_smooth()
    {
      // Return the number of post-smoothing iterations to be done on each
      // level of the hierarchy
      return Npost_smooth;
    } // End of npost_smooth
 
    unsigned& npre_smooth()
    {
      // Return the number of pre-smoothing iterations to be done on each
      // level of the hierarchy
      return Npre_smooth;
    } // End of npre_smooth
 
    void pre_smooth(const unsigned& level)
    {
      // Run pre-smoother 'max_iter' times
      Pre_smoothers_storage_pt[level]->complex_smoother_solve(
        Rhs_mg_vectors_storage[level], X_mg_vectors_storage[level]);
 
      // Calculate the residual vector on this level
      residual_norm(level);
    } // End of pre_smooth
 
    void post_smooth(const unsigned& level)
    {
      // Run post-smoother 'max_iter' times
      Post_smoothers_storage_pt[level]->complex_smoother_solve(
        Rhs_mg_vectors_storage[level], X_mg_vectors_storage[level]);
 
      // Calculate the residual vector on this level
      residual_norm(level);
    } // End of post_smooth
 
    double residual_norm(const unsigned& level)
    {
      // Loop over the real and imaginary part of the residual vector on
      // the given level
      for (unsigned j = 0; j < 2; j++)
      {
        // And zero the entries of residual
        Residual_mg_vectors_storage[level][j].initialise(0.0);
      }
 
      // Return the norm of the residual
      return residual_norm(level, Residual_mg_vectors_storage[level]);
    } // End of residual_norm
 
    double residual_norm(const unsigned& level, Vector<DoubleVector>& residual);
 
    void setup_coarsest_level_structures();
 
    // The result is placed in X_mg
    void direct_solve()
    {
      // Concatenate the vectors in X_mg into one vector, coarsest_x_mg
      DoubleVectorHelpers::concatenate(X_mg_vectors_storage[Nlevel - 1],
                                       Coarsest_x_mg);
 
      // Concatenate the vectors in Rhs_mg into one vector, coarsest_rhs_mg
      DoubleVectorHelpers::concatenate(Rhs_mg_vectors_storage[Nlevel - 1],
                                       Coarsest_rhs_mg);
 
      // Get solution by direct solve:
      Coarsest_matrix_mg_pt->solve(Coarsest_rhs_mg, Coarsest_x_mg);
 
      // Split the solution vector into a vector of DoubleVectors and store it
      DoubleVectorHelpers::split(Coarsest_x_mg,
                                 X_mg_vectors_storage[Nlevel - 1]);
    } // End of direct_solve
 
    void interpolation_matrix_set(const unsigned& level,
                                  double* value,
                                  int* col_index,
                                  int* row_st,
                                  unsigned& ncol,
                                  unsigned& nnz)
    {
      // Dynamically allocate the interpolation matrix
      Interpolation_matrices_storage_pt[level] = new CRDoubleMatrix;
 
      // Build the matrix
      Interpolation_matrices_storage_pt[level]->build_without_copy(
        ncol, nnz, value, col_index, row_st);
 
    } // End of interpolation_matrix_set
 
    void interpolation_matrix_set(const unsigned& level,
                                  Vector<double>& value,
                                  Vector<int>& col_index,
                                  Vector<int>& row_st,
                                  unsigned& ncol,
                                  unsigned& nrow)
    {
      // Dynamically allocate the interpolation matrix
      Interpolation_matrices_storage_pt[level] = new CRDoubleMatrix;
 
      // Make the distribution pointer
      LinearAlgebraDistribution* dist_pt = new LinearAlgebraDistribution(
        Mg_hierarchy_pt[level]->communicator_pt(), nrow, false);
 
#ifdef PARANOID
#ifdef OOMPH_HAS_MPI
      // Set up the warning messages
      std::string warning_message =
        "Setup of interpolation matrix distribution ";
      warning_message += "has not been tested with MPI.";
 
      // If we're not running the code in serial
      if (dist_pt->communicator_pt()->nproc() > 1)
      {
        // Throw a warning
        OomphLibWarning(
          warning_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
#endif
 
      // Build the matrix itself
      Interpolation_matrices_storage_pt[level]->build(
        dist_pt, ncol, value, col_index, row_st);
 
      // Delete the newly created distribution pointer
      delete dist_pt;
 
      // Make it a null pointer
      dist_pt = 0;
 
    } // End of interpolation_matrix_set
 
    void set_restriction_matrices_as_interpolation_transposes()
    {
      for (unsigned i = 0; i < Nlevel - 1; i++)
      {
        // Dynamically allocate the restriction matrix
        Restriction_matrices_storage_pt[i] = new CRDoubleMatrix;
 
        // Set the restriction matrix to be the transpose of the
        // interpolation matrix
        Interpolation_matrices_storage_pt[i]->get_matrix_transpose(
          Restriction_matrices_storage_pt[i]);
      }
    } // End of set_restriction_matrices_as_interpolation_transposes
 
    void restrict_residual(const unsigned& level);
 
    void interpolate_and_correct(const unsigned& level);
 
    void level_up_local_coord_of_node(const int& son_type, Vector<double>& s);
 
    void setup_interpolation_matrices();
 
    void setup_interpolation_matrices_unstructured();
 
    void setup_transfer_matrices();
 
    void full_setup();
 
    void preconditioner_solve(const DoubleVector& r, DoubleVector& z)
    {
      // Split up the RHS vector into DoubleVectors, whose entries are
      // arranged to match the matrix blocks and assign it
      this->get_block_vectors(r, Rhs_mg_vectors_storage[0]);
 
      // Split up the solution vector into DoubleVectors, whose entries are
      // arranged to match the matrix blocks and assign it
      this->get_block_vectors(z, X_mg_vectors_storage[0]);
 
      // Run the MG method and assign the solution to z
      this->mg_solve(X_mg_vectors_storage[0]);
 
      // Copy solution in block vectors block_z back to z
      this->return_block_vectors(X_mg_vectors_storage[0], z);
 
      // Only output if the V-cycle output isn't suppressed
      if (!(this->Suppress_v_cycle_output))
      {
        // Notify user that the hierarchy of levels is complete
        oomph_info << "\n=========="
                   << "Multigrid Preconditioner Solve Complete"
                   << "=========" << std::endl;
      }
 
      // Only enable and assign the stream pointer again if we originally
      // suppressed everything otherwise it won't be set yet
      if (this->Suppress_all_output)
      {
        // Now enable the stream pointer again
        oomph_info.stream_pt() = this->Stream_pt;
      }
    } // End of preconditioner_solve
 
    unsigned iterations() const
    {
      // Return the number of V-cycles which have been done
      return V_cycle_counter;
    } // End of iterations
 
    using Preconditioner::setup;
 
  private:
    PreSmootherFactoryFctPt Pre_smoother_factory_function_pt;
 
    PostSmootherFactoryFctPt Post_smoother_factory_function_pt;
 
    void mg_solve(Vector<DoubleVector>& result);
 
    void block_preconditioner_self_test();
 
    void setup()
    {
      // Run the full setup
      this->full_setup();
 
      // Only enable and assign the stream pointer again if we originally
      // suppressed everything otherwise it won't be set yet
      if (this->Suppress_all_output)
      {
        // Now enable the stream pointer again
        oomph_info.stream_pt() = this->Stream_pt;
      }
    } // End of setup
 
    void setup_mg_hierarchy();
 
    void setup_mg_structures();
 
    void maximum_edge_width(const unsigned& level, double& h);
 
    void setup_smoothers();
 
    HelmholtzMGProblem* Mg_problem_pt;
 
    Vector<HelmholtzMGProblem*> Mg_hierarchy_pt;
 
    Vector<Vector<CRDoubleMatrix*>> Mg_matrices_storage_pt;
 
    CRDoubleMatrix* Coarsest_matrix_mg_pt;
 
    DoubleVector Coarsest_x_mg;
 
    DoubleVector Coarsest_rhs_mg;
 
    Vector<CRDoubleMatrix*> Interpolation_matrices_storage_pt;
 
    Vector<CRDoubleMatrix*> Restriction_matrices_storage_pt;
 
    Vector<Vector<DoubleVector>> X_mg_vectors_storage;
 
    Vector<Vector<DoubleVector>> Rhs_mg_vectors_storage;
 
    Vector<Vector<DoubleVector>> Residual_mg_vectors_storage;
 
    Vector<HelmholtzSmoother*> Pre_smoothers_storage_pt;
 
    Vector<HelmholtzSmoother*> Post_smoothers_storage_pt;
 
    Vector<double> Max_edge_width;
 
    double Wavenumber;
 
    double Tolerance;
 
    unsigned Nlevel;
 
    unsigned Npre_smooth;
 
    unsigned Npost_smooth;
 
    unsigned Nvcycle;
 
    unsigned V_cycle_counter;
 
    bool Doc_time;
 
    bool Suppress_v_cycle_output;
 
    bool Suppress_all_output;
 
    bool Has_been_setup;
 
    bool Has_been_solved;
 
    std::ostream* Stream_pt;
 
    double Alpha_shift;
  };
 
  //========================================================================
  //========================================================================
  template<unsigned DIM>
  double HelmholtzMGPreconditioner<DIM>::residual_norm(
    const unsigned& level, Vector<DoubleVector>& residual)
  {
    // Number of rows in each block vector
    unsigned n_rows = X_mg_vectors_storage[level][0].nrow();
 
#ifdef PARANOID
    // PARANOID check - if the residual vector doesn't have length 2 it cannot
    // be used here since we need two vectors corresponding to the real and
    // imaginary part of the residual
    if (residual.size() != 2)
    {
      // Throw an error if the residual vector doesn't have the correct length
      throw OomphLibError("This residual vector must have length 2. ",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
    if (residual[0].nrow() != residual[1].nrow())
    {
      // Create an output stream
      std::ostringstream error_message_stream;
 
      // Store the error message
      error_message_stream << "Residual[0] has length: " << residual[0].nrow()
                           << "\nResidual[1] has length: " << residual[1].nrow()
                           << "\nThis method requires that the constituent "
                           << "DoubleVectors in residual have the same length. "
                           << std::endl;
 
      // Throw an error
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Loop over the block vectors
    for (unsigned i = 0; i < 2; i++)
    {
      // Start by setting the distribution of the residuals vector if
      // it is not set up
      if (!residual[i].distribution_built())
      {
        // Set up distribution
        LinearAlgebraDistribution dist(
          Mg_hierarchy_pt[level]->communicator_pt(), n_rows, false);
 
        // Build the distribution
        residual[i].build(&dist, 0.0);
      }
      // Otherwise just zero the entries of residual
      else
      {
#ifdef PARANOID
        // PARANOID check - if the residuals are distributed then this method
        // cannot be used, a distributed residuals can only be assembled by
        // get_jacobian(...) for CRDoubleMatrices
        if (residual[i].distributed())
        {
          throw OomphLibError(
            "This method can only assemble a non-distributed residuals vector ",
            OOMPH_CURRENT_FUNCTION,
            OOMPH_EXCEPTION_LOCATION);
        }
#endif
 
        // And zero the entries of residual
        residual[i].initialise(0.0);
      }
    }
 
    // Store the pointer to the distribution of Matrix_real_pt (the same as
    // Matrix_imag_pt presumably so we only need to use one)
    LinearAlgebraDistribution* dist_pt =
      Mg_matrices_storage_pt[level][0]->distribution_pt();
 
    // Create 4 temporary vectors to store the various matrix-vector products
    // required. The appropriate combination has been appended to the name of
    // the vector:
    // Vector to store A_r*x_r:
    DoubleVector temp_vec_rr(dist_pt, 0.0);
 
    // Vector to store A_r*x_c:
    DoubleVector temp_vec_rc(dist_pt, 0.0);
 
    // Vector to store A_c*x_r:
    DoubleVector temp_vec_cr(dist_pt, 0.0);
 
    // Vector to store A_c*x_c:
    DoubleVector temp_vec_cc(dist_pt, 0.0);
 
    // We can't delete the distribution pointer because the Jacobian on the
    // finest level is using it but we can make it a null pointer
    dist_pt = 0;
 
    // Calculate the different combinations of A*x (or A*e depending on the
    // level in the heirarchy) in the complex case:
    // A_r*x_r:
    Mg_matrices_storage_pt[level][0]->multiply(X_mg_vectors_storage[level][0],
                                               temp_vec_rr);
 
    // A_r*x_c:
    Mg_matrices_storage_pt[level][0]->multiply(X_mg_vectors_storage[level][1],
                                               temp_vec_rc);
 
    // A_c*x_r:
    Mg_matrices_storage_pt[level][1]->multiply(X_mg_vectors_storage[level][0],
                                               temp_vec_cr);
 
    // A_c*x_c:
    Mg_matrices_storage_pt[level][1]->multiply(X_mg_vectors_storage[level][1],
                                               temp_vec_cc);
 
    // Real part of the residual:
    residual[0] = Rhs_mg_vectors_storage[level][0];
    residual[0] -= temp_vec_rr;
    residual[0] += temp_vec_cc;
 
    // Imaginary part of the residual:
    residual[1] = Rhs_mg_vectors_storage[level][1];
    residual[1] -= temp_vec_rc;
    residual[1] -= temp_vec_cr;
 
    // Get the residual norm of the real part of the residual vector
    double norm_r = residual[0].norm();
 
    // Get the residual norm of the imaginary part of the residual vector
    double norm_c = residual[1].norm();
 
    // Return the true norm of the residual vector which depends on the
    // norm of the real part and the norm of the imaginary part
    return sqrt(norm_r * norm_r + norm_c * norm_c);
  }
 
  //=====================================================================
  //=====================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::block_preconditioner_self_test()
  {
    // Start clock
    clock_t t_bl_start = clock();
 
#ifdef PARANOID
    if (Mg_hierarchy_pt[0]->mesh_pt() == 0)
    {
      std::stringstream err;
      err << "Please set pointer to mesh using set_bulk_helmholtz_mesh(...).\n";
      throw OomphLibError(
        err.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // The preconditioner works with one mesh; set it! Since we only use the
    // block preconditioner on the finest level, we use the mesh from that level
    this->set_nmesh(1);
 
    // Elements in actual pml layer are trivially wrapped versions of
    // their bulk counterparts. Technically they are different
    // elements so we have to allow different element types
    bool allow_different_element_types_in_mesh = true;
    this->set_mesh(
      0, Mg_problem_pt->mesh_pt(), allow_different_element_types_in_mesh);
 
#ifdef PARANOID
    // This preconditioner only works for 2 dof types
    unsigned n_dof_types = this->ndof_types();
    if (n_dof_types != 2)
    {
      std::stringstream tmp;
      tmp << "This preconditioner only works for problems with 2 dof types\n"
          << "Yours has " << n_dof_types;
      throw OomphLibError(
        tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Set up the generic block look up scheme
    this->block_setup();
 
    // Extract the number of blocks.
    unsigned nblock_types = this->nblock_types();
#ifdef PARANOID
    if (nblock_types != 2)
    {
      std::stringstream tmp;
      tmp << "There are supposed to be two block types.\n"
          << "Yours has " << nblock_types;
      throw OomphLibError(
        tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // This is how the 2x2 block matrices are extracted. We retain the sanity
    // check (i.e. the diagonals are the same and the off-diagonals are
    // negatives of each other in PARANOID mode. Otherwise we only extract 2
    // matrices
    DenseMatrix<CRDoubleMatrix*> block_pt(nblock_types, nblock_types, 0);
    for (unsigned i = 0; i < nblock_types; i++)
    {
      for (unsigned j = 0; j < nblock_types; j++)
      {
        // we want...
        block_pt(i, j) = new CRDoubleMatrix;
        this->get_block(i, j, *block_pt(i, j));
      }
    }
 
    // Check that diagonal matrices are the same
    //------------------------------------------
    {
      unsigned nnz1 = block_pt(0, 0)->nnz();
      unsigned nnz2 = block_pt(1, 1)->nnz();
      if (nnz1 != nnz2)
      {
        std::stringstream tmp;
        tmp << "nnz of diagonal blocks doesn't match: " << nnz1
            << " != " << nnz2 << std::endl;
        throw OomphLibError(
          tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
      unsigned nrow1 = block_pt(0, 0)->nrow();
      unsigned nrow2 = block_pt(1, 1)->nrow();
      if (nrow1 != nrow2)
      {
        std::stringstream tmp;
        tmp << "nrow of diagonal blocks doesn't match: " << nrow1
            << " != " << nrow2 << std::endl;
        throw OomphLibError(
          tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
 
      // Check entries
      bool fail = false;
      double tol = 1.0e-15;
      std::stringstream tmp;
 
      // Check row starts
      for (unsigned i = 0; i < nrow1 + 1; i++)
      {
        if (block_pt(0, 0)->row_start()[i] != block_pt(1, 1)->row_start()[i])
        {
          fail = true;
          tmp << "Row starts of diag matrices don't match for row " << i
              << " : " << block_pt(0, 0)->row_start()[i] << " "
              << block_pt(1, 1)->row_start()[i] << " " << std::endl;
        }
      }
 
      // Check values and column indices
      for (unsigned i = 0; i < nnz1; i++)
      {
        if (block_pt(0, 0)->column_index()[i] !=
            block_pt(1, 1)->column_index()[i])
        {
          fail = true;
          tmp << "Column  of diag matrices indices don't match for entry " << i
              << " : " << block_pt(0, 0)->column_index()[i] << " "
              << block_pt(1, 1)->column_index()[i] << " " << std::endl;
        }
 
        if (fabs(block_pt(0, 0)->value()[i] - block_pt(1, 1)->value()[i]) > tol)
        {
          fail = true;
          tmp << "Values of diag matrices don't match for entry " << i << " : "
              << block_pt(0, 0)->value()[i] << " " << block_pt(1, 1)->value()[i]
              << " " << std::endl;
        }
      }
      if (fail)
      {
        throw OomphLibError(
          tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
    }
 
    // Check that off-diagonal matrices are negatives
    //-----------------------------------------------
    {
      unsigned nnz1 = block_pt(0, 1)->nnz();
      unsigned nnz2 = block_pt(1, 0)->nnz();
      if (nnz1 != nnz2)
      {
        std::stringstream tmp;
        tmp << "nnz of diagonal blocks doesn't match: " << nnz1
            << " != " << nnz2 << std::endl;
        throw OomphLibError(
          tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
      unsigned nrow1 = block_pt(0, 1)->nrow();
      unsigned nrow2 = block_pt(1, 0)->nrow();
      if (nrow1 != nrow2)
      {
        std::stringstream tmp;
        tmp << "nrow of off-diagonal blocks doesn't match: " << nrow1
            << " != " << nrow2 << std::endl;
        throw OomphLibError(
          tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
 
 
      // Check entries
      bool fail = false;
      double tol = 1.0e-15;
      std::stringstream tmp;
 
      // Check row starts
      for (unsigned i = 0; i < nrow1 + 1; i++)
      {
        if (block_pt(0, 1)->row_start()[i] != block_pt(1, 0)->row_start()[i])
        {
          fail = true;
          tmp << "Row starts of off-diag matrices don't match for row " << i
              << " : " << block_pt(0, 1)->row_start()[i] << " "
              << block_pt(1, 0)->row_start()[i] << " " << std::endl;
        }
      }
 
      // Check values and column indices
      for (unsigned i = 0; i < nnz1; i++)
      {
        if (block_pt(0, 1)->column_index()[i] !=
            block_pt(1, 0)->column_index()[i])
        {
          fail = true;
          tmp << "Column indices of off-diag matrices don't match for entry "
              << i << " : " << block_pt(0, 1)->column_index()[i] << " "
              << block_pt(1, 0)->column_index()[i] << " " << std::endl;
        }
 
        if (fabs(block_pt(0, 1)->value()[i] + block_pt(1, 0)->value()[i]) > tol)
        {
          fail = true;
          tmp << "Values of off-diag matrices aren't negatives of "
              << "each other for entry " << i << " : "
              << block_pt(0, 1)->value()[i] << " " << block_pt(1, 0)->value()[i]
              << " " << std::endl;
        }
      }
      if (fail)
      {
        throw OomphLibError(
          tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
    }
 
    // Clean up
    for (unsigned i = 0; i < nblock_types; i++)
    {
      for (unsigned j = 0; j < nblock_types; j++)
      {
        delete block_pt(i, j);
        block_pt(i, j) = 0;
      }
    }
 
    // Stop clock
    clock_t t_bl_end = clock();
    double total_setup_time = double(t_bl_end - t_bl_start) / CLOCKS_PER_SEC;
    oomph_info << "CPU time for block preconditioner self-test [sec]: "
               << total_setup_time << "\n"
               << std::endl;
  }
 
  //===================================================================
  //===================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::full_setup()
  {
#ifdef OOMPH_HAS_MPI
    // Make sure that this is running in serial. Can't guarantee it'll
    // work when the problem is distributed over several processors
    if (MPI_Helpers::communicator_pt()->nproc() > 1)
    {
      // Throw a warning
      OomphLibWarning("Can't guarantee the MG solver will work in parallel!",
                      OOMPH_CURRENT_FUNCTION,
                      OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Initialise the timer start variable
    double t_fs_start = 0.0;
 
    // If we're allowed to output
    if (!Suppress_all_output)
    {
      // Start the timer
      t_fs_start = TimingHelpers::timer();
 
      // Notify user that the hierarchy of levels is complete
      oomph_info
        << "\n========Starting Setup of Multigrid Preconditioner========"
        << std::endl;
 
      // Notify user that the hierarchy of levels is complete
      oomph_info
        << "\nStarting the full setup of the Helmholtz multigrid solver."
        << std::endl;
    }
 
#ifdef PARANOID
    // PARANOID check - Make sure the dimension of the solver matches the
    // dimension of the elements used in the problems mesh
    if (dynamic_cast<FiniteElement*>(Mg_problem_pt->mesh_pt()->element_pt(0))
          ->dim() != DIM)
    {
      // Create an error message
      std::string err_strng = "The dimension of the elements used in the mesh ";
      err_strng += "does not match the dimension of the solver.";
 
      // Throw the error
      throw OomphLibError(
        err_strng, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
    // PARANOID check - The elements of the bulk mesh must all be refineable
    // elements otherwise we cannot deal with this
    if (Mg_problem_pt->mg_bulk_mesh_pt() != 0)
    {
      // Find the number of elements in the bulk mesh
      unsigned n_elements = Mg_problem_pt->mg_bulk_mesh_pt()->nelement();
 
      // Loop over the elements in the mesh and ensure that they are
      // all refineable elements
      for (unsigned el_counter = 0; el_counter < n_elements; el_counter++)
      {
        // Upcast global mesh element to a refineable element
        RefineableElement* el_pt = dynamic_cast<RefineableElement*>(
          Mg_problem_pt->mg_bulk_mesh_pt()->element_pt(el_counter));
 
        // Check if the upcast worked or not; if el_pt is a null pointer the
        // element is not refineable
        if (el_pt == 0)
        {
          // Create an output steam
          std::ostringstream error_message_stream;
 
          // Store the error message
          error_message_stream << "Element in bulk mesh could not be upcast to "
                               << "a refineable element." << std::endl;
 
          // Throw an error
          throw OomphLibError(error_message_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      } // for (unsigned el_counter=0;el_counter<n_elements;el_counter++)
    }
    // If the provided bulk mesh pointer is a null pointer
    else
    {
      // Create an output steam
      std::ostringstream error_message_stream;
 
      // Store the error message
      error_message_stream
        << "The provided bulk mesh pointer is a null pointer. " << std::endl;
 
      // Throw an error
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    } // if (Mg_problem_pt->mg_bulk_mesh_pt()!=0)
#endif
 
    // If this is not the first Newton step then we will already have things
    // in storage. If this is the case, delete them
    clean_up_memory();
 
    // Resize the Mg_hierarchy_pt vector
    Mg_hierarchy_pt.resize(1, 0);
 
    // Set the pointer to the finest level as the first entry in Mg_hierarchy_pt
    Mg_hierarchy_pt[0] = Mg_problem_pt;
 
    // Create the hierarchy of levels
    setup_mg_hierarchy();
 
    // Set up the interpolation and restriction matrices
    setup_transfer_matrices();
 
    // Set up the data structures on each level, i.e. the system matrix,
    // LHS and RHS vectors and smoothers
    setup_mg_structures();
 
    // Set up the smoothers on all of the levels
    setup_smoothers();
 
    // Loop over all of the coarser levels
    for (unsigned i = 1; i < Nlevel; i++)
    {
      // Delete the i-th coarse-grid HelmholtzMGProblem object
      delete Mg_hierarchy_pt[i];
 
      // Set it to be a null pointer
      Mg_hierarchy_pt[i] = 0;
    }
 
    // Indicate that the full setup has been completed
    Has_been_setup = true;
 
    // If we're allowed to output to the screen
    if (!Suppress_all_output)
    {
      // Output the time taken to complete the full setup
      double t_fs_end = TimingHelpers::timer();
      double full_setup_time = t_fs_end - t_fs_start;
 
      // Output the CPU time
      oomph_info << "\nCPU time for full setup [sec]: " << full_setup_time
                 << std::endl;
 
      // Notify user that the hierarchy of levels is complete
      oomph_info << "\n==============="
                 << "Multigrid Full Setup Complete"
                 << "==============\n"
                 << std::endl;
    } // if (!Suppress_all_output)
  } // End of full_setup
 
  //===================================================================
  //===================================================================
  // Function to set up the hierachy of levels.
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::setup_mg_hierarchy()
  {
    // Initialise the timer start variable
    double t_m_start = 0.0;
 
    // Notify the user if it is allowed
    if (!Suppress_all_output)
    {
      // Notify user of progress
      oomph_info << "\n==============="
                 << "Creating Multigrid Hierarchy"
                 << "===============\n"
                 << std::endl;
 
      // Start clock
      t_m_start = TimingHelpers::timer();
    }
 
    // Create a bool to indicate whether or not we could create an unrefined
    // copy. This bool will be assigned the value FALSE when the current copy
    // is the last level of the multigrid hierarchy
    bool managed_to_create_unrefined_copy = true;
 
    // Now keep making copies and try to make an unrefined copy of
    // the mesh
    unsigned level = 0;
 
    // Set up all of the levels by making a completely unrefined copy
    // of the problem using the function make_new_problem
    while (managed_to_create_unrefined_copy)
    {
      // Make a new object of the same type as the derived problem
      HelmholtzMGProblem* new_problem_pt = Mg_problem_pt->make_new_problem();
 
      // Do anything that needs to be done before we can refine the mesh
      new_problem_pt->actions_before_adapt();
 
      // To create the next level in the hierarchy we need to create a mesh
      // which matches the refinement of the current problem and then unrefine
      // the mesh. This can alternatively be done using the function
      // refine_base_mesh_as_in_reference_mesh_minus_one which takes a
      // reference mesh to do precisely the above with
      managed_to_create_unrefined_copy =
        new_problem_pt->mg_bulk_mesh_pt()
          ->refine_base_mesh_as_in_reference_mesh_minus_one(
            Mg_hierarchy_pt[level]->mg_bulk_mesh_pt());
 
      // If we were able to unrefine the problem on the current level
      // then add the unrefined problem to a vector of the levels
      if (managed_to_create_unrefined_copy)
      {
        // Another level has been created so increment the level counter
        level++;
 
        // If the documentation of everything has not been suppressed
        // then tell the user we managed to create another level
        if (!Suppress_all_output)
        {
          // Notify user that unrefinement was successful
          oomph_info << "Success! Level " << level
                     << " has been created:" << std::endl;
        }
 
        // Do anything that needs to be done after refinement
        new_problem_pt->actions_after_adapt();
 
        // Do the equation numbering for the new problem
        oomph_info << "\n - Number of equations: "
                   << new_problem_pt->assign_eqn_numbers() << "\n"
                   << std::endl;
 
        // Add the new problem pointer onto the vector of MG levels
        // and increment the value of level by 1
        Mg_hierarchy_pt.push_back(new_problem_pt);
      }
      // If we weren't able to create an unrefined copy
      else
      {
        // Delete the new problem
        delete new_problem_pt;
 
        // Make it a null pointer
        new_problem_pt = 0;
 
        // Assign the number of levels to Nlevel
        Nlevel = Mg_hierarchy_pt.size();
 
        // If we're allowed to document then tell the user we've reached
        // the coarsest level of the hierarchy
        if (!Suppress_all_output)
        {
          // Notify the user
          oomph_info << "Reached the coarsest level! "
                     << "Number of levels: " << Nlevel << std::endl;
        }
      } // if (managed_to_create_unrefined_copy)
    } // while (managed_to_create_unrefined_copy)
 
    //------------------------------------------------------------------
    // Given that we know the number of levels in the hierarchy we can
    // resize the vectors which will store all the information required
    // for our solver:
    //------------------------------------------------------------------
    // Resize the vector storing all of the system matrices
    Mg_matrices_storage_pt.resize(Nlevel);
 
    // Resize the vector storing all of the solution vectors (X_mg)
    X_mg_vectors_storage.resize(Nlevel);
 
    // Resize the vector storing all of the RHS vectors (Rhs_mg)
    Rhs_mg_vectors_storage.resize(Nlevel);
 
    // Resize the vector storing all of the residual vectors
    Residual_mg_vectors_storage.resize(Nlevel);
 
    // Allocate space for the Max_edge_width vector, we only need the value
    // of h on the levels where we use a smoother
    Max_edge_width.resize(Nlevel - 1, 0.0);
 
    // Allocate space for the pre-smoother storage vector (remember, we do
    // not need a smoother on the coarsest level; we use a direct solve there)
    Pre_smoothers_storage_pt.resize(Nlevel - 1, 0);
 
    // Allocate space for the post-smoother storage vector (remember, we do
    // not need a smoother on the coarsest level; we use a direct solve there)
    Post_smoothers_storage_pt.resize(Nlevel - 1, 0);
 
    // Resize the vector storing all of the interpolation matrices
    Interpolation_matrices_storage_pt.resize(Nlevel - 1, 0);
 
    // Resize the vector storing all of the restriction matrices
    Restriction_matrices_storage_pt.resize(Nlevel - 1, 0);
 
    // If we're allowed to output information to the screen
    if (!Suppress_all_output)
    {
      // Stop clock
      double t_m_end = TimingHelpers::timer();
      double total_setup_time = double(t_m_end - t_m_start);
      oomph_info
        << "\nCPU time for creation of hierarchy of MG problems [sec]: "
        << total_setup_time << std::endl;
 
      // Notify user that the hierarchy of levels is complete
      oomph_info << "\n==============="
                 << "Hierarchy Creation Complete"
                 << "================\n"
                 << std::endl;
    }
  } // End of setup_mg_hierarchy
 
  //===================================================================
  //===================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::setup_transfer_matrices()
  {
    // Initialise the timer start variable
    double t_r_start = 0.0;
 
    // Notify the user (if we're allowed)
    if (!Suppress_all_output)
    {
      // Notify user of progress
      oomph_info << "Creating the transfer matrices." << std::endl;
 
      // Start the clock!
      t_r_start = TimingHelpers::timer();
    }
 
    // Using full weighting so use setup_interpolation_matrices.
    // Note: There are two methods to choose from here, the ideal choice is
    // setup_interpolation_matrices() but that requires a refineable mesh base
    if (dynamic_cast<TreeBasedRefineableMeshBase*>(
          Mg_problem_pt->mg_bulk_mesh_pt()))
    {
      setup_interpolation_matrices();
    }
    // If the mesh is unstructured we have to use the locate_zeta function
    // to set up the interpolation matrices
    else
    {
      setup_interpolation_matrices_unstructured();
    }
 
    // Loop over all levels that will be assigned a restriction matrix
    set_restriction_matrices_as_interpolation_transposes();
 
    // If we're allowed
    if (!Suppress_all_output)
    {
      // Stop the clock
      double t_r_end = TimingHelpers::timer();
      double total_G_setup_time = double(t_r_end - t_r_start);
      oomph_info << "\nCPU time for transfer matrices setup [sec]: "
                 << total_G_setup_time << std::endl;
 
      // Notify user that the hierarchy of levels is complete
      oomph_info
        << "\n============Transfer Matrices Setup Complete=============="
        << "\n"
        << std::endl;
    }
  } // End of setup_transfer_matrices function
 
  //===================================================================
  //===================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::setup_mg_structures()
  {
    // Initialise the timer start variable
    double t_m_start = 0.0;
 
    // Start the clock (if we're allowed to time things)
    if (!Suppress_all_output)
    {
      // Start the clock
      t_m_start = TimingHelpers::timer();
    }
 
    // Calculate the wavenumber value:
    //--------------------------------
    // Reset the value of Wavenumber
    Wavenumber = 0.0;
 
    // Upcast the first element in the bulk mesh
    PMLHelmholtzEquations<DIM>* pml_helmholtz_el_pt =
      dynamic_cast<PMLHelmholtzEquations<DIM>*>(
        Mg_problem_pt->mg_bulk_mesh_pt()->element_pt(0));
 
    // Grab the k_squared value from the element pointer and square root.
    // Note, we assume the wavenumber is the same everywhere in the mesh
    // and it is also the same on every level.
    Wavenumber = sqrt(pml_helmholtz_el_pt->k_squared());
 
    // We don't need the pointer anymore so make it a null pointer but don't
    // delete the underlying element data
    pml_helmholtz_el_pt = 0;
 
    // Set up the system matrix and accompanying vectors on each level:
    //-----------------------------------------------------------------
    // Loop over each level and extract the system matrix, solution vector
    // right-hand side vector and residual vector (to store the value of r=b-Ax)
    for (unsigned i = 0; i < Nlevel; i++)
    {
      // If we're allowed to output
      if (!Suppress_all_output)
      {
        // Output the level we're working on
        oomph_info << "Setting up MG structures on level: " << i << "\n"
                   << std::endl;
      }
 
      // Resize the solution and RHS vector
      unsigned n_dof_per_block = Mg_hierarchy_pt[i]->ndof() / 2;
 
      // Create the linear algebra distribution to build the vectors
      LinearAlgebraDistribution* dist_pt = new LinearAlgebraDistribution(
        Mg_hierarchy_pt[i]->communicator_pt(), n_dof_per_block, false);
 
#ifdef PARANOID
#ifdef OOMPH_HAS_MPI
      // Set up the warning messages
      std::string warning_message = "Setup of distribution has not been ";
      warning_message += "tested with MPI.";
 
      // If we're not running the code in serial
      if (dist_pt->communicator_pt()->nproc() > 1)
      {
        // Throw a warning
        OomphLibWarning(
          warning_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
#endif
 
      // Create storage for the i-th level system matrix:
      //-------------------------------------------------
      // Resize the i-th entry of the matrix storage vector to contain two
      // CRDoubleMatrix pointers
      Mg_matrices_storage_pt[i].resize(2, 0);
 
      // Loop over the real and imaginary part
      for (unsigned j = 0; j < 2; j++)
      {
        // Dynamically allocate a new CRDoubleMatrix
        Mg_matrices_storage_pt[i][j] = new CRDoubleMatrix;
      }
 
      // Build the matrix distribution (real part)
      Mg_matrices_storage_pt[i][0]->build(dist_pt);
 
      // Build the matrix distribution (imaginary part)
      Mg_matrices_storage_pt[i][1]->build(dist_pt);
 
      // Create storage for the i-th level solution vector:
      //---------------------------------------------------
      // Resize the i-th level solution vector to contain two DoubleVectors
      X_mg_vectors_storage[i].resize(2);
 
      // Build the approximate solution (real part)
      X_mg_vectors_storage[i][0].build(dist_pt);
 
      // Build the approximate solution (imaginary part)
      X_mg_vectors_storage[i][1].build(dist_pt);
 
      // Create storage for the i-th level RHS vector:
      //----------------------------------------------
      // Resize the i-th level RHS vector to contain two DoubleVectors
      Rhs_mg_vectors_storage[i].resize(2);
 
      // Build the point source function (real part)
      Rhs_mg_vectors_storage[i][0].build(dist_pt);
 
      // Build the point source function (imaginary part)
      Rhs_mg_vectors_storage[i][1].build(dist_pt);
 
      // Create storage for the i-th level residual vector:
      //---------------------------------------------------
      // Resize the i-th level residual vector to contain two DoubleVectors
      Residual_mg_vectors_storage[i].resize(2);
 
      // Build the residual vector, r=b-Ax (real part)
      Residual_mg_vectors_storage[i][0].build(dist_pt);
 
      // Build the residual vector, r=b-Ax (imaginary part)
      Residual_mg_vectors_storage[i][1].build(dist_pt);
 
      // Delete the distribution pointer
      delete dist_pt;
 
      // Make it a null pointer
      dist_pt = 0;
 
      // Compute system matrix on the current level. On the finest level of the
      // hierarchy the system matrix is given by the complex-shifted Laplacian
      // preconditioner. On the subsequent levels the Galerkin approximation
      // is used to give us a coarse-grid representation of the problem
      if (i == 0)
      {
        // Initialise the timer start variable
        double t_jac_start = 0.0;
 
        // If we're allowed to output things
        if (!Suppress_all_output)
        {
          // Start timer for Jacobian setup
          t_jac_start = TimingHelpers::timer();
        }
 
#ifdef PARANOID
        // Make sure the block preconditioner returns the correct sort of matrix
        block_preconditioner_self_test();
#endif
 
        // The preconditioner works with one mesh; set it!
        this->set_nmesh(1);
 
        // Elements in actual pml layer are trivially wrapped versions of their
        // bulk counterparts. Technically they are different elements so we have
        // to allow different element types
        bool allow_different_element_types_in_mesh = true;
        this->set_mesh(0,
                       Mg_hierarchy_pt[0]->mesh_pt(),
                       allow_different_element_types_in_mesh);
 
#ifdef PARANOID
        // Find the number of dof types we're dealing with
        unsigned n_dof_types = this->ndof_types();
 
        // This preconditioner only works for 2 dof types
        if (n_dof_types != 2)
        {
          // Create an error message
          std::stringstream tmp;
          tmp
            << "This preconditioner only works for problems with 2 dof types\n"
            << "Yours has " << n_dof_types;
 
          // Throw an error
          throw OomphLibError(
            tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
        }
#endif
 
        // If we're not using a value of zero for the alpha shift
        if (Alpha_shift != 0.0)
        {
          //------------------------------------------------------------------
          // Set the damping in all of the PML elements to create the complex-
          // shifted Laplacian preconditioner:
          //------------------------------------------------------------------
          // Create a pointer which will contain the value of the shift given
          // by Alpha_shift
          double* alpha_shift_pt = new double(Alpha_shift);
 
          // Calculate the number of elements in the mesh
          unsigned n_element = Mg_hierarchy_pt[0]->mesh_pt()->nelement();
 
          // To grab the static variable used to set the value of alpha we first
          // need to get an element of type PMLHelmholtzEquations (we
          // arbitrarily chose the first element in the mesh)
          PMLHelmholtzEquations<DIM>* el_pt =
            dynamic_cast<PMLHelmholtzEquations<DIM>*>(
              Mg_hierarchy_pt[0]->mesh_pt()->element_pt(0));
 
          // Now grab the pointer from the element
          static double* default_physical_constant_value_pt = el_pt->alpha_pt();
 
          // Loop over all of the elements
          for (unsigned i_el = 0; i_el < n_element; i_el++)
          {
            // Upcast from a GeneralisedElement to a PmlHelmholtzElement
            PMLHelmholtzEquations<DIM>* el_pt =
              dynamic_cast<PMLHelmholtzEquations<DIM>*>(
                Mg_hierarchy_pt[0]->mesh_pt()->element_pt(i_el));
 
            // Make the internal element alpha pointer point to Alpha_shift (the
            // chosen value of the shift to be applied to the problem)
            el_pt->alpha_pt() = alpha_shift_pt;
          }
 
          //---------------------------------------------------------------------
          // We want to solve the problem with the modified Jacobian but the
          // Preconditioner will have a handle to pointer which points to the
          // system matrix. If we wish to use the block preconditioner to
          // extract the appropriate blocks of the matrix we need to assign it.
          // To avoid losing the original system matrix we will create a
          // temporary pointer to it which will be reassigned after we have use
          // the block preconditioner to extract the blocks of the shifted
          // matrix:
          //---------------------------------------------------------------------
          // Create a temporary pointer to the Jacobian
          CRDoubleMatrix* jacobian_pt = this->matrix_pt();
 
          // Create a new CRDoubleMatrix to hold the shifted Jacobian matrix
          CRDoubleMatrix* shifted_jacobian_pt = new CRDoubleMatrix;
 
          // Allocate space for the residuals
          DoubleVector residuals;
 
          // Get the residuals vector and the shifted Jacobian. Note, we do
          // not need to assign the residuals vector; we're simply using
          // MG as a preconditioner
          Mg_hierarchy_pt[0]->get_jacobian(residuals, *shifted_jacobian_pt);
 
          // Replace the current matrix used in Preconditioner by the new matrix
          this->set_matrix_pt(shifted_jacobian_pt);
 
          // Set up the generic block look up scheme
          this->block_setup();
 
          // Extract the number of blocks.
          unsigned nblock_types = this->nblock_types();
 
#ifdef PARANOID
          // PARANOID check - there must only be two block types
          if (nblock_types != 2)
          {
            // Create the error message
            std::stringstream tmp;
            tmp << "There are supposed to be two block types.\nYours has "
                << nblock_types << std::endl;
 
            // Throw an error
            throw OomphLibError(
              tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
          }
#endif
 
          // Store the level
          unsigned level = 0;
 
          // Loop over the rows of the block matrix
          for (unsigned i_row = 0; i_row < nblock_types; i_row++)
          {
            // Fix the column index
            unsigned j_col = 0;
 
            // Extract the required blocks, i.e. the first column
            this->get_block(
              i_row, j_col, *Mg_matrices_storage_pt[level][i_row]);
          }
 
          // The blocks have been extracted from the shifted Jacobian therefore
          // we no longer have any use for it
          delete shifted_jacobian_pt;
 
          // Now the appropriate blocks have been extracted from the shifted
          // Jacobian we reset the matrix pointer in Preconditioner to the
          // original Jacobian so the linear solver isn't affected
          this->set_matrix_pt(jacobian_pt);
 
          //--------------------------------------------------------
          // Reassign the damping factor in all of the PML elements:
          //--------------------------------------------------------
          // Loop over all of the elements
          for (unsigned i_el = 0; i_el < n_element; i_el++)
          {
            // Upcast from a GeneralisedElement to a PmlHelmholtzElement
            PMLHelmholtzEquations<DIM>* el_pt =
              dynamic_cast<PMLHelmholtzEquations<DIM>*>(
                Mg_hierarchy_pt[0]->mesh_pt()->element_pt(i_el));
 
            // Set the value of alpha
            el_pt->alpha_pt() = default_physical_constant_value_pt;
          }
 
          // We've finished using the alpha_shift_pt pointer so delete it
          // as it was dynamically allocated
          delete alpha_shift_pt;
 
          // Make it a null pointer
          alpha_shift_pt = 0;
        }
        // If the value of the shift is zero then we use the original
        // Jacobian matrix
        else
        {
          // The Jacobian has already been provided so now we just need to set
          // up the generic block look up scheme
          this->block_setup();
 
          // Extract the number of blocks.
          unsigned nblock_types = this->nblock_types();
 
#ifdef PARANOID
          // If there are not only two block types we have a problem
          if (nblock_types != 2)
          {
            std::stringstream tmp;
            tmp << "There are supposed to be two block types.\n"
                << "Yours has " << nblock_types;
            throw OomphLibError(
              tmp.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
          }
#endif
 
          // Store the level
          unsigned level = 0;
 
          // Loop over the rows of the block matrix
          for (unsigned i_row = 0; i_row < nblock_types; i_row++)
          {
            // Fix the column index (since we only want the first column)
            unsigned j_col = 0;
 
            // Extract the required blocks
            this->get_block(
              i_row, j_col, *Mg_matrices_storage_pt[level][i_row]);
          }
        } // if (Alpha_shift!=0.0) else
 
        if (!Suppress_all_output)
        {
          // Document the time taken
          double t_jac_end = TimingHelpers::timer();
          double jacobian_setup_time = t_jac_end - t_jac_start;
          oomph_info << " - Time for setup of Jacobian block matrix [sec]: "
                     << jacobian_setup_time << "\n"
                     << std::endl;
        }
      }
      // If we're not dealing with the finest level we're dealing with a
      // coarse-grid problem
      else
      {
        // Initialise the timer start variable
        double t_gal_start = 0.0;
 
        // If we're allowed
        if (!Suppress_all_output)
        {
          // Start timer for Galerkin matrix calculation
          t_gal_start = TimingHelpers::timer();
        }
 
        //---------------------------------------------------------------------
        // The system matrix on the coarser levels must be formed using the
        // Galerkin approximation which we do by calculating the product
        // A^2h = I^2h_h * A^h * I^h_2h, i.e. the coarser version of the
        // finer grid system matrix is formed by multiplying by the (fine grid)
        // restriction matrix from the left and the (fine grid) interpolation
        // matrix from the left. Fortunately, since the restriction and
        // interpolation acts on the real and imaginary parts separately we
        // can calculate the real and imaginary parts of the Galerkin
        // approximation separately.
        //---------------------------------------------------------------------
 
        //----------------------------------------------
        // Real component of the Galerkin approximation:
        //----------------------------------------------
        // First we need to calculate A^h * I^h_2h which we store as A^2h
        Mg_matrices_storage_pt[i - 1][0]->multiply(
          *Interpolation_matrices_storage_pt[i - 1],
          *Mg_matrices_storage_pt[i][0]);
 
        // Now calculate I^2h_h * (A^h * I^h_2h) where the quantity in brackets
        // was just calculated. This updates A^2h to give us the true
        // Galerkin approximation to the finer grid matrix
        Restriction_matrices_storage_pt[i - 1]->multiply(
          *Mg_matrices_storage_pt[i][0], *Mg_matrices_storage_pt[i][0]);
 
        //---------------------------------------------------
        // Imaginary component of the Galerkin approximation:
        //---------------------------------------------------
        // First we need to calculate A^h * I^h_2h which we store as A^2h
        Mg_matrices_storage_pt[i - 1][1]->multiply(
          *Interpolation_matrices_storage_pt[i - 1],
          *Mg_matrices_storage_pt[i][1]);
 
        // Now calculate I^2h_h * (A^h * I^h_2h) where the quantity in brackets
        // was just calculated. This updates A^2h to give us the true
        // Galerkin approximation to the finer grid matrix
        Restriction_matrices_storage_pt[i - 1]->multiply(
          *Mg_matrices_storage_pt[i][1], *Mg_matrices_storage_pt[i][1]);
 
        // If the user did not choose to suppress everything
        if (!Suppress_all_output)
        {
          // End timer for Galerkin matrix calculation
          double t_gal_end = TimingHelpers::timer();
 
          // Calculate setup time
          double galerkin_matrix_calculation_time = t_gal_end - t_gal_start;
 
          // Document the time taken
          oomph_info << " - Time for system block matrix formation using the "
                     << "Galerkin approximation [sec]: "
                     << galerkin_matrix_calculation_time << "\n"
                     << std::endl;
        }
      } // if (i==0) else
 
      // We only
      if (i < Nlevel - 1)
      {
        // Find the maximum edge width, h:
        //--------------------------------
        // Initialise the timer start variable
        double t_para_start = 0.0;
 
        // If we're allowed to output things
        if (!Suppress_all_output)
        {
          // Start timer for parameter calculation on the i-th level
          t_para_start = TimingHelpers::timer();
        }
 
        // Calculate the i-th entry of Wavenumber and Max_edge_width
        maximum_edge_width(i, Max_edge_width[i]);
 
        // If the user did not choose to suppress everything
        if (!Suppress_all_output)
        {
          // End timer for Galerkin matrix calculation
          double t_para_end = TimingHelpers::timer();
 
          // Calculate setup time
          double parameter_calculation_time = t_para_end - t_para_start;
 
          // Document the time taken
          oomph_info << " - Time for maximum edge width calculation [sec]: "
                     << parameter_calculation_time << "\n"
                     << std::endl;
        }
      } // if (i<Nlevel-1)
    } // for (unsigned i=0;i<Nlevel;i++)
 
    // The last system matrix that needs to be setup is the fully expanded
    // version of the system matrix on the coarsest level. This is needed
    // for the direct solve on the coarsest level
    setup_coarsest_level_structures();
 
    // If we're allowed to output
    if (!Suppress_all_output)
    {
      // Stop clock
      double t_m_end = TimingHelpers::timer();
      double total_setup_time = double(t_m_end - t_m_start);
      oomph_info << "CPU time for setup of MG structures [sec]: "
                 << total_setup_time << std::endl;
 
      // Notify user that the hierarchy of levels is complete
      oomph_info << "\n============"
                 << "Multigrid Structures Setup Complete"
                 << "===========\n"
                 << std::endl;
    }
  } // End of setup_mg_structures
 
  //=========================================================================
  //=========================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::setup_coarsest_level_structures()
  {
    // Start timer
    double t_cm_start = TimingHelpers::timer();
 
    //---------------------------------------------------
    // Extract information from the constituent matrices:
    //---------------------------------------------------
 
    // Grab the real and imaginary matrix parts from storage
    CRDoubleMatrix* real_matrix_pt = Mg_matrices_storage_pt[Nlevel - 1][0];
    CRDoubleMatrix* imag_matrix_pt = Mg_matrices_storage_pt[Nlevel - 1][1];
 
    // Number of nonzero entries in A_r
    unsigned nnz_r = real_matrix_pt->nnz();
    unsigned nnz_c = imag_matrix_pt->nnz();
 
    // Calculate the total number of rows (and thus columns) in the real matrix
    unsigned n_rows_r = real_matrix_pt->nrow();
 
    // Acquire access to the value, row_start and column_index arrays from
    // the real and imaginary portions of the full matrix.
    // Real part:
    const double* value_r_pt = real_matrix_pt->value();
    const int* row_start_r_pt = real_matrix_pt->row_start();
    const int* column_index_r_pt = real_matrix_pt->column_index();
 
    // Imaginary part:
    const double* value_c_pt = imag_matrix_pt->value();
    const int* row_start_c_pt = imag_matrix_pt->row_start();
    const int* column_index_c_pt = imag_matrix_pt->column_index();
 
#ifdef PARANOID
    // PARANOID check - make sure the matrices have the same number of rows
    // to ensure they are compatible
    if (real_matrix_pt->nrow() != imag_matrix_pt->nrow())
    {
      std::ostringstream error_message_stream;
      error_message_stream << "The real and imaginary matrices do not have "
                           << "compatible sizes";
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
    // PARANOID check - make sure the matrices have the same number of columns
    // to ensure they are compatible
    if (real_matrix_pt->ncol() != imag_matrix_pt->ncol())
    {
      std::ostringstream error_message_stream;
      error_message_stream << "The real and imaginary matrices do not have "
                           << "compatible sizes";
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Calculate the total number of nonzeros in the full matrix
    unsigned nnz = 2 * (nnz_r + nnz_c);
 
    // Allocate space for the row_start and column_index vectors associated with
    // the complete matrix
    Vector<double> value(nnz, 0.0);
    Vector<int> column_index(nnz, 0);
    Vector<int> row_start(2 * n_rows_r + 1, 0);
 
    //----------------------------
    // Build the row start vector:
    //----------------------------
 
    // Loop over the rows of the row_start vector. This is decomposed into
    // two parts. The first (n_rows_r+1) entries are found by computing
    // the entry-wise addition of row_start_r+row_start_c. The remaining
    // entries are almost the same as the first (n_rows_r+1). The only
    // distinction is that we need to shift the values of the entries by
    // the number of nonzeros in the top half of A. This is obvious from
    // observing the structure of the complete matrix.
 
    // Loop over the rows in the top half:
    for (unsigned i = 0; i < n_rows_r; i++)
    {
      // Set the i-th entry in the row start vector
      row_start[i] = *(row_start_r_pt + i) + *(row_start_c_pt + i);
    }
 
    // Loop over the rows in the bottom half:
    for (unsigned i = n_rows_r; i < 2 * n_rows_r; i++)
    {
      // Set the i-th entry in the row start vector (bottom half)
      row_start[i] = row_start[i - n_rows_r] + (nnz_r + nnz_c);
    }
 
    // Set the last entry in the row start vector
    row_start[2 * n_rows_r] = nnz;
 
    //-----------------------------------------
    // Build the column index and value vector:
    //-----------------------------------------
 
    // Variable to store the index of the nonzero
    unsigned i_nnz = 0;
 
    // Loop over the top half of the complete matrix
    for (unsigned i = 0; i < n_rows_r; i++)
    {
      // Calculate the number of nonzeros in the i-th row of A_r
      unsigned i_row_r_nnz = *(row_start_r_pt + i + 1) - *(row_start_r_pt + i);
 
      // Calculate the number of nonzeros in the i-th row of A_c
      unsigned i_row_c_nnz = *(row_start_c_pt + i + 1) - *(row_start_c_pt + i);
 
      // The index of the first entry in the i-th row of A_r
      unsigned i_first_entry_r = *(row_start_r_pt + i);
 
      // The index of the first entry in the i-th row of A_c
      unsigned i_first_entry_c = *(row_start_c_pt + i);
 
      // Loop over the number of nonzeros in the row associated with A_r
      for (unsigned j = 0; j < i_row_r_nnz; j++)
      {
        // Assign the column index of the j-th entry in the i-th row of A_r
        // to the column_index vector
        column_index[i_nnz] = *(column_index_r_pt + i_first_entry_r + j);
 
        // Assign the corresponding entry in the value vector
        value[i_nnz] = *(value_r_pt + i_first_entry_r + j);
 
        // Increment the value of i_nnz
        i_nnz++;
      }
 
      // Loop over the number of nonzeros in the row associated with -A_c
      for (unsigned j = 0; j < i_row_c_nnz; j++)
      {
        // Assign the column index of the j-th entry in the i-th row of -A_c
        // to the column_index vector
        column_index[i_nnz] =
          *(column_index_c_pt + i_first_entry_c + j) + n_rows_r;
 
        // Assign the corresponding entry in the value vector
        value[i_nnz] = -*(value_c_pt + i_first_entry_c + j);
 
        // Increment the value of i_nnz
        i_nnz++;
      }
    } // for (unsigned i=0;i<n_rows_r;i++)
 
    // Loop over the bottom half of the complete matrix
    for (unsigned i = n_rows_r; i < 2 * n_rows_r; i++)
    {
      // Calculate the number of nonzeros in row i of A_r
      unsigned i_row_r_nnz =
        *(row_start_r_pt + i - n_rows_r + 1) - *(row_start_r_pt + i - n_rows_r);
 
      // Calculate the number of nonzeros in row i of A_c
      unsigned i_row_c_nnz =
        *(row_start_c_pt + i - n_rows_r + 1) - *(row_start_c_pt + i - n_rows_r);
 
      // Get the index of the first entry in the i-th row of A_r
      unsigned i_first_entry_r = *(row_start_r_pt + i - n_rows_r);
 
      // Get the index of the first entry in the i-th row of A_c
      unsigned i_first_entry_c = *(row_start_c_pt + i - n_rows_r);
 
      // Loop over the number of nonzeros in the row associated with A_c
      for (unsigned j = 0; j < i_row_c_nnz; j++)
      {
        // Assign the column index of the j-th entry in the i-th row of A_c
        // to the column_index vector
        column_index[i_nnz] = *(column_index_c_pt + i_first_entry_c + j);
 
        // Assign the corresponding entry in the value vector
        value[i_nnz] = *(value_c_pt + i_first_entry_c + j);
 
        // Increment the value of i_nnz
        i_nnz++;
      }
 
      // Loop over the number of nonzeros in the row associated with A_r
      for (unsigned j = 0; j < i_row_r_nnz; j++)
      {
        // Assign the column index of the j-th entry in the i-th row of A_r
        // to the column_index vector
        column_index[i_nnz] =
          *(column_index_r_pt + i_first_entry_r + j) + n_rows_r;
 
        // Assign the corresponding entry in the value vector
        value[i_nnz] = *(value_r_pt + i_first_entry_r + j);
 
        // Increment the value of i_nnz
        i_nnz++;
      }
    } // for (unsigned i=n_rows_r;i<2*n_rows_r;i++)
 
    //-----------------------
    // Build the full matrix:
    //-----------------------
 
    // Allocate space for a CRDoubleMatrix
    Coarsest_matrix_mg_pt = new CRDoubleMatrix;
 
    // Make the distribution pointer
    LinearAlgebraDistribution* dist_pt = new LinearAlgebraDistribution(
      Mg_hierarchy_pt[Nlevel - 1]->communicator_pt(), 2 * n_rows_r, false);
 
    // First, we need to build the matrix. Making use of its particular
    // structure we know that there are 2*n_rows_r columns in this matrix.
    // The remaining information has just been sorted out
    Coarsest_matrix_mg_pt->build(
      dist_pt, 2 * n_rows_r, value, column_index, row_start);
 
    // Build the distribution of associated solution vector
    Coarsest_x_mg.build(dist_pt);
 
    // Build the distribution of associated RHS vector
    Coarsest_rhs_mg.build(dist_pt);
 
    // Delete the associated distribution pointer
    delete dist_pt;
 
    // Summarise setup
    double t_cm_end = TimingHelpers::timer();
    double total_setup_time = double(t_cm_end - t_cm_start);
    oomph_info << " - Time for formation of the full matrix "
               << "on the coarsest level [sec]: " << total_setup_time << "\n"
               << std::endl;
  } // End of setup_coarsest_matrix_mg
 
 
  //==========================================================================
  //==========================================================================
  template<>
  void HelmholtzMGPreconditioner<2>::maximum_edge_width(const unsigned& level,
                                                        double& h)
  {
    // Create a pointer to the "bulk" mesh
    TreeBasedRefineableMeshBase* bulk_mesh_pt =
      Mg_hierarchy_pt[level]->mg_bulk_mesh_pt();
 
    // Reset the value of h
    h = 0.0;
 
    // Find out how many nodes there are along one edge of the first element.
    // We assume here that all elements have the same number of nodes
    unsigned nnode_1d =
      dynamic_cast<FiniteElement*>(bulk_mesh_pt->element_pt(0))->nnode_1d();
 
    // Sort out corner node IDs:
    //--------------------------
    // Initialise a vector to store local node IDs of the corners
    Vector<unsigned> corner_node_id(4, 0);
 
    // Identify the local node ID of the first corner
    corner_node_id[0] = 0;
 
    // Identify the local node ID of the second corner
    corner_node_id[1] = nnode_1d - 1;
 
    // Identify the local node ID of the third corner
    corner_node_id[2] = nnode_1d * nnode_1d - 1;
 
    // Identify the local node ID of the fourth corner
    corner_node_id[3] = nnode_1d * (nnode_1d - 1);
 
    // Create storage for the nodal information:
    //------------------------------------------
    // Pointer to the first corner node on the j-th edge
    Node* first_node_pt = 0;
 
    // Pointer to the second corner node on the j-th edge
    Node* second_node_pt = 0;
 
    // Vector to store the (Eulerian) position of the first corner node
    // along a given edge of the element
    Vector<double> first_node_x(2, 0.0);
 
    // Vector to store the (Eulerian) position of the second corner node
    // along a given edge of the element
    Vector<double> second_node_x(2, 0.0);
 
    // Calculate h:
    //-------------
    // Find out how many elements there are in the bulk mesh
    unsigned n_element = bulk_mesh_pt->nelement();
 
    // Store a pointer which will point to a given element in the bulk mesh
    FiniteElement* el_pt = 0;
 
    // Initialise a dummy variable to compare with h
    double test_h = 0.0;
 
    // Store the number of edges in a 2D quad element
    unsigned n_edge = 4;
 
    // Loop over all of the elements in the bulk mesh
    for (unsigned i = 0; i < n_element; i++)
    {
      // Upcast the pointer to the i-th element to a FiniteElement pointer
      el_pt = dynamic_cast<FiniteElement*>(bulk_mesh_pt->element_pt(i));
 
      // Loop over the edges of the element
      for (unsigned j = 0; j < n_edge; j++)
      {
        // Get the local node ID of the first corner node on the j-th edge
        first_node_pt = el_pt->node_pt(corner_node_id[j]);
 
        // Get the local node ID of the second corner node on the j-th edge
        second_node_pt = el_pt->node_pt(corner_node_id[(j + 1) % 4]);
 
        // Obtain the (Eulerian) position of the first corner node
        for (unsigned n = 0; n < 2; n++)
        {
          // Grab the n-th coordinate of this node
          first_node_x[n] = first_node_pt->x(n);
        }
 
        // Obtain the (Eulerian) position of the second corner node
        for (unsigned n = 0; n < 2; n++)
        {
          // Grab the n-th coordinate of this node
          second_node_x[n] = second_node_pt->x(n);
        }
 
        // Reset the value of test_h
        test_h = 0.0;
 
        // Calculate the norm of (second_node_x-first_node_x)
        for (unsigned n = 0; n < 2; n++)
        {
          // Update the value of test_h
          test_h += pow(second_node_x[n] - first_node_x[n], 2.0);
        }
 
        // Square root the value of h
        test_h = sqrt(test_h);
 
        // Check if the value of test_h is greater than h
        if (test_h > h)
        {
          // If the value of test_h is greater than h then store it
          h = test_h;
        }
      } // for (unsigned j=0;j<n_edge;j++)
    } // for (unsigned i=0;i<n_element;i++)
  } // End of maximum_edge_width
 
  //==========================================================================
  //==========================================================================
  template<>
  void HelmholtzMGPreconditioner<3>::maximum_edge_width(const unsigned& level,
                                                        double& h)
  {
    // Create a pointer to the "bulk" mesh
    TreeBasedRefineableMeshBase* bulk_mesh_pt =
      Mg_hierarchy_pt[level]->mg_bulk_mesh_pt();
 
    //--------------------------------
    // Find the maximum edge width, h:
    //--------------------------------
    // Reset the value of h
    h = 0.0;
 
    // Find out how many nodes there are along one edge of the first element.
    // We assume here that all elements have the same number of nodes
    unsigned nnode_1d =
      dynamic_cast<FiniteElement*>(bulk_mesh_pt->element_pt(0))->nnode_1d();
 
    // Sort out corner node IDs:
    //--------------------------
    // Grab the octree pointer from the first element in the bulk mesh
    OcTree* octree_pt =
      dynamic_cast<RefineableQElement<3>*>(bulk_mesh_pt->element_pt(0))
        ->octree_pt();
 
    // Initialise a vector to store local node IDs of the corners
    Vector<unsigned> corner_node_id(8, 0);
 
    // Identify the local node ID of the first corner
    corner_node_id[0] =
      octree_pt->vertex_to_node_number(OcTreeNames::LDB, nnode_1d);
 
    // Identify the local node ID of the second corner
    corner_node_id[1] =
      octree_pt->vertex_to_node_number(OcTreeNames::RDB, nnode_1d);
 
    // Identify the local node ID of the third corner
    corner_node_id[2] =
      octree_pt->vertex_to_node_number(OcTreeNames::LUB, nnode_1d);
 
    // Identify the local node ID of the fourth corner
    corner_node_id[3] =
      octree_pt->vertex_to_node_number(OcTreeNames::RUB, nnode_1d);
 
    // Identify the local node ID of the fifth corner
    corner_node_id[4] =
      octree_pt->vertex_to_node_number(OcTreeNames::LDF, nnode_1d);
 
    // Identify the local node ID of the sixth corner
    corner_node_id[5] =
      octree_pt->vertex_to_node_number(OcTreeNames::RDF, nnode_1d);
 
    // Identify the local node ID of the seventh corner
    corner_node_id[6] =
      octree_pt->vertex_to_node_number(OcTreeNames::LUF, nnode_1d);
 
    // Identify the local node ID of the eighth corner
    corner_node_id[7] =
      octree_pt->vertex_to_node_number(OcTreeNames::RUF, nnode_1d);
 
    // Sort out the local node ID pairs:
    //----------------------------------
    // Store the number of edges in a 3D cubic element
    unsigned n_edge = 12;
 
    // Create a vector to store the pairs of adjacent nodes
    Vector<std::pair<unsigned, unsigned>> edge_node_pair(n_edge);
 
    // First edge
    edge_node_pair[0] = std::make_pair(corner_node_id[0], corner_node_id[1]);
 
    // Second edge
    edge_node_pair[1] = std::make_pair(corner_node_id[0], corner_node_id[2]);
 
    // Third edge
    edge_node_pair[2] = std::make_pair(corner_node_id[0], corner_node_id[4]);
 
    // Fourth edge
    edge_node_pair[3] = std::make_pair(corner_node_id[1], corner_node_id[3]);
 
    // Fifth edge
    edge_node_pair[4] = std::make_pair(corner_node_id[1], corner_node_id[5]);
 
    // Sixth edge
    edge_node_pair[5] = std::make_pair(corner_node_id[2], corner_node_id[3]);
 
    // Seventh edge
    edge_node_pair[6] = std::make_pair(corner_node_id[2], corner_node_id[6]);
 
    // Eighth edge
    edge_node_pair[7] = std::make_pair(corner_node_id[3], corner_node_id[7]);
 
    // Ninth edge
    edge_node_pair[8] = std::make_pair(corner_node_id[4], corner_node_id[5]);
 
    // Tenth edge
    edge_node_pair[9] = std::make_pair(corner_node_id[4], corner_node_id[6]);
 
    // Eleventh edge
    edge_node_pair[10] = std::make_pair(corner_node_id[5], corner_node_id[7]);
 
    // Twelfth edge
    edge_node_pair[11] = std::make_pair(corner_node_id[6], corner_node_id[7]);
 
    // Create storage for the nodal information:
    //------------------------------------------
    // Pointer to the first corner node on the j-th edge
    Node* first_node_pt = 0;
 
    // Pointer to the second corner node on the j-th edge
    Node* second_node_pt = 0;
 
    // Vector to store the (Eulerian) position of the first corner node
    // along a given edge of the element
    Vector<double> first_node_x(3, 0.0);
 
    // Vector to store the (Eulerian) position of the second corner node
    // along a given edge of the element
    Vector<double> second_node_x(3, 0.0);
 
    // Calculate h:
    //-------------
    // Find out how many elements there are in the bulk mesh
    unsigned n_element = bulk_mesh_pt->nelement();
 
    // Store a pointer which will point to a given element in the bulk mesh
    FiniteElement* el_pt = 0;
 
    // Initialise a dummy variable to compare with h
    double test_h = 0.0;
 
    // Loop over all of the elements in the bulk mesh
    for (unsigned i = 0; i < n_element; i++)
    {
      // Upcast the pointer to the i-th element to a FiniteElement pointer
      el_pt = dynamic_cast<FiniteElement*>(bulk_mesh_pt->element_pt(i));
 
      // Loop over the edges of the element
      for (unsigned j = 0; j < n_edge; j++)
      {
        // Get the local node ID of the first corner node on the j-th edge
        first_node_pt = el_pt->node_pt(edge_node_pair[j].first);
 
        // Get the local node ID of the second corner node on the j-th edge
        second_node_pt = el_pt->node_pt(edge_node_pair[j].second);
 
        // Obtain the (Eulerian) position of the first corner node
        for (unsigned n = 0; n < 3; n++)
        {
          // Grab the n-th coordinate of this node
          first_node_x[n] = first_node_pt->x(n);
        }
 
        // Obtain the (Eulerian) position of the second corner node
        for (unsigned n = 0; n < 3; n++)
        {
          // Grab the n-th coordinate of this node
          second_node_x[n] = second_node_pt->x(n);
        }
 
        // Reset the value of test_h
        test_h = 0.0;
 
        // Calculate the norm of (second_node_x-first_node_x)
        for (unsigned n = 0; n < 3; n++)
        {
          // Update the value of test_h
          test_h += pow(second_node_x[n] - first_node_x[n], 2.0);
        }
 
        // Square root the value of h
        test_h = sqrt(test_h);
 
        // Check if the value of test_h is greater than h
        if (test_h > h)
        {
          // If the value of test_h is greater than h then store it
          h = test_h;
        }
      } // for (unsigned j=0;j<n_edge;j++)
    } // for (unsigned i=0;i<n_element;i++)
  } // End of maximum_edge_width
 
  //=========================================================================
  //=========================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::setup_smoothers()
  {
    // Initialise the timer start variable
    double t_m_start = 0.0;
 
    // Start the clock (if we're allowed to time things)
    if (!Suppress_all_output)
    {
      // Notify user
      oomph_info << "Starting the setup of all smoothers.\n" << std::endl;
 
      // Start the clock
      t_m_start = TimingHelpers::timer();
    }
 
    // Loop over the levels and assign the pre- and post-smoother pointers
    for (unsigned i = 0; i < Nlevel - 1; i++)
    {
      // If the pre-smoother factory function pointer hasn't been assigned
      // then we simply create a new instance of the ComplexDampedJacobi
      // smoother which is the default pre-smoother
      if (0 == Pre_smoother_factory_function_pt)
      {
        // If the value of kh is strictly less than 0.5 then use damped Jacobi
        // as a smoother on this level otherwise use complex GMRES as a smoother
        // [see Elman et al."A multigrid method enhanced by Krylov subspace
        // iteration for discrete Helmholtz equations", p.1305]
        if (Wavenumber * Max_edge_width[i] < 0.5)
        {
          // Make a new ComplexDampedJacobi object and assign its pointer
          ComplexDampedJacobi<CRDoubleMatrix>* damped_jacobi_pt =
            new ComplexDampedJacobi<CRDoubleMatrix>;
 
          // Assign the pre-smoother pointer
          Pre_smoothers_storage_pt[i] = damped_jacobi_pt;
 
          // Make the smoother calculate the value of omega and store it
          damped_jacobi_pt->calculate_omega(Wavenumber, Max_edge_width[i]);
        }
        else
        {
          // Make a new ComplexGMRES object and assign its pointer
          ComplexGMRES<CRDoubleMatrix>* gmres_pt =
            new ComplexGMRES<CRDoubleMatrix>;
 
          // Assign the pre-smoother pointer
          Pre_smoothers_storage_pt[i] = gmres_pt;
        }
      }
      // Otherwise we use the pre-smoother factory function pointer to
      // generate a new pre-smoother
      else
      {
        // Get a pointer to an object of the same type as the pre-smoother
        Pre_smoothers_storage_pt[i] = (*Pre_smoother_factory_function_pt)();
      } // if (0==Pre_smoother_factory_function_pt)
 
      // If the post-smoother factory function pointer hasn't been assigned
      // then we simply create a new instance of the ComplexDampedJacobi
      // smoother which is the default post-smoother
      if (0 == Post_smoother_factory_function_pt)
      {
        // If the value of kh is strictly less than 0.52 then use damped Jacobi
        // as a smoother on this level otherwise use complex GMRES as a smoother
        // [see Elman et al."A multigrid method enhanced by Krylov subspace
        // iteration for discrete Helmholtz equations", p.1295]
        if (Wavenumber * Max_edge_width[i] < 0.5)
        {
          // Make a new ComplexDampedJacobi object and assign its pointer
          ComplexDampedJacobi<CRDoubleMatrix>* damped_jacobi_pt =
            new ComplexDampedJacobi<CRDoubleMatrix>;
 
          // Assign the pre-smoother pointer
          Post_smoothers_storage_pt[i] = damped_jacobi_pt;
 
          // Make the smoother calculate the value of omega and store it
          damped_jacobi_pt->calculate_omega(Wavenumber, Max_edge_width[i]);
        }
        else
        {
          // Make a new ComplexGMRES object and assign its pointer
          ComplexGMRES<CRDoubleMatrix>* gmres_pt =
            new ComplexGMRES<CRDoubleMatrix>;
 
          // Assign the pre-smoother pointer
          Post_smoothers_storage_pt[i] = gmres_pt;
        }
      }
      // Otherwise we use the post-smoother factory function pointer to
      // generate a new post-smoother
      else
      {
        // Get a pointer to an object of the same type as the post-smoother
        Post_smoothers_storage_pt[i] = (*Post_smoother_factory_function_pt)();
      }
    } // if (0==Post_smoother_factory_function_pt)
 
    // Set the tolerance for the pre- and post-smoothers. The norm of the
    // solution which is compared against the tolerance is calculated
    // differently to the multigrid solver. To ensure that the smoother
    // continues to solve for the prescribed number of iterations we
    // lower the tolerance
    for (unsigned i = 0; i < Nlevel - 1; i++)
    {
      // Set the tolerance of the i-th level pre-smoother
      Pre_smoothers_storage_pt[i]->tolerance() = 1.0e-16;
 
      // Set the tolerance of the i-th level post-smoother
      Post_smoothers_storage_pt[i]->tolerance() = 1.0e-16;
    }
 
    // Set the number of pre- and post-smoothing iterations in each
    // pre- and post-smoother
    for (unsigned i = 0; i < Nlevel - 1; i++)
    {
      // Set the number of pre-smoothing iterations as the value of Npre_smooth
      Pre_smoothers_storage_pt[i]->max_iter() = Npre_smooth;
 
      // Set the number of pre-smoothing iterations as the value of Npost_smooth
      Post_smoothers_storage_pt[i]->max_iter() = Npost_smooth;
    }
 
    // Complete the setup of all of the smoothers
    for (unsigned i = 0; i < Nlevel - 1; i++)
    {
      // Pass a pointer to the system matrix on the i-th level to the i-th
      // level pre-smoother
      Pre_smoothers_storage_pt[i]->complex_smoother_setup(
        Mg_matrices_storage_pt[i]);
 
      // Pass a pointer to the system matrix on the i-th level to the i-th
      // level post-smoother
      Post_smoothers_storage_pt[i]->complex_smoother_setup(
        Mg_matrices_storage_pt[i]);
    }
 
    // Set up the distributions of each smoother
    for (unsigned i = 0; i < Nlevel - 1; i++)
    {
      // Get the number of dofs on the i-th level of the hierarchy.
      // This will be the same as the size of the solution vector
      // associated with the i-th level
      unsigned n_dof = X_mg_vectors_storage[i][0].nrow();
 
      // Create a LinearAlgebraDistribution which will be passed to the
      // linear solver
      LinearAlgebraDistribution dist(
        Mg_hierarchy_pt[i]->communicator_pt(), n_dof, false);
 
#ifdef PARANOID
#ifdef OOMPH_HAS_MPI
      // Set up the warning messages
      std::string warning_message =
        "Setup of pre- and post-smoother distribution ";
      warning_message += "has not been tested with MPI.";
 
      // If we're not running the code in serial
      if (dist.communicator_pt()->nproc() > 1)
      {
        // Throw a warning
        OomphLibWarning(
          warning_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
#endif
 
      // Build the distribution of the pre-smoother
      Pre_smoothers_storage_pt[i]->build_distribution(dist);
 
      // Build the distribution of the post-smoother
      Post_smoothers_storage_pt[i]->build_distribution(dist);
    }
 
    // Disable the smoother output
    if (!Doc_time)
    {
      // Disable time documentation from the smoothers and the SuperLU linear
      // solver (this latter is only done on the coarsest level where it is
      // required)
      disable_smoother_and_superlu_doc_time();
    }
 
    // If we're allowed to output
    if (!Suppress_all_output)
    {
      // Stop clock
      double t_m_end = TimingHelpers::timer();
      double total_setup_time = double(t_m_end - t_m_start);
      oomph_info << "CPU time for setup of smoothers on all levels [sec]: "
                 << total_setup_time << std::endl;
 
      // Notify user that the extraction is complete
      oomph_info << "\n=================="
                 << "Smoother Setup Complete"
                 << "=================" << std::endl;
    }
  } // End of setup_smoothers
 
 
  //===================================================================
  //===================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::setup_interpolation_matrices()
  {
    // Variable to hold the number of sons in an element
    unsigned n_sons;
 
    // Number of son elements
    if (DIM == 2)
    {
      n_sons = 4;
    }
    else if (DIM == 3)
    {
      n_sons = 8;
    }
    else
    {
      std::ostringstream error_message_stream;
      error_message_stream << "DIM should be 2 or 3 not " << DIM << std::endl;
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
#ifdef PARANOID
    // PARANOID check - for our implementation we demand that the first
    // submesh is the refineable bulk mesh that is provided by the function
    // mg_bulk_mesh_pt. Since each mesh in the hierarchy will be set up
    // in the same manner through the problem constructor we only needed
    // to check the finest level mesh
    if (Mg_hierarchy_pt[0]->mesh_pt(0) != Mg_hierarchy_pt[0]->mg_bulk_mesh_pt())
    {
      // Create an output stream
      std::ostringstream error_message_stream;
 
      // Create the error message
      error_message_stream << "MG solver requires the first submesh be the "
                           << "refineable bulk mesh provided by the user."
                           << std::endl;
 
      // Throw the error message
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Vector of local coordinates in the element
    Vector<double> s(DIM, 0.0);
 
    // Vector to contain the (Eulerian) spatial location of the fine node.
    // This will only be used if we have a PML layer attached to the mesh
    // which will require the use of locate_zeta functionality
    Vector<double> fine_node_position(DIM, 0.0);
 
    // Find the number of nodes in an element in the mesh. Since this
    // quantity will be the same in all meshes we can precompute it here
    // using the original problem pointer
    unsigned nnod_element =
      dynamic_cast<FiniteElement*>(Mg_problem_pt->mesh_pt()->element_pt(0))
        ->nnode();
 
    // Initialise a null pointer which will be used to point to an object
    // of the class MeshAsGeomObject
    MeshAsGeomObject* coarse_mesh_from_obj_pt = 0;
 
    // Loop over each level (apart from the coarsest level; an interpolation
    // matrix and thus a restriction matrix is not needed here), with 0 being
    // the finest level and Nlevel-1 being the coarsest level
    for (unsigned level = 0; level < Nlevel - 1; level++)
    {
      // Store the ID of the fine level
      unsigned fine_level = level;
 
      // Store the ID of the coarse level
      unsigned coarse_level = level + 1;
 
      // Make a pointer to the mesh on the finer level and dynamic_cast
      // it as an object of the refineable mesh class.
      Mesh* ref_fine_mesh_pt = Mg_hierarchy_pt[fine_level]->mesh_pt();
 
      // Make a pointer to the mesh on the coarse level and dynamic_cast
      // it as an object of the refineable mesh class
      Mesh* ref_coarse_mesh_pt = Mg_hierarchy_pt[coarse_level]->mesh_pt();
 
      // Access information about the number of elements in the fine mesh
      // from the pointer to the fine mesh (to loop over the rows of the
      // interpolation matrix)
      unsigned fine_n_element = ref_fine_mesh_pt->nelement();
 
      // Find the number of elements in the bulk mesh
      unsigned n_bulk_mesh_element =
        Mg_hierarchy_pt[fine_level]->mg_bulk_mesh_pt()->nelement();
 
      // If there are elements apart from those in the bulk mesh
      // then we require locate_zeta functionality. For this reason
      // we have to assign a value to the MeshAsGeomObject pointer
      // which we do by creating a MeshAsGeomObject from the coarse
      // mesh pointer
      if (fine_n_element > n_bulk_mesh_element)
      {
        // Assign the pointer to coarse_mesh_from_obj_pt
        coarse_mesh_from_obj_pt =
          new MeshAsGeomObject(Mg_hierarchy_pt[coarse_level]->mesh_pt());
      }
 
      // Find the number of dofs in the fine mesh
      unsigned n_fine_dof = Mg_hierarchy_pt[fine_level]->ndof();
 
      // Find the number of dofs in the coarse mesh
      unsigned n_coarse_dof = Mg_hierarchy_pt[coarse_level]->ndof();
 
      // To get the number of rows in the interpolation matrix we divide
      // the number of dofs on each level by 2 since there are 2 dofs at
      // each node in the mesh corresponding to the real and imaginary
      // part of the solution:
 
      // Get the numbers of rows in the interpolation matrix.
      unsigned n_row = n_fine_dof / 2.0;
 
      // Get the numbers of columns in the interpolation matrix
      unsigned n_col = n_coarse_dof / 2.0;
 
      // Mapping relating the pointers to related elements in the coarse
      // and fine meshes: coarse_mesh_element_pt[fine_mesh_element_pt]. If
      // the element in the fine mesh has been unrefined between these two
      // levels, this map returns the father element in the coarsened mesh.
      // If this element in the fine mesh has not been unrefined, the map
      // returns the pointer to the same-sized equivalent element in the
      // coarsened mesh.
      std::map<RefineableQElement<DIM>*, RefineableQElement<DIM>*>
        coarse_mesh_reference_element_pt;
 
      // Variable to count the number of elements in the fine mesh
      unsigned e_fine = 0;
 
      // Variable to count the number of elements in the coarse mesh
      unsigned e_coarse = 0;
 
      // While loop over fine elements (while because we're incrementing the
      // counter internally if the element was unrefined...). We only bother
      // going until we've covered all of the elements in the bulk mesh
      // because once we reach the PML layer (if there is one) there will be
      // no tree structure to match in between levels as PML elements are
      // simply regenerated once the bulk mesh has been refined.
      while (e_fine < n_bulk_mesh_element)
      {
        // Pointer to element in fine mesh
        RefineableQElement<DIM>* el_fine_pt =
          dynamic_cast<RefineableQElement<DIM>*>(
            ref_fine_mesh_pt->finite_element_pt(e_fine));
 
#ifdef PARANOID
        // Make sure the coarse level element pointer is not a null pointer. If
        // it is something has gone wrong
        if (e_coarse > ref_coarse_mesh_pt->nelement() - 1)
        {
          // Create an output stream
          std::ostringstream error_message_stream;
 
          // Create an error message
          error_message_stream
            << "The coarse level mesh has " << ref_coarse_mesh_pt->nelement()
            << " elements but the coarse\nelement loop "
            << "is looking at the " << e_coarse << "-th element!" << std::endl;
 
          // Throw the error message
          throw OomphLibError(error_message_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
#endif
 
        // Pointer to element in coarse mesh
        RefineableQElement<DIM>* el_coarse_pt =
          dynamic_cast<RefineableQElement<DIM>*>(
            ref_coarse_mesh_pt->finite_element_pt(e_coarse));
 
        // If the levels are different then the element in the fine
        // mesh has been unrefined between these two levels
        if (el_fine_pt->tree_pt()->level() > el_coarse_pt->tree_pt()->level())
        {
          // The element in the fine mesh has been unrefined between these two
          // levels. Hence it and its three brothers (ASSUMED to be stored
          // consecutively in the Mesh's vector of pointers to its constituent
          // elements -- we'll check this!) share the same father element in
          // the coarse mesh, currently pointed to by el_coarse_pt.
          for (unsigned i = 0; i < n_sons; i++)
          {
            // Set mapping to father element in coarse mesh
            coarse_mesh_reference_element_pt
              [dynamic_cast<RefineableQElement<DIM>*>(
                ref_fine_mesh_pt->finite_element_pt(e_fine))] = el_coarse_pt;
 
            // Increment counter for elements in fine mesh
            e_fine++;
          }
        }
        // The element in the fine mesh has not been unrefined between
        // these two levels, so the reference element is the same-sized
        // equivalent element in the coarse mesh
        else if (el_fine_pt->tree_pt()->level() ==
                 el_coarse_pt->tree_pt()->level())
        {
          // The element in the fine mesh has not been unrefined between these
          // two levels
          coarse_mesh_reference_element_pt
            [dynamic_cast<RefineableQElement<DIM>*>(
              ref_fine_mesh_pt->finite_element_pt(e_fine))] = el_coarse_pt;
 
          // Increment counter for elements in fine mesh
          e_fine++;
        }
        // If the element has been unrefined between levels. Not something
        // we can deal with at the moment (although it would be relatively
        // simply to implement...).
        // Option 1: Find the son elements (from the coarse mesh) associated
        // with the father element (from the fine mesh) and extract the
        // appropriate nodal values to find the nodal values in the father
        // element.
        // Option 2: Use the function
        //                       unrefine_uniformly();
        // to ensure that the coarser meshes really only have father elements
        // or the element itself.
        else
        {
          // Create an output stream
          std::ostringstream error_message_stream;
 
          // Create an error message
          error_message_stream << "Element unrefined between levels! Can't "
                               << "handle this case yet..." << std::endl;
 
          // Throw the error message
          throw OomphLibError(error_message_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        } // if (el_fine_pt->tree_pt()->level()!=...)
 
        // Increment counter for elements in coarse mesh
        e_coarse++;
      } // while(e_fine<n_bulk_mesh_element)
 
      // To allow an update of a row only once we use STL vectors for bools
      std::vector<bool> contribution_made(n_row, false);
 
      // Create the row start vector whose i-th entry will contain the index
      // in column_start where the entries in the i-th row of the matrix start
      Vector<int> row_start(n_row + 1);
 
      // Create the column index vector whose entries will store the column
      // index of each contribution, i.e. the global equation of the coarse
      // mesh node which supplies a contribution. We don't know how many
      // entries this will have so we dynamically allocate entries at run time
      Vector<int> column_index;
 
      // Create the value vector which will be used to store the nonzero
      // entries in the CRDoubleMatrix. We don't know how many entries this
      // will have either so we dynamically allocate its entries at run time
      Vector<double> value;
 
      // The value of index will tell us which row of the interpolation matrix
      // we're working on in the following for loop
      // DRAIG: Should this worry us? We assume that every node we cover is
      // the next node in the mesh (we loop over the elements and the nodes
      // inside that). This does work but it may not work for some meshes
      unsigned index = 0;
 
      // New loop to go over each element in the fine mesh
      for (unsigned k = 0; k < fine_n_element; k++)
      {
        // Set a pointer to the element in the fine mesh
        RefineableQElement<DIM>* el_fine_pt =
          dynamic_cast<RefineableQElement<DIM>*>(
            ref_fine_mesh_pt->finite_element_pt(k));
 
        // Initialise a null pointer which will point to the corresponding
        // coarse mesh element. If we're in the bulk mesh, this will point
        // to the parent element of the fine mesh element or itself (if
        // there was no refinement between the levels). If, however, we're
        // in the PML layer then this will point to element that encloses
        // the fine mesh PML element
        RefineableQElement<DIM>* el_coarse_pt = 0;
 
        // Variable to store the son type of the fine mesh element with
        // respect to the corresponding coarse mesh element (set to OMEGA
        // if no unrefinement took place)
        int son_type = 0;
 
        // If we're in the bulk mesh we can assign the coarse mesh element
        // pointer right now using the map coarse_mesh_reference_element_pt
        if (k < n_bulk_mesh_element)
        {
          // Get the reference element (either the father element or the
          // same-sized element) in the coarse mesh
          el_coarse_pt = coarse_mesh_reference_element_pt[el_fine_pt];
 
          // Check if the elements are different on both levels (i.e. to check
          // if any unrefinement took place)
          if (el_fine_pt->tree_pt()->level() !=
              el_coarse_pt->tree_pt()->level())
          {
            // If the element was refined then we need the son type
            son_type = el_fine_pt->tree_pt()->son_type();
          }
          else
          {
            // If the element is the same in both meshes
            son_type = Tree::OMEGA;
          }
        } // if (k<n_bulk_mesh_element)
 
        // Loop through all the nodes in an element in the fine mesh
        for (unsigned i = 0; i < nnod_element; i++)
        {
          // Row number in interpolation matrix: Global equation number
          // of the d.o.f. stored at this node in the fine element
          int ii = el_fine_pt->node_pt(i)->eqn_number(0);
 
          // Check whether or not the node is a proper d.o.f.
          if (ii >= 0)
          {
            // Only assign values to the given row of the interpolation matrix
            // if they haven't already been assigned. Notice, we check if the
            // (ii/2)-th entry has been set. This is because there are two dofs
            // at each node so dividing by two will give us the row number
            // associated with this node
            if (contribution_made[ii / 2] == false)
            {
              // The value of index was initialised when we allocated space
              // for the three vectors to store the CRDoubleMatrix. We use
              // index to go through the entries of the row_start vector.
              // The next row starts at the value.size() (draw out the entries
              // in value if this doesn't make sense noting that the storage
              // for the vector 'value' is dynamically allocated)
              row_start[index] = value.size();
 
              // If we're in the PML layer then we need to find the parent
              // element associated with a given node using locate_zeta.
              // Unfortunately, if this is the case and a given local node in
              // the fine mesh element lies on the interface between two
              // elements (E1) and (E2) in the coarse mesh then either element
              // will be returned. Suppose, for instance, a pointer to (E1) is
              // returned. If the next local node lies in the (E2) then the
              // contributions which we pick up from the local nodes in (E1)
              // will most likely be incorrect while the column index (the
              // global equation number of the coarse mesh node) corresponding
              // to the given contribution will definitely be incorrect. If
              // we're not using linear quad elements we can get around this by
              // using locate_zeta carefully by identifying a node which is
              // internal to the fine mesh element. This will allow us to
              // correctly obtain the corresponding element in the coarse mesh
              if (k >= n_bulk_mesh_element)
              {
                // Get the (Eulerian) spatial location of the i-th local node
                // in the fine mesh element
                el_fine_pt->node_pt(i)->position(fine_node_position);
 
                // Initalise a null pointer to point to an object of the class
                // GeomObject
                GeomObject* el_pt = 0;
 
                // Get the reference element (either the father element or the
                // same-sized element) in the coarse-mesh PML layer using
                // the locate_zeta functionality
                coarse_mesh_from_obj_pt->locate_zeta(
                  fine_node_position, el_pt, s);
 
                // Upcast the GeomElement object to a RefineableQElement object
                el_coarse_pt = dynamic_cast<RefineableQElement<DIM>*>(el_pt);
              }
              else
              {
                // Calculate the local coordinates of the given node
                el_fine_pt->local_coordinate_of_node(i, s);
 
                // Find the local coordinates s, of the present node, in the
                // reference element in the coarse mesh, given the element's
                // son_type (e.g. SW,SE... )
                level_up_local_coord_of_node(son_type, s);
              }
 
              // Allocate space for shape functions in the coarse mesh
              Shape psi(el_coarse_pt->nnode());
 
              // Set the shape function in the reference element
              el_coarse_pt->shape(s, psi);
 
              // Auxiliary storage
              std::map<unsigned, double> contribution;
              Vector<unsigned> keys;
 
              // Find the number of nodes in an element in the coarse mesh
              unsigned nnod_coarse = el_coarse_pt->nnode();
 
              // Loop through all the nodes in the reference element
              for (unsigned j = 0; j < nnod_coarse; j++)
              {
                // Column number in interpolation matrix: Global equation
                // number of the d.o.f. stored at this node in the coarse
                // element
                int jj = el_coarse_pt->node_pt(j)->eqn_number(0);
 
                // If the value stored at this node is pinned or hanging
                if (jj < 0)
                {
                  // Hanging node: In this case we need to accumulate the
                  // contributions from the master nodes
                  if (el_coarse_pt->node_pt(j)->is_hanging())
                  {
                    // Find the number of master nodes of the hanging
                    // the node in the reference element
                    HangInfo* hang_info_pt =
                      el_coarse_pt->node_pt(j)->hanging_pt();
                    unsigned nmaster = hang_info_pt->nmaster();
 
                    // Loop over the master nodes
                    for (unsigned i_master = 0; i_master < nmaster; i_master++)
                    {
                      // Set up a pointer to the master node
                      Node* master_node_pt =
                        hang_info_pt->master_node_pt(i_master);
 
                      // The column number in the interpolation matrix: the
                      // global equation number of the d.o.f. stored at this
                      // master node for the coarse element
                      int master_jj = master_node_pt->eqn_number(0);
 
                      // Is the master node a proper d.o.f.?
                      if (master_jj >= 0)
                      {
                        // If the weight of the master node is non-zero
                        if (psi(j) * hang_info_pt->master_weight(i_master) !=
                            0.0)
                        {
                          contribution[master_jj / 2] +=
                            psi(j) * hang_info_pt->master_weight(i_master);
                        }
                      } // if (master_jj>=0)
                    } // for (unsigned i_master=0;i_master<nmaster;i_master++)
                  } // if (el_coarse_pt->node_pt(j)->is_hanging())
                }
                // In the case that the node is not pinned or hanging
                else
                {
                  // If we can get a nonzero contribution from the shape
                  // function
                  if (psi(j) != 0.0)
                  {
                    contribution[jj / 2] += psi(j);
                  }
                } // if (jj<0) else
              } // for (unsigned j=0;j<nnod_coarse;j++)
 
              // Put the contributions into the value vector
              for (std::map<unsigned, double>::iterator it =
                     contribution.begin();
                   it != contribution.end();
                   ++it)
              {
                if (it->second != 0)
                {
                  // The first entry in contribution is the column index
                  column_index.push_back(it->first);
 
                  // The second entry in contribution is the value of the
                  // contribution
                  value.push_back(it->second);
                }
              } // for (std::map<unsigned,double>::iterator it=...)
 
              // Increment the value of index by 1 to indicate that we have
              // finished the row, or equivalently, covered another global
              // node in the fine mesh
              index++;
 
              // Change the entry in contribution_made to true now to indicate
              // that the row has been filled
              contribution_made[ii / 2] = true;
            } // if(contribution_made[ii/2]==false)
          } // if (ii>=0)
        } // for(unsigned i=0;i<nnod_element;i++)
      } // for (unsigned k=0;k<fine_n_element;k++)
 
      // Set the last entry in the row_start vector
      row_start[n_row] = value.size();
 
      // Set the interpolation matrix to be that formed as the CRDoubleMatrix
      // using the vectors value, row_start, column_index and the value
      // of fine_n_unknowns and coarse_n_unknowns
      interpolation_matrix_set(
        level, value, column_index, row_start, n_col, n_row);
    } // for (unsigned level=0;level<Nlevel-1;level++)
  } // End of setup_interpolation_matrices
 
  //===================================================================
  //===================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<
    DIM>::setup_interpolation_matrices_unstructured()
  {
#ifdef PARANOID
    if ((DIM != 2) && (DIM != 3))
    {
      std::ostringstream error_message_stream;
      error_message_stream << "DIM should be 2 or 3 not " << DIM << std::endl;
      throw OomphLibError(error_message_stream.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Vector of local coordinates in the element
    Vector<double> s(DIM, 0.0);
 
    // Loop over each level (apart from the coarsest level; an interpolation
    // matrix and thus a restriction matrix is not needed here), with 0 being
    // the finest level and Nlevel-1 being the coarsest level
    for (unsigned level = 0; level < Nlevel - 1; level++)
    {
      // Assign values to a couple of variables to help out
      unsigned coarse_level = level + 1;
      unsigned fine_level = level;
 
      // Make a pointer to the mesh on the finer level and dynamic_cast
      // it as an object of the refineable mesh class
      Mesh* ref_fine_mesh_pt = Mg_hierarchy_pt[fine_level]->mg_bulk_mesh_pt();
 
      // To use the locate zeta functionality the coarse mesh must be of the
      // type MeshAsGeomObject
      MeshAsGeomObject* coarse_mesh_from_obj_pt =
        new MeshAsGeomObject(Mg_hierarchy_pt[coarse_level]->mg_bulk_mesh_pt());
 
      // Access information about the number of degrees of freedom
      // from the pointers to the problem on each level
      unsigned coarse_n_unknowns = Mg_hierarchy_pt[coarse_level]->ndof();
      unsigned fine_n_unknowns = Mg_hierarchy_pt[fine_level]->ndof();
 
      // Make storage vectors to form the interpolation matrix using a
      // condensed row matrix (CRDoubleMatrix). The i-th value of the vector
      // row_start contains entries which tells us at which entry of the
      // vector column_start we start on the i-th row of the actual matrix.
      // The entries of column_start indicate the column position of each
      // non-zero entry in every row. This runs through the first row
      // from left to right then the second row (again, left to right)
      // and so on until the end. The entries in value are the entries in
      // the interpolation matrix. The vector column_start has the same length
      // as the vector value.
      Vector<double> value;
      Vector<int> column_index;
      Vector<int> row_start(fine_n_unknowns + 1);
 
      // Vector to contain the (Eulerian) spatial location of the fine node
      Vector<double> fine_node_position(DIM);
 
      // Find the number of nodes in the mesh
      unsigned n_node_fine_mesh = ref_fine_mesh_pt->nnode();
 
      // Loop over the unknowns in the mesh
      for (unsigned i_fine_node = 0; i_fine_node < n_node_fine_mesh;
           i_fine_node++)
      {
        // Set a pointer to the i_fine_unknown-th node in the fine mesh
        Node* fine_node_pt = ref_fine_mesh_pt->node_pt(i_fine_node);
 
        // Get the global equation number
        int i_fine = fine_node_pt->eqn_number(0);
 
        // If the node is a proper d.o.f.
        if (i_fine >= 0)
        {
          // Row number in interpolation matrix: Global equation number
          // of the d.o.f. stored at this node in the fine element
          row_start[i_fine] = value.size();
 
          // Get the (Eulerian) spatial location of the fine node
          fine_node_pt->position(fine_node_position);
 
          // Create a null pointer to the GeomObject class
          GeomObject* el_pt = 0;
 
          // Get the reference element (either the father element or the
          // same-sized element) in the coarse mesh using locate_zeta
          coarse_mesh_from_obj_pt->locate_zeta(fine_node_position, el_pt, s);
 
          // Upcast GeomElement as a FiniteElement
          FiniteElement* el_coarse_pt = dynamic_cast<FiniteElement*>(el_pt);
 
          // Find the number of nodes in the element
          unsigned n_node = el_coarse_pt->nnode();
 
          // Allocate space for shape functions in the coarse mesh
          Shape psi(n_node);
 
          // Calculate the geometric shape functions at local coordinate s
          el_coarse_pt->shape(s, psi);
 
          // Auxiliary storage
          std::map<unsigned, double> contribution;
          Vector<unsigned> keys;
 
          // Loop through all the nodes in the (coarse mesh) element containing
          // the node pointed to by fine_node_pt (fine mesh)
          for (unsigned j_node = 0; j_node < n_node; j_node++)
          {
            // Get the j_coarse_unknown-th node in the coarse element
            Node* coarse_node_pt = el_coarse_pt->node_pt(j_node);
 
            // Column number in interpolation matrix: Global equation number of
            // the d.o.f. stored at this node in the coarse element
            int j_coarse = coarse_node_pt->eqn_number(0);
 
            // If the value stored at this node is pinned or hanging
            if (j_coarse < 0)
            {
              // Hanging node: In this case we need to accumulate the
              // contributions from the master nodes
              if (el_coarse_pt->node_pt(j_node)->is_hanging())
              {
                // Find the number of master nodes of the hanging
                // the node in the reference element
                HangInfo* hang_info_pt = coarse_node_pt->hanging_pt();
                unsigned nmaster = hang_info_pt->nmaster();
 
                // Loop over the master nodes
                for (unsigned i_master = 0; i_master < nmaster; i_master++)
                {
                  // Set up a pointer to the master node
                  Node* master_node_pt = hang_info_pt->master_node_pt(i_master);
 
                  // The column number in the interpolation matrix: the
                  // global equation number of the d.o.f. stored at this master
                  // node for the coarse element
                  int master_jj = master_node_pt->eqn_number(0);
 
                  // Is the master node a proper d.o.f.?
                  if (master_jj >= 0)
                  {
                    // If the weight of the master node is non-zero
                    if (psi(j_node) * hang_info_pt->master_weight(i_master) !=
                        0.0)
                    {
                      contribution[master_jj] +=
                        psi(j_node) * hang_info_pt->master_weight(i_master);
                    }
                  } // End of if statement (check it's not a boundary node)
                } // End of the loop over the master nodes
              } // End of the if statement for only hanging nodes
            } // End of the if statement for pinned or hanging nodes
            // In the case that the node is not pinned or hanging
            else
            {
              // If we can get a nonzero contribution from the shape function
              // at the j_node-th node in the element
              if (psi(j_node) != 0.0)
              {
                contribution[j_coarse] += psi(j_node);
              }
            } // End of the if-else statement (check if the node was
              // pinned/hanging)
          } // Finished loop over the nodes j in the reference element (coarse)
 
          // Put the contributions into the value vector
          for (std::map<unsigned, double>::iterator it = contribution.begin();
               it != contribution.end();
               ++it)
          {
            if (it->second != 0)
            {
              value.push_back(it->second);
              column_index.push_back(it->first);
            }
          } // End of putting contributions into the value vector
        } // End check (whether or not the fine node was a d.o.f.)
      } // End of the for-loop over nodes in the fine mesh
 
      // Set the last entry of row_start
      row_start[fine_n_unknowns] = value.size();
 
      // Set the interpolation matrix to be that formed as the CRDoubleMatrix
      // using the vectors value, row_start, column_index and the value
      // of fine_n_unknowns and coarse_n_unknowns
      interpolation_matrix_set(level,
                               value,
                               column_index,
                               row_start,
                               coarse_n_unknowns,
                               fine_n_unknowns);
    } // End of loop over each level
  } // End of setup_interpolation_matrices_unstructured
 
  //=========================================================================
  //=========================================================================
  template<>
  void HelmholtzMGPreconditioner<3>::level_up_local_coord_of_node(
    const int& son_type, Vector<double>& s)
  {
    // If the element is unrefined between the levels the local coordinate
    // of the node in one element is the same as that in the other element
    // therefore we only need to perform calculations if the levels are
    // different (i.e. son_type is not OMEGA)
    if (son_type != Tree::OMEGA)
    {
      // Scale the local coordinate from the range [-1,1]x[-1,1]x[-1,1]
      // to the range [0,1]x[0,1]x[0,1] to match the width of the local
      // coordinate range of the fine element from the perspective of
      // the father element. This then simply requires a shift of the
      // coordinates to match which type of son element we're dealing with
      s[0] = (s[0] + 1.0) / 2.0;
      s[1] = (s[1] + 1.0) / 2.0;
      s[2] = (s[2] + 1.0) / 2.0;
 
      // Cases: The son_type determines how the local coordinates should be
      // shifted to give the local coordinates in the coarse mesh element
      switch (son_type)
      {
        case OcTreeNames::LDF:
          s[0] -= 1;
          s[1] -= 1;
          break;
 
        case OcTreeNames::LDB:
          s[0] -= 1;
          s[1] -= 1;
          s[2] -= 1;
          break;
 
        case OcTreeNames::LUF:
          s[0] -= 1;
          break;
 
        case OcTreeNames::LUB:
          s[0] -= 1;
          s[2] -= 1;
          break;
 
        case OcTreeNames::RDF:
          s[1] -= 1;
          break;
 
        case OcTreeNames::RDB:
          s[1] -= 1;
          s[2] -= 1;
          break;
 
        case OcTreeNames::RUF:
          break;
 
        case OcTreeNames::RUB:
          s[2] -= 1;
          break;
      }
    } // if son_type!=Tree::OMEGA
  } // End of level_up_local_coord_of_node
 
  //=========================================================================
  //=========================================================================
  template<>
  void HelmholtzMGPreconditioner<2>::level_up_local_coord_of_node(
    const int& son_type, Vector<double>& s)
  {
    // If the element is unrefined between the levels the local coordinate
    // of the node in one element is the same as that in the other element
    // therefore we only need to perform calculations if the levels are
    // different (i.e. son_type is not OMEGA)
    if (son_type != Tree::OMEGA)
    {
      // Scale the local coordinate from the range [-1,1]x[-1,1] to the range
      // [0,1]x[0,1] to match the width of the local coordinate range of the
      // fine element from the perspective of the father element. This
      // then simply requires a shift of the coordinates to match which type
      // of son element we're dealing with
      s[0] = (s[0] + 1.0) / 2.0;
      s[1] = (s[1] + 1.0) / 2.0;
 
      // Cases: The son_type determines how the local coordinates should be
      // shifted to give the local coordinates in the coarse mesh element
      switch (son_type)
      {
          // If we're dealing with the bottom-left element we need to shift
          // the range [0,1]x[0,1] to [-1,0]x[-1,0]
        case QuadTreeNames::SW:
          s[0] -= 1;
          s[1] -= 1;
          break;
 
          // If we're dealing with the bottom-right element we need to shift
          // the range [0,1]x[0,1] to [0,1]x[-1,0]
        case QuadTreeNames::SE:
          s[1] -= 1;
          break;
 
          // If we're dealing with the top-right element we need to shift the
          // range [0,1]x[0,1] to [0,1]x[0,1], i.e. nothing needs to be done
        case QuadTreeNames::NE:
          break;
 
          // If we're dealing with the top-left element we need to shift
          // the range [0,1]x[0,1] to [-1,0]x[0,1]
        case QuadTreeNames::NW:
          s[0] -= 1;
          break;
      }
    } // if son_type!=Tree::OMEGA
  } // End of level_up_local_coord_of_node
 
  //===================================================================
  //===================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::restrict_residual(const unsigned& level)
  {
#ifdef PARANOID
    // Check to make sure we can actually restrict the vector
    if (!(level < Nlevel - 1))
    {
      // Throw an error
      throw OomphLibError("Input level exceeds the possible parameter choice.",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Multiply the real part of the residual vector by the restriction
    // matrix on the level-th level
    Restriction_matrices_storage_pt[level]->multiply(
      Residual_mg_vectors_storage[level][0],
      Rhs_mg_vectors_storage[level + 1][0]);
 
    // Multiply the imaginary part of the residual vector by the restriction
    // matrix on the level-th level
    Restriction_matrices_storage_pt[level]->multiply(
      Residual_mg_vectors_storage[level][1],
      Rhs_mg_vectors_storage[level + 1][1]);
 
  } // End of restrict_residual
 
  //===================================================================
  //===================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::interpolate_and_correct(
    const unsigned& level)
  {
#ifdef PARANOID
    // Check to make sure we can actually restrict the vector
    if (!(level > 0))
    {
      // Throw an error
      throw OomphLibError("Input level exceeds the possible parameter choice.",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Build distribution of a temporary vector (real part)
    DoubleVector temp_soln_r(
      X_mg_vectors_storage[level - 1][0].distribution_pt());
 
    // Build distribution of a temporary vector (imaginary part)
    DoubleVector temp_soln_c(
      X_mg_vectors_storage[level - 1][1].distribution_pt());
 
    // Interpolate the solution vector
    Interpolation_matrices_storage_pt[level - 1]->multiply(
      X_mg_vectors_storage[level][0], temp_soln_r);
 
    // Interpolate the solution vector
    Interpolation_matrices_storage_pt[level - 1]->multiply(
      X_mg_vectors_storage[level][1], temp_soln_c);
 
    // Update the real part of the solution
    X_mg_vectors_storage[level - 1][0] += temp_soln_r;
 
    // Update the imaginary part of the solution
    X_mg_vectors_storage[level - 1][1] += temp_soln_c;
 
  } // End of interpolate_and_correct
 
  //===================================================================
  //===================================================================
  template<unsigned DIM>
  void HelmholtzMGPreconditioner<DIM>::mg_solve(Vector<DoubleVector>& result)
  {
    // If we're allowed to time things
    double t_mg_start = 0.0;
    if (!Suppress_v_cycle_output)
    {
      // Start the clock!
      t_mg_start = TimingHelpers::timer();
    }
 
    // Current level
    unsigned finest_level = 0;
 
    // Initialise the V-cycle counter
    V_cycle_counter = 0;
 
    // Calculate the norm of the residual then output
    double normalised_residual_norm = residual_norm(finest_level);
    if (!Suppress_v_cycle_output)
    {
      oomph_info << "\nResidual on finest level for V-cycle: "
                 << normalised_residual_norm << std::endl;
    }
 
    // Outer loop over V-cycles
    //-------------------------
    // While the tolerance is not met and the maximum number of
    // V-cycles has not been completed
    while ((normalised_residual_norm > Tolerance) &&
           (V_cycle_counter != Nvcycle))
    {
      // If the user did not wish to suppress the V-cycle output
      if (!Suppress_v_cycle_output)
      {
        // Output the V-cycle counter
        oomph_info << "\nStarting V-cycle: " << V_cycle_counter << std::endl;
      }
 
      //---------------------------------------------------------------------
      // Loop downwards over all levels that have coarser levels beneath them
      //---------------------------------------------------------------------
      for (unsigned i = 0; i < Nlevel - 1; i++)
      {
        // Initialise X_mg and Residual_mg to 0.0 except for the finest level
        // since X_mg contains the current approximation to the solution and
        // Residual_mg contains the RHS vector on the finest level
        if (i != 0)
        {
          // Initialise the real part of the solution vector
          X_mg_vectors_storage[i][0].initialise(0.0);
 
          // Initialise the imaginary part of the solution vector
          X_mg_vectors_storage[i][1].initialise(0.0);
 
          // Initialise the real part of the residual vector
          Residual_mg_vectors_storage[i][0].initialise(0.0);
 
          // Initialise the imaginary part of the residual vector
          Residual_mg_vectors_storage[i][1].initialise(0.0);
        }
 
        // Perform a few pre-smoothing steps and return vector that contains
        // the residuals of the linear system at this level.
        pre_smooth(i);
 
        // Restrict the residual to the next coarser mesh and
        // assign it to the RHS vector at that level
        restrict_residual(i);
 
      } // Moving down the V-cycle
 
      //-----------------------------------------------------------
      // Reached the lowest level: Do a direct solve, using the RHS
      // vector obtained by restriction from above.
      //-----------------------------------------------------------
      direct_solve();
 
      //---------------------------------------------------------------
      // Loop upwards over all levels that have finer levels above them
      //---------------------------------------------------------------
      for (unsigned i = Nlevel - 1; i > 0; i--)
      {
        // Interpolate solution at current level onto
        // next finer mesh and correct the solution x at that level
        interpolate_and_correct(i);
 
        // Perform a few post-smoothing steps (ignore
        // vector that contains the residuals of the linear system
        // at this level)
        post_smooth(i - 1);
      }
 
      // Set counter for number of cycles (for doc)
      V_cycle_counter++;
 
      // Calculate the new residual norm then output (if allowed)
      normalised_residual_norm = residual_norm(finest_level);
 
      // Print the residual on the finest level
      if (!Suppress_v_cycle_output)
      {
        oomph_info << "Residual on finest level of V-cycle: "
                   << normalised_residual_norm << std::endl;
      }
    } // End of the V-cycles
 
    // Copy the solution into the result vector
    result = X_mg_vectors_storage[finest_level];
 
    // Need an extra line space if V-cycle output is suppressed
    if (!Suppress_v_cycle_output)
    {
      oomph_info << std::endl;
    }
 
    // If all output is to be suppressed
    if (!Suppress_all_output)
    {
      // Output number of V-cycles taken to solve
      if (normalised_residual_norm < Tolerance)
      {
        Has_been_solved = true;
      }
    } // if (!Suppress_all_output)
 
    // If the V-cycle output isn't suppressed
    if (!Suppress_v_cycle_output)
    {
      // Stop the clock
      double t_mg_end = TimingHelpers::timer();
      double total_G_setup_time = double(t_mg_end - t_mg_start);
      oomph_info << "CPU time for MG solver [sec]: " << total_G_setup_time
                 << std::endl;
    }
  } // end of mg_solve
 
} // End of namespace oomph
 
#endif