]> https://gitweb.dealii.org/ - dealii.git/commitdiff
Add new test case (an interesting one).
authorWolfgang Bangerth <bangerth@math.tamu.edu>
Fri, 24 Apr 1998 08:12:43 +0000 (08:12 +0000)
committerWolfgang Bangerth <bangerth@math.tamu.edu>
Fri, 24 Apr 1998 08:12:43 +0000 (08:12 +0000)
git-svn-id: https://svn.dealii.org/trunk@189 0785d39b-7218-0410-832d-ea1e28bc413d

deal.II/deal.II/Attic/examples/error-estimation/error-estimation.cc [new file with mode: 0644]
tests/big-tests/error-estimation/error-estimation.cc [new file with mode: 0644]

diff --git a/deal.II/deal.II/Attic/examples/error-estimation/error-estimation.cc b/deal.II/deal.II/Attic/examples/error-estimation/error-estimation.cc
new file mode 100644 (file)
index 0000000..5ea3622
--- /dev/null
@@ -0,0 +1,382 @@
+/* $Id$ */
+
+#include <grid/tria.h>
+#include <grid/dof.h>
+#include <grid/tria_accessor.h>
+#include <grid/dof_accessor.h>
+#include <grid/tria_iterator.h>
+#include <grid/tria_boundary.h>
+#include <grid/dof_constraints.h>
+#include <basic/function.h>
+#include <basic/data_io.h>
+#include <fe/fe_lib.h>
+#include <fe/quadrature_lib.h>
+#include <numerics/base.h>
+#include <numerics/assembler.h>
+
+
+#include <map.h>
+#include <fstream.h>
+#include <cmath>
+#include <string>
+extern "C" {
+#  include <stdlib.h>
+}
+
+
+
+
+template <int dim>
+class PoissonEquation :  public Equation<dim> {
+  public:
+    PoissonEquation (const Function<dim> &rhs) :
+                   Equation<dim>(1),
+                   right_hand_side (rhs)  {};
+
+    virtual void assemble (dFMatrix            &cell_matrix,
+                          dVector             &rhs,
+                          const FEValues<dim> &fe_values,
+                          const Triangulation<dim>::cell_iterator &cell) const;
+    virtual void assemble (dFMatrix            &cell_matrix,
+                          const FEValues<dim> &fe_values,
+                          const Triangulation<dim>::cell_iterator &cell) const;
+    virtual void assemble (dVector             &rhs,
+                          const FEValues<dim> &fe_values,
+                          const Triangulation<dim>::cell_iterator &cell) const;
+  protected:
+    const Function<dim> &right_hand_side;
+};
+
+
+
+
+
+
+template <int dim>
+class PoissonProblem : public ProblemBase<dim> {
+  public:
+    PoissonProblem ();
+
+    void clear ();
+    void create_new ();
+    void run (unsigned int level);
+    void print_history () const;
+    
+  protected:
+    Triangulation<dim> *tria;
+    DoFHandler<dim>    *dof;
+    
+    Function<dim>      *rhs;
+    Function<dim>      *boundary_values;
+
+    vector<double> l1_error, l2_error, linfty_error, h1_seminorm_error, h1_error;
+    vector<int>    n_dofs;
+};
+
+
+
+
+
+/**
+  Right hand side constructed such that the exact solution is
+  $x*y*exp(-(x**2+y**2)*10)$.
+  */
+template <int dim>
+class RHSPoly : public Function<dim> {
+  public:
+                                    /**
+                                     * Return the value of the function
+                                     * at the given point.
+                                     */
+    virtual double operator () (const Point<dim> &p) const;
+};
+
+
+
+template <int dim>
+class Solution : public Function<dim> {
+  public:
+                                    /**
+                                     * Return the value of the function
+                                     * at the given point.
+                                     */
+    virtual double operator () (const Point<dim> &p) const;
+                                    /**
+                                     * Return the gradient of the function
+                                     * at the given point.
+                                     */
+    virtual Point<dim> gradient (const Point<dim> &p) const;
+};
+
+
+
+
+double RHSPoly<2>::operator () (const Point<2> &p) const {
+  return (120.-400.*p.square())*p(0)*p(1)*exp(-10.*p.square());
+};
+
+
+
+double Solution<2>::operator () (const Point<2> &p) const {
+  return p(0)*p(1)*exp(-10*p.square());
+};
+
+
+Point<2> Solution<2>::gradient (const Point<2> &p) const {
+  return Point<2> ((1-20.*p(0)*p(0))*p(1)*exp(-10*p.square()),
+                  (1-20.*p(1)*p(1))*p(0)*exp(-10*p.square()));
+};
+
+  
+
+
+
+
+void PoissonEquation<2>::assemble (dFMatrix            &cell_matrix,
+                                  dVector             &rhs,
+                                  const FEValues<2>   &fe_values,
+                                  const Triangulation<2>::cell_iterator &) const {
+  for (unsigned int point=0; point<fe_values.n_quadrature_points; ++point)
+    for (unsigned int i=0; i<fe_values.total_dofs; ++i) 
+      {
+       for (unsigned int j=0; j<fe_values.total_dofs; ++j)
+         cell_matrix(i,j) += (fe_values.shape_grad(i,point) *
+                              fe_values.shape_grad(j,point)) *
+                             fe_values.JxW(point);
+       rhs(i) += fe_values.shape_value(i,point) *
+                 right_hand_side(fe_values.quadrature_point(point)) *
+                 fe_values.JxW(point);
+      };
+};
+
+
+
+template <int dim>
+void PoissonEquation<dim>::assemble (dFMatrix            &,
+                                    const FEValues<dim> &,
+                                    const Triangulation<dim>::cell_iterator &) const {
+  Assert (false, ExcPureVirtualFunctionCalled());
+};
+
+
+
+template <int dim>
+void PoissonEquation<dim>::assemble (dVector             &,
+                                    const FEValues<dim> &,
+                                    const Triangulation<dim>::cell_iterator &) const {
+  Assert (false, ExcPureVirtualFunctionCalled());
+};
+
+
+
+
+
+
+
+
+
+template <int dim>
+PoissonProblem<dim>::PoissonProblem () :
+               tria(0), dof(0), rhs(0), boundary_values(0) {};
+
+
+
+
+template <int dim>
+void PoissonProblem<dim>::clear () {
+  if (tria != 0) {
+    delete tria;
+    tria = 0;
+  };
+  
+  if (dof != 0) {
+    delete dof;
+    dof = 0;
+  };
+
+  if (rhs != 0) 
+    {
+      delete rhs;
+      rhs = 0;
+    };
+
+  if (boundary_values != 0) 
+    {
+      delete boundary_values;
+      boundary_values = 0;
+    };
+
+  ProblemBase<dim>::clear ();
+};
+
+
+
+
+template <int dim>
+void PoissonProblem<dim>::create_new () {
+  clear ();
+  
+  tria = new Triangulation<dim>();
+  dof = new DoFHandler<dim> (tria);
+  set_tria_and_dof (tria, dof);
+};
+
+
+
+
+
+
+template <int dim>
+void PoissonProblem<dim>::run (const unsigned int level) {
+  create_new ();
+  
+  cout << "Refinement level = " << level
+       << endl;
+  
+  cout << "    Making grid... ";
+  tria->create_hypercube (-1,+1);
+  tria->refine_global (level);
+  cout << tria->n_active_cells() << " active cells." << endl;
+
+  rhs             = new RHSPoly<dim>();
+  boundary_values = new Solution<dim> ();
+  
+  
+  FELinear<dim>                   fe;
+  PoissonEquation<dim>            equation (*rhs);
+  QGauss3<dim>                    quadrature;
+  
+  cout << "    Distributing dofs... "; 
+  dof->distribute_dofs (fe);
+  cout << dof->n_dofs() << " degrees of freedom." << endl;
+  n_dofs.push_back (dof->n_dofs());
+
+  cout << "    Assembling matrices..." << endl;
+  UpdateFields update_flags = UpdateFields(update_q_points  | update_gradients |
+                                          update_jacobians | update_JxW_values);
+  
+  ProblemBase<dim>::DirichletBC dirichlet_bc;
+  dirichlet_bc[0] = boundary_values;
+  assemble (equation, quadrature, fe, update_flags, dirichlet_bc);
+
+  cout << "    Solving..." << endl;
+  solve ();
+
+  Solution<dim> sol;
+  dVector       l1_error_per_cell, l2_error_per_cell, linfty_error_per_cell;
+  dVector       h1_seminorm_error_per_cell, h1_error_per_cell;
+  QGauss3<dim>  q;
+  
+  cout << "    Calculating L1 error... ";
+  integrate_difference (sol, l1_error_per_cell, q, fe, L1_norm);
+  cout << l1_error_per_cell.l1_norm() << endl;
+  l1_error.push_back (l1_error_per_cell.l1_norm());
+
+  cout << "    Calculating L2 error... ";
+  integrate_difference (sol, l2_error_per_cell, q, fe, L2_norm);
+  cout << l2_error_per_cell.l2_norm() << endl;
+  l2_error.push_back (l2_error_per_cell.l2_norm());
+
+  cout << "    Calculating L-infinity error... ";
+  integrate_difference (sol, linfty_error_per_cell, q, fe, Linfty_norm);
+  cout << linfty_error_per_cell.linfty_norm() << endl;
+  linfty_error.push_back (linfty_error_per_cell.linfty_norm());
+  
+  cout << "    Calculating H1-seminorm error... ";
+  integrate_difference (sol, h1_seminorm_error_per_cell, q, fe, H1_seminorm);
+  cout << h1_seminorm_error_per_cell.l2_norm() << endl;
+  h1_seminorm_error.push_back (h1_seminorm_error_per_cell.l2_norm());
+
+  cout << "    Calculating H1 error... ";
+  integrate_difference (sol, h1_error_per_cell, q, fe, H1_norm);
+  cout << h1_error_per_cell.l2_norm() << endl;
+  h1_error.push_back (h1_error_per_cell.l2_norm());
+
+  if (level<=5) 
+    {
+      dVector l1_error_per_dof, l2_error_per_dof, linfty_error_per_dof;
+      dVector h1_seminorm_error_per_dof, h1_error_per_dof;
+      dof->distribute_cell_to_dof_vector (l1_error_per_cell, l1_error_per_dof);
+      dof->distribute_cell_to_dof_vector (l2_error_per_cell, l2_error_per_dof);
+      dof->distribute_cell_to_dof_vector (linfty_error_per_cell, linfty_error_per_dof);
+      dof->distribute_cell_to_dof_vector (h1_seminorm_error_per_cell, h1_seminorm_error_per_dof);
+      dof->distribute_cell_to_dof_vector (h1_error_per_cell, h1_error_per_dof);
+
+      string filename = "gnuplot.";
+      filename += ('0'+level);
+      cout << "    Writing error plots to <" << filename << ">..." << endl;
+      
+      DataOut<dim> out;
+      ofstream gnuplot(filename.c_str());
+      fill_data (out);
+      out.add_data_vector (l1_error_per_dof, "L1-Error");
+      out.add_data_vector (l2_error_per_dof, "L2-Error");
+      out.add_data_vector (linfty_error_per_dof, "Linfty-Error");
+      out.add_data_vector (h1_seminorm_error_per_dof, "H1-seminorm-Error");
+      out.add_data_vector (h1_error_per_dof, "H1-Error");
+      out.write_gnuplot (gnuplot);
+      gnuplot.close ();
+    }
+  else
+    cout << "    Not writing error as grid." << endl;
+  
+  cout << endl;
+};
+
+
+template <int dim>
+void PoissonProblem<dim>::print_history () const {
+  ofstream out("gnuplot.history");
+  out << "# n_dofs    l1_error l2_error linfty_error h1_seminorm_error h1_error"
+      << endl;
+  for (unsigned int i=0; i<n_dofs.size(); ++i)
+    out << n_dofs[i]
+       << "    "
+       << l1_error[i] << "  "
+       << l2_error[i] << "  "
+       << linfty_error[i] << "  "
+       << h1_seminorm_error[i] << "  "
+       << h1_error[i] << endl;
+
+  double average_l1=0,
+        average_l2=0,
+     average_linfty=0,
+    average_h1_semi=0,
+        average_h1=0;
+  for (unsigned int i=1; i<n_dofs.size(); ++i) 
+    {
+      average_l1 += l1_error[i]/l1_error[i-1];
+      average_l2 += l2_error[i]/l2_error[i-1];
+      average_linfty += linfty_error[i]/linfty_error[i-1];
+      average_h1_semi += h1_seminorm_error[i]/h1_seminorm_error[i-1];
+      average_h1 += h1_error[i]/h1_error[i-1];
+    };
+
+  average_l1 /= (l1_error.size()-1);
+  average_l2 /= (l1_error.size()-1);
+  average_linfty /= (l1_error.size()-1);
+  average_h1_semi /= (l1_error.size()-1);
+  average_h1 /= (l1_error.size()-1);
+
+  cout << "Average error reduction rates for h->h/2:" << endl;
+  cout << "    L1 error         : " << 1./average_l1 << endl
+       << "    L2 error         : " << 1./average_l2 << endl
+       << "    Linfty error     : " << 1./average_linfty << endl
+       << "    H1 seminorm error: " << 1./average_h1_semi << endl
+       << "    H1 error         : " << 1./average_h1 << endl;
+};
+
+
+
+
+int main () {
+  PoissonProblem<2> problem;
+
+  for (unsigned int level=3; level<7; ++level)
+    problem.run (level);
+
+  cout << endl << "Printing convergence history to <gnuplot.history>..." << endl;
+  problem.print_history ();
+
+  return 0;
+};
diff --git a/tests/big-tests/error-estimation/error-estimation.cc b/tests/big-tests/error-estimation/error-estimation.cc
new file mode 100644 (file)
index 0000000..5ea3622
--- /dev/null
@@ -0,0 +1,382 @@
+/* $Id$ */
+
+#include <grid/tria.h>
+#include <grid/dof.h>
+#include <grid/tria_accessor.h>
+#include <grid/dof_accessor.h>
+#include <grid/tria_iterator.h>
+#include <grid/tria_boundary.h>
+#include <grid/dof_constraints.h>
+#include <basic/function.h>
+#include <basic/data_io.h>
+#include <fe/fe_lib.h>
+#include <fe/quadrature_lib.h>
+#include <numerics/base.h>
+#include <numerics/assembler.h>
+
+
+#include <map.h>
+#include <fstream.h>
+#include <cmath>
+#include <string>
+extern "C" {
+#  include <stdlib.h>
+}
+
+
+
+
+template <int dim>
+class PoissonEquation :  public Equation<dim> {
+  public:
+    PoissonEquation (const Function<dim> &rhs) :
+                   Equation<dim>(1),
+                   right_hand_side (rhs)  {};
+
+    virtual void assemble (dFMatrix            &cell_matrix,
+                          dVector             &rhs,
+                          const FEValues<dim> &fe_values,
+                          const Triangulation<dim>::cell_iterator &cell) const;
+    virtual void assemble (dFMatrix            &cell_matrix,
+                          const FEValues<dim> &fe_values,
+                          const Triangulation<dim>::cell_iterator &cell) const;
+    virtual void assemble (dVector             &rhs,
+                          const FEValues<dim> &fe_values,
+                          const Triangulation<dim>::cell_iterator &cell) const;
+  protected:
+    const Function<dim> &right_hand_side;
+};
+
+
+
+
+
+
+template <int dim>
+class PoissonProblem : public ProblemBase<dim> {
+  public:
+    PoissonProblem ();
+
+    void clear ();
+    void create_new ();
+    void run (unsigned int level);
+    void print_history () const;
+    
+  protected:
+    Triangulation<dim> *tria;
+    DoFHandler<dim>    *dof;
+    
+    Function<dim>      *rhs;
+    Function<dim>      *boundary_values;
+
+    vector<double> l1_error, l2_error, linfty_error, h1_seminorm_error, h1_error;
+    vector<int>    n_dofs;
+};
+
+
+
+
+
+/**
+  Right hand side constructed such that the exact solution is
+  $x*y*exp(-(x**2+y**2)*10)$.
+  */
+template <int dim>
+class RHSPoly : public Function<dim> {
+  public:
+                                    /**
+                                     * Return the value of the function
+                                     * at the given point.
+                                     */
+    virtual double operator () (const Point<dim> &p) const;
+};
+
+
+
+template <int dim>
+class Solution : public Function<dim> {
+  public:
+                                    /**
+                                     * Return the value of the function
+                                     * at the given point.
+                                     */
+    virtual double operator () (const Point<dim> &p) const;
+                                    /**
+                                     * Return the gradient of the function
+                                     * at the given point.
+                                     */
+    virtual Point<dim> gradient (const Point<dim> &p) const;
+};
+
+
+
+
+double RHSPoly<2>::operator () (const Point<2> &p) const {
+  return (120.-400.*p.square())*p(0)*p(1)*exp(-10.*p.square());
+};
+
+
+
+double Solution<2>::operator () (const Point<2> &p) const {
+  return p(0)*p(1)*exp(-10*p.square());
+};
+
+
+Point<2> Solution<2>::gradient (const Point<2> &p) const {
+  return Point<2> ((1-20.*p(0)*p(0))*p(1)*exp(-10*p.square()),
+                  (1-20.*p(1)*p(1))*p(0)*exp(-10*p.square()));
+};
+
+  
+
+
+
+
+void PoissonEquation<2>::assemble (dFMatrix            &cell_matrix,
+                                  dVector             &rhs,
+                                  const FEValues<2>   &fe_values,
+                                  const Triangulation<2>::cell_iterator &) const {
+  for (unsigned int point=0; point<fe_values.n_quadrature_points; ++point)
+    for (unsigned int i=0; i<fe_values.total_dofs; ++i) 
+      {
+       for (unsigned int j=0; j<fe_values.total_dofs; ++j)
+         cell_matrix(i,j) += (fe_values.shape_grad(i,point) *
+                              fe_values.shape_grad(j,point)) *
+                             fe_values.JxW(point);
+       rhs(i) += fe_values.shape_value(i,point) *
+                 right_hand_side(fe_values.quadrature_point(point)) *
+                 fe_values.JxW(point);
+      };
+};
+
+
+
+template <int dim>
+void PoissonEquation<dim>::assemble (dFMatrix            &,
+                                    const FEValues<dim> &,
+                                    const Triangulation<dim>::cell_iterator &) const {
+  Assert (false, ExcPureVirtualFunctionCalled());
+};
+
+
+
+template <int dim>
+void PoissonEquation<dim>::assemble (dVector             &,
+                                    const FEValues<dim> &,
+                                    const Triangulation<dim>::cell_iterator &) const {
+  Assert (false, ExcPureVirtualFunctionCalled());
+};
+
+
+
+
+
+
+
+
+
+template <int dim>
+PoissonProblem<dim>::PoissonProblem () :
+               tria(0), dof(0), rhs(0), boundary_values(0) {};
+
+
+
+
+template <int dim>
+void PoissonProblem<dim>::clear () {
+  if (tria != 0) {
+    delete tria;
+    tria = 0;
+  };
+  
+  if (dof != 0) {
+    delete dof;
+    dof = 0;
+  };
+
+  if (rhs != 0) 
+    {
+      delete rhs;
+      rhs = 0;
+    };
+
+  if (boundary_values != 0) 
+    {
+      delete boundary_values;
+      boundary_values = 0;
+    };
+
+  ProblemBase<dim>::clear ();
+};
+
+
+
+
+template <int dim>
+void PoissonProblem<dim>::create_new () {
+  clear ();
+  
+  tria = new Triangulation<dim>();
+  dof = new DoFHandler<dim> (tria);
+  set_tria_and_dof (tria, dof);
+};
+
+
+
+
+
+
+template <int dim>
+void PoissonProblem<dim>::run (const unsigned int level) {
+  create_new ();
+  
+  cout << "Refinement level = " << level
+       << endl;
+  
+  cout << "    Making grid... ";
+  tria->create_hypercube (-1,+1);
+  tria->refine_global (level);
+  cout << tria->n_active_cells() << " active cells." << endl;
+
+  rhs             = new RHSPoly<dim>();
+  boundary_values = new Solution<dim> ();
+  
+  
+  FELinear<dim>                   fe;
+  PoissonEquation<dim>            equation (*rhs);
+  QGauss3<dim>                    quadrature;
+  
+  cout << "    Distributing dofs... "; 
+  dof->distribute_dofs (fe);
+  cout << dof->n_dofs() << " degrees of freedom." << endl;
+  n_dofs.push_back (dof->n_dofs());
+
+  cout << "    Assembling matrices..." << endl;
+  UpdateFields update_flags = UpdateFields(update_q_points  | update_gradients |
+                                          update_jacobians | update_JxW_values);
+  
+  ProblemBase<dim>::DirichletBC dirichlet_bc;
+  dirichlet_bc[0] = boundary_values;
+  assemble (equation, quadrature, fe, update_flags, dirichlet_bc);
+
+  cout << "    Solving..." << endl;
+  solve ();
+
+  Solution<dim> sol;
+  dVector       l1_error_per_cell, l2_error_per_cell, linfty_error_per_cell;
+  dVector       h1_seminorm_error_per_cell, h1_error_per_cell;
+  QGauss3<dim>  q;
+  
+  cout << "    Calculating L1 error... ";
+  integrate_difference (sol, l1_error_per_cell, q, fe, L1_norm);
+  cout << l1_error_per_cell.l1_norm() << endl;
+  l1_error.push_back (l1_error_per_cell.l1_norm());
+
+  cout << "    Calculating L2 error... ";
+  integrate_difference (sol, l2_error_per_cell, q, fe, L2_norm);
+  cout << l2_error_per_cell.l2_norm() << endl;
+  l2_error.push_back (l2_error_per_cell.l2_norm());
+
+  cout << "    Calculating L-infinity error... ";
+  integrate_difference (sol, linfty_error_per_cell, q, fe, Linfty_norm);
+  cout << linfty_error_per_cell.linfty_norm() << endl;
+  linfty_error.push_back (linfty_error_per_cell.linfty_norm());
+  
+  cout << "    Calculating H1-seminorm error... ";
+  integrate_difference (sol, h1_seminorm_error_per_cell, q, fe, H1_seminorm);
+  cout << h1_seminorm_error_per_cell.l2_norm() << endl;
+  h1_seminorm_error.push_back (h1_seminorm_error_per_cell.l2_norm());
+
+  cout << "    Calculating H1 error... ";
+  integrate_difference (sol, h1_error_per_cell, q, fe, H1_norm);
+  cout << h1_error_per_cell.l2_norm() << endl;
+  h1_error.push_back (h1_error_per_cell.l2_norm());
+
+  if (level<=5) 
+    {
+      dVector l1_error_per_dof, l2_error_per_dof, linfty_error_per_dof;
+      dVector h1_seminorm_error_per_dof, h1_error_per_dof;
+      dof->distribute_cell_to_dof_vector (l1_error_per_cell, l1_error_per_dof);
+      dof->distribute_cell_to_dof_vector (l2_error_per_cell, l2_error_per_dof);
+      dof->distribute_cell_to_dof_vector (linfty_error_per_cell, linfty_error_per_dof);
+      dof->distribute_cell_to_dof_vector (h1_seminorm_error_per_cell, h1_seminorm_error_per_dof);
+      dof->distribute_cell_to_dof_vector (h1_error_per_cell, h1_error_per_dof);
+
+      string filename = "gnuplot.";
+      filename += ('0'+level);
+      cout << "    Writing error plots to <" << filename << ">..." << endl;
+      
+      DataOut<dim> out;
+      ofstream gnuplot(filename.c_str());
+      fill_data (out);
+      out.add_data_vector (l1_error_per_dof, "L1-Error");
+      out.add_data_vector (l2_error_per_dof, "L2-Error");
+      out.add_data_vector (linfty_error_per_dof, "Linfty-Error");
+      out.add_data_vector (h1_seminorm_error_per_dof, "H1-seminorm-Error");
+      out.add_data_vector (h1_error_per_dof, "H1-Error");
+      out.write_gnuplot (gnuplot);
+      gnuplot.close ();
+    }
+  else
+    cout << "    Not writing error as grid." << endl;
+  
+  cout << endl;
+};
+
+
+template <int dim>
+void PoissonProblem<dim>::print_history () const {
+  ofstream out("gnuplot.history");
+  out << "# n_dofs    l1_error l2_error linfty_error h1_seminorm_error h1_error"
+      << endl;
+  for (unsigned int i=0; i<n_dofs.size(); ++i)
+    out << n_dofs[i]
+       << "    "
+       << l1_error[i] << "  "
+       << l2_error[i] << "  "
+       << linfty_error[i] << "  "
+       << h1_seminorm_error[i] << "  "
+       << h1_error[i] << endl;
+
+  double average_l1=0,
+        average_l2=0,
+     average_linfty=0,
+    average_h1_semi=0,
+        average_h1=0;
+  for (unsigned int i=1; i<n_dofs.size(); ++i) 
+    {
+      average_l1 += l1_error[i]/l1_error[i-1];
+      average_l2 += l2_error[i]/l2_error[i-1];
+      average_linfty += linfty_error[i]/linfty_error[i-1];
+      average_h1_semi += h1_seminorm_error[i]/h1_seminorm_error[i-1];
+      average_h1 += h1_error[i]/h1_error[i-1];
+    };
+
+  average_l1 /= (l1_error.size()-1);
+  average_l2 /= (l1_error.size()-1);
+  average_linfty /= (l1_error.size()-1);
+  average_h1_semi /= (l1_error.size()-1);
+  average_h1 /= (l1_error.size()-1);
+
+  cout << "Average error reduction rates for h->h/2:" << endl;
+  cout << "    L1 error         : " << 1./average_l1 << endl
+       << "    L2 error         : " << 1./average_l2 << endl
+       << "    Linfty error     : " << 1./average_linfty << endl
+       << "    H1 seminorm error: " << 1./average_h1_semi << endl
+       << "    H1 error         : " << 1./average_h1 << endl;
+};
+
+
+
+
+int main () {
+  PoissonProblem<2> problem;
+
+  for (unsigned int level=3; level<7; ++level)
+    problem.run (level);
+
+  cout << endl << "Printing convergence history to <gnuplot.history>..." << endl;
+  problem.print_history ();
+
+  return 0;
+};

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