]> https://gitweb.dealii.org/ - dealii.git/commitdiff
Complete program, and major part of documentation.
authorRalf Hartmann <Ralf.Hartmann@dlr.de>
Tue, 20 Nov 2001 16:23:23 +0000 (16:23 +0000)
committerRalf Hartmann <Ralf.Hartmann@dlr.de>
Tue, 20 Nov 2001 16:23:23 +0000 (16:23 +0000)
git-svn-id: https://svn.dealii.org/trunk@5220 0785d39b-7218-0410-832d-ea1e28bc413d

deal.II/examples/step-12/step-12.cc

index 741eb6c9ab6f958059fbb10f6973f14c38ca6c13..5bea1272c8d40f0ba7f203092fd941c2b92f510e 100644 (file)
 #include <numerics/data_out.h>
 #include <grid/grid_out.h>
 #include <grid/grid_refinement.h>
-#include <numerics/error_estimator.h>
 
+                                // This is the first new file. It
+                                // declares the MappingQ1 class that
+                                // gives the standard bilinear
+                                // mapping. For bilinear mappings use
+                                // an object of this class rather
+                                // than an object of the MappingQ(1)
+                                // class, as the MappingQ1 class is
+                                // optimized due to the
+                                // pre-knowledge of the actual
+                                // polynomial degree 1.
 #include <fe/mapping_q1.h>
+
+                                // Here the discontinuous finite
+                                // elements are defined. They are
+                                // used as all other finite elements.
 #include <fe/fe_dgq.h>
-#include <lac/precondition_block.h>
-#include <lac/solver_richardson.h>
 
+                                // We are going to use the simplest
+                                // possible solver, called richardson
+                                // iteration, that represents a simple
+                                // defect correction. This, in
+                                // combination with a block SSOR
+                                // preconditioner (defined in
+                                // precondition_block.h), that uses
+                                // the special block matrix structur
+                                // of system matrices arising from DG
+                                // discretizations.
+#include <lac/solver_richardson.h>
+#include <lac/precondition_block.h>
 
-#include <fstream>
+                                // We are going to use gradients as
+                                // refinement indicator.
+#include <numerics/derivative_approximation.h>
 
 
-template <int dim>
-class Beta
-{
-  public:
-    Beta () {};
+                                // Finally we do some time comparison
+                                // using the ``Timer'' class.
+#include <base/timer.h>
 
-    void value_list (const std::vector<Point<dim> > &points,
-                    std::vector<Point<dim> > &values) const;
-};
+                                // And this again is C++:
+#include <fstream>
 
 
+                                // First we define the class
+                                // representing the equation-specific
+                                // functions. Both classes, ``RHS''
+                                // and ``BoundaryValues'', are
+                                // derived from the Function
+                                // class. Only the ``value_list''
+                                // function are implemented because
+                                // only lists of function values are
+                                // computed rather than single
+                                // values.
 template <int dim>
 class RHS:  public Function<dim>
 {
@@ -56,90 +88,66 @@ class RHS:  public Function<dim>
 
 
 template <int dim>
-class BoundaryFunction:  public Function<dim>
+class BoundaryValues:  public Function<dim>
 {
   public:
-    BoundaryFunction() {};
+    BoundaryValues() {};
     
     virtual void value_list (const std::vector<Point<dim> > &points,
                             std::vector<double> &values,
                             const unsigned int component=0) const;
 };
 
+
+                                // The class ``Beta'' that represents
+                                // the vector valued flow field of
+                                // the linear transport equation is
+                                // not derived from the Function
+                                // class as we prefer to get function
+                                // values of type ``Point'' rather
+                                // than of type
+                                // ``Vector<double>''. This, because
+                                // there exist scalar products
+                                // between ``Point'' and ``Point'' as
+                                // well as between ``Point'' and
+                                // ``Tensor'', simplifying terms like
+                                // $\beta\cdot n$ and
+                                // $\beta\cdot\nabla v$.
 template <int dim>
-class DGAssembler
+class Beta
 {
   public:
-    DGAssembler() {};
+    Beta () {};
 
-    void assemble_cell_term(const FEValuesBase<dim>& fe_v,
-                           FullMatrix<double> &cell_matrix,
-                           Vector<double> &cell_vector);
-    
-    void assemble_face_term(const FEFaceValuesBase<dim>& fe_v,
-                           const FEFaceValuesBase<dim>& fe_v_neighbor,
-                           FullMatrix<double> &cell_matrix,
-                           FullMatrix<double> &cell_inflow_matrix,
-                           Vector<double> &cell_vector);
-    
-  private:
-    Beta<dim> beta_function;
-    RHS<dim> rhs_function;
-    BoundaryFunction<dim> boundary_function;
+    void value_list (const std::vector<Point<dim> > &points,
+                    std::vector<Point<dim> > &values) const;
 };
 
-                                // The main class is again almost
-                                // unchanged. Two additions, however,
-                                // are made: we have added the
-                                // ``refine'' function, which is used
-                                // to adaptively refine the grid
-                                // (instead of the global refinement
-                                // in the previous examples), and a
-                                // variable which will hold the
-                                // constraints associated to the
-                                // hanging nodes.
+
+                                // The implementation of the
+                                // ``value_list'' functions of these
+                                // classes are rather simple.  For
+                                // simplicity the right hand side is
+                                // set to be zero.
 template <int dim>
-class TransportProblem
+void RHS<dim>::value_list(const std::vector<Point<dim> > &,
+                         std::vector<double> &values,
+                         const unsigned int) const
 {
-  public:
-    TransportProblem ();
-    ~TransportProblem ();
-
-    void run ();
-    
-  private:
-    void setup_system ();
-    void assemble_system ();
-    void solve ();
-    void refine_grid ();
-    void output_results (const unsigned int cycle) const;
-
-    Triangulation<dim>   triangulation;
-    MappingQ1<dim>       mapping;
-    
-                                    // We need a finite element
-                                    // again. This time, we will want
-                                    // to use quadratic polynomials
-                                    // (but this is only specified in
-                                    // the constructor):
-    FE_DGQ<dim>          fe;
-    DoFHandler<dim>      dof_handler;
-
-    SparsityPattern      sparsity_pattern;
-    SparseMatrix<double> system_matrix;
-    
-    Vector<double>       solution;
-    Vector<double>       right_hand_side;
-
-    DGAssembler<dim>     dg_assembler;
-};
-
+  for (unsigned int i=0; i<values.size(); ++i)
+    values[i]=0;
+}
 
+                                // The flow field is chosen to be
+                                // circular, anticlockwise, and with
+                                // the origin as midpoint.
 template <>
 void Beta<2>::value_list(const std::vector<Point<2> > &points,
                         std::vector<Point<2> > &values) const
 {
-  Assert(values.size()==points.size(), ExcDimensionMismatch(values.size(),points.size()));
+  Assert(values.size()==points.size(),
+        ExcDimensionMismatch(values.size(),points.size()));
+
   for (unsigned int i=0; i<points.size(); ++i)
     {
       const Point<2> &p=points[i];
@@ -151,27 +159,23 @@ void Beta<2>::value_list(const std::vector<Point<2> > &points,
     }
 }
 
-
-
-
-template <int dim>
-void RHS<dim>::value_list(const std::vector<Point<dim> > &,
-                         std::vector<double> &values,
-                         const unsigned int) const
-{
-  for (unsigned int i=0; i<values.size(); ++i)
-    values[i]=0;
-}
-
-
-
-
+                                // Hence the inflow boundary of the
+                                // unit square [0,1]^2 are the right
+                                // and the lower boundaries. We
+                                // prescribe discontinuous boundary
+                                // values 1 and 0 on the x-axis and
+                                // value 0 on the right boundary. The
+                                // values of this function on the
+                                // outflow boundaries will not be
+                                // used within the DG scheme.
 template <int dim>
-void BoundaryFunction<dim>::value_list(const std::vector<Point<dim> > &points,
+void BoundaryValues<dim>::value_list(const std::vector<Point<dim> > &points,
                                       std::vector<double> &values,
                                       const unsigned int) const
 {
-  Assert(values.size()==points.size(), ExcDimensionMismatch(values.size(),points.size()));
+  Assert(values.size()==points.size(),
+        ExcDimensionMismatch(values.size(),points.size()));
+
   for (unsigned int i=0; i<values.size(); ++i)
     {
       if (points[i](0)<0.5)
@@ -181,29 +185,87 @@ void BoundaryFunction<dim>::value_list(const std::vector<Point<dim> > &points,
     }
 }
 
+                                // Next we define the equation-
+                                // dependent and DG-method-dependent
+                                // class ``DGTransportEquation''. Its
+                                // member functions were already
+                                // mentioned in the Introduction and
+                                // will be explained
+                                // below. Furthermore it includes
+                                // objects of the previously defined
+                                // ``Beta'', ``RHS'' and
+                                // ``BoundaryValues'' function
+                                // classes.
+template <int dim>
+class DGTransportEquation
+{
+  public:
+    DGTransportEquation() {};
 
+    void assemble_cell_term(const FEValues<dim>& fe_v,
+                           FullMatrix<double> &u_v_matrix,
+                           Vector<double> &cell_vector);
+    
+    void assemble_face_term1(const FEFaceValuesBase<dim>& fe_v,
+                            const FEFaceValuesBase<dim>& fe_v_neighbor,
+                            FullMatrix<double> &u_v_matrix,
+                            FullMatrix<double> &un_v_matrix,
+                            Vector<double> &cell_vector);
+
+    void assemble_face_term2(const FEFaceValuesBase<dim>& fe_v,
+                            const FEFaceValuesBase<dim>& fe_v_neighbor,
+                            FullMatrix<double> &u_v_matrix,
+                            FullMatrix<double> &un_v_matrix,
+                            FullMatrix<double> &u_vn_matrix,
+                            FullMatrix<double> &un_vn_matrix,
+                            Vector<double> &cell_vector);
+  private:
+    Beta<dim> beta_function;
+    RHS<dim> rhs_function;
+    BoundaryValues<dim> boundary_function;
+};
 
-
+                                // ``u_v_matrix'' is a cell matrix,
+                                // i.e. for a DG method of degree 1,
+                                // it is of size 4 times 4, and
+                                // ``cell_vector'' is of size 4.
+                                // When this function is invoked,
+                                // ``fe_v'' was reinited with the
+                                // current cell before and includes
+                                // all shape values needed.
 template <int dim>
-void DGAssembler<dim>::assemble_cell_term(const FEValuesBase<dim>& fe_v,
-                                         FullMatrix<double> &cell_matrix,
-                                         Vector<double> &cell_vector)
+void DGTransportEquation<dim>::assemble_cell_term(
+  const FEValues<dim>& fe_v,
+  FullMatrix<double> &u_v_matrix,
+  Vector<double> &cell_vector)
 {
+                                  // First we ask ``fe_v'' for the
+                                  // shape grads, shape values and
+                                  // quadrature weights,
   const vector<vector<Tensor<1,2> > > &grad_v = fe_v.get_shape_grads ();
   const FullMatrix<double> &v = fe_v.get_shape_values ();
   const vector<double> &JxW = fe_v.get_JxW_values ();
 
+                                  // Then the flow field beta and the
+                                  // ``rhs_function'' are evaluated at
+                                  // the quadrature points,
   vector<Point<dim> > beta (fe_v.n_quadrature_points);
   vector<double> rhs (fe_v.n_quadrature_points);
   
   beta_function.value_list (fe_v.get_quadrature_points(), beta);
   rhs_function.value_list (fe_v.get_quadrature_points(), rhs);
   
+                                  // and the cell matrix and cell
+                                  // vector are assembled as in
+                                  // previous tutorial steps.  Here,
+                                  // the terms $-(u,\beta\cdot\nabla
+                                  // v)_K$ and $(f,v)_K$ are
+                                  // assembled.
   for (unsigned int point=0; point<fe_v.n_quadrature_points; ++point)
     for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i) 
       {
        for (unsigned int j=0; j<fe_v.dofs_per_cell; ++j)
-         cell_matrix(i,j) -= beta[point]*grad_v[i][point]*
+         u_v_matrix(i,j) -= beta[point]*grad_v[i][point]*
                              v(j,point) *
                              JxW[point];
        
@@ -212,40 +274,89 @@ void DGAssembler<dim>::assemble_cell_term(const FEValuesBase<dim>& fe_v,
 }
 
 
+                                // The ``assemble_face_term1''
+                                // function assembles the face terms
+                                // corresponding to the first version
+                                // of the DG method, cf. above. Then,
+                                // the face terms are given as a sum
+                                // of integrals over all cell
+                                // boundaries.
+                                //
+                                // When this function is invoked,
+                                // ``fe_v'' and ``fe_v_neighbor'' are
+                                // already reinited with the current
+                                // cell and the neighoring cell,
+                                // respectively, as well as with the
+                                // current face. Hence they provide
+                                // the inner and outer shape values
+                                // on the face.
+                                //
+                                // In addition to the cell matrix
+                                // ``u_v_matrix'' and the
+                                // ``cell_vector'' this function has
+                                // got a new argument
+                                // ``un_v_matrix'', that stores
+                                // contributions to the system matrix
+                                // that are based on outer values of
+                                // u, see $\hat u_h$ in the
+                                // Introduction, and inner values of
+                                // v, see $v_h$. Here we note that
+                                // ``un'' is the short notation for
+                                // ``u_neighbor'' and represents
+                                // $\hat u_h$.
 template <int dim>
-void DGAssembler<dim>::assemble_face_term(const FEFaceValuesBase<dim>& fe_v,
-                                         const FEFaceValuesBase<dim>& fe_v_neighbor,      
-                                         FullMatrix<double> &cell_matrix,
-                                         FullMatrix<double> &cell_inflow_matrix,
-                                         Vector<double> &cell_vector)
+void DGTransportEquation<dim>::assemble_face_term1(
+  const FEFaceValuesBase<dim>& fe_v,
+  const FEFaceValuesBase<dim>& fe_v_neighbor,      
+  FullMatrix<double> &u_v_matrix,
+  FullMatrix<double> &un_v_matrix,
+  Vector<double> &cell_vector)
 {
-  DoFHandler<dim>::face_iterator face=fe_v.get_face();
-  
+                                  // Again, we ask the FEValues
+                                  // objects for the shape values and
+                                  // the quadrature weights
   const FullMatrix<double> &v = fe_v.get_shape_values ();
   const FullMatrix<double> &v_neighbor = fe_v_neighbor.get_shape_values ();  
   const vector<double> &JxW = fe_v.get_JxW_values ();
+                                  // but also for the normals.
   const vector<Point<dim> > &normals = fe_v.get_normal_vectors ();
 
+                                  // We also evaluate the flow field
+                                  // at the quadrature points
   vector<Point<dim> > beta (fe_v.n_quadrature_points);
-  vector<double> g(fe_v.n_quadrature_points);
   
   beta_function.value_list (fe_v.get_quadrature_points(), beta);
 
+                                  // and the boundary values if the
+                                  // current face belongs to the
+                                  // boundary.
+  vector<double> g(fe_v.n_quadrature_points);
+  DoFHandler<dim>::face_iterator face=fe_v.get_face();
   if (face->at_boundary())
     boundary_function.value_list (fe_v.get_quadrature_points(), g);
 
+                                  // Then we assemble the cell matrix
+                                  // and cell vector according to the
+                                  // DG method given in the
+                                  // introduction.
   for (unsigned int point=0; point<fe_v.n_quadrature_points; ++point)
     {
       double beta_n=beta[point] * normals[point];
       if (beta_n>0)
+                                        // The term $(\beta\cdot n
+                                        // u,v)_{\partial K_+}$,
        for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i)
          for (unsigned int j=0; j<fe_v.dofs_per_cell; ++j)
-           cell_matrix(i,j) += beta_n *
-                               v(j,point) *
-                               v(i,point) *
-                               JxW[point];
+           u_v_matrix(i,j) += beta_n *
+                              v(j,point) *
+                              v(i,point) *
+                              JxW[point];
       else
        {
+                                          // at the boundary the term
+                                          // $(\beta\cdot n
+                                          // g,v)_{\partial
+                                          // K_-\cap\partial\Omega}$,
          if (face->at_boundary())
            for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i)
              cell_vector(i) -= beta_n *
@@ -253,26 +364,200 @@ void DGAssembler<dim>::assemble_face_term(const FEFaceValuesBase<dim>& fe_v,
                                v(i,point) *
                                JxW[point];
          else
+                                            // and on inner faces the
+                                            // term $(\beta\cdot n
+                                            // \hat u,v)_{\partial
+                                            // K_-}$
            for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i)
              for (unsigned int k=0; k<fe_v_neighbor.dofs_per_cell; ++k)
-               cell_inflow_matrix(i,k) += beta_n *
-                                          v_neighbor(k,point) *
-                                          v(i,point) *
-                                          JxW[point];
+               un_v_matrix(i,k) += beta_n *
+                                   v_neighbor(k,point) *
+                                   v(i,point) *
+                                   JxW[point];
        }
     }
 }
 
+                                // Now we look at the assembling
+                                // function that assembles the face
+                                // terms corresponding to the second
+                                // version of the DG method,
+                                // cf. above. Then, the face terms
+                                // are given as a sum of integrals
+                                // over all faces.  Here we need two
+                                // additional cell matrices
+                                // ``u_vn_matrix'' and
+                                // ``un_vn_matrix'' that will store
+                                // contributions due to terms
+                                // involving u and vn as well as un
+                                // and vn.
+template <int dim>
+void DGTransportEquation<dim>::assemble_face_term2(
+  const FEFaceValuesBase<dim>& fe_v,
+  const FEFaceValuesBase<dim>& fe_v_neighbor,      
+  FullMatrix<double> &u_v_matrix,
+  FullMatrix<double> &un_v_matrix,
+  FullMatrix<double> &u_vn_matrix,
+  FullMatrix<double> &un_vn_matrix,
+  Vector<double> &cell_vector)
+{
+                                  // the first few lines are the same
+  const FullMatrix<double> &v = fe_v.get_shape_values ();
+  const FullMatrix<double> &v_neighbor = fe_v_neighbor.get_shape_values ();  
+  const vector<double> &JxW = fe_v.get_JxW_values ();
+  const vector<Point<dim> > &normals = fe_v.get_normal_vectors ();
+
+  vector<Point<dim> > beta (fe_v.n_quadrature_points);
   
+  beta_function.value_list (fe_v.get_quadrature_points(), beta);
+
+  vector<double> g(fe_v.n_quadrature_points);
+  DoFHandler<dim>::face_iterator face=fe_v.get_face();
+  if (face->at_boundary())
+    boundary_function.value_list (fe_v.get_quadrature_points(), g);
+
+  for (unsigned int point=0; point<fe_v.n_quadrature_points; ++point)
+    {
+      double beta_n=beta[point] * normals[point];
+      if (beta_n>0)
+       {
+                                          // This terms we've already seen,
+         for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i)
+           for (unsigned int j=0; j<fe_v.dofs_per_cell; ++j)
+             u_v_matrix(i,j) += beta_n *
+                                v(j,point) *
+                                v(i,point) *
+                                JxW[point];
+
+                                          // on inner faces we
+                                          // additionally have the
+                                          // term $(\beta\cdot n
+                                          // u,\hat v)_{\partial K_+},
+         if (!face->at_boundary())
+           for (unsigned int k=0; k<fe_v_neighbor.dofs_per_cell; ++k)
+             for (unsigned int j=0; j<fe_v.dofs_per_cell; ++j)
+               u_vn_matrix(k,j) -= beta_n *
+                                   v(j,point) *
+                                   v_neighbor(k,point) *
+                                   JxW[point];
+       }
+      else
+       {
+                                          // this one we already know,
+         if (face->at_boundary())
+           for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i)
+             cell_vector(i) -= beta_n *
+                               g[point] *
+                               v(i,point) *
+                               JxW[point];
+         else
+           {
+                                              // this one also,
+             for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i)
+               for (unsigned int l=0; l<fe_v_neighbor.dofs_per_cell; ++l)
+                 un_v_matrix(i,l) += beta_n *
+                                     v_neighbor(l,point) *
+                                     v(i,point) *
+                                     JxW[point];
+
+                                              // and this is another
+                                              // new one:
+                                              // $(\beta\cdot n \hat
+                                              // u,\hat v)_{\partial
+                                              // K_-}$.
+             for (unsigned int k=0; k<fe_v_neighbor.dofs_per_cell; ++k)
+               for (unsigned int l=0; l<fe_v_neighbor.dofs_per_cell; ++l)
+                 un_vn_matrix(k,l) -= beta_n *
+                                      v_neighbor(l,point) *
+                                      v_neighbor(k,point) *
+                                      JxW[point];
+           }
+       }
+    }
+}
+
+
+                                // After these preparations, we
+                                // proceed with the main part of this
+                                // program. The main class, here
+                                // called ``DGMethod'' is basically
+                                // the main class of step 6. One of
+                                // the differences is that there's no
+                                // ConstraintMatrix object. This is,
+                                // because there are no hanging nodes
+                                // in DG discretizations.
+template <int dim>
+class DGMethod
+{
+  public:
+    DGMethod ();
+    ~DGMethod ();
+
+    void run ();
+    
+  private:
+    void setup_system ();
+    void assemble_system1 ();
+    void assemble_system2 ();
+    void solve (Vector<double> &solution);
+    void refine_grid ();
+    void output_results (const unsigned int cycle) const;
+    
+    Triangulation<dim>   triangulation;
+    MappingQ1<dim>       mapping;
+    
+                                    // Furthermore we want to
+                                    // use DG elements of degree 1
+                                    // (but this is only specified in
+                                    // the constructor):
+    FE_DGQ<dim>          fe;
+    DoFHandler<dim>      dof_handler;
+
+    SparsityPattern      sparsity_pattern;
+    SparseMatrix<double> system_matrix;
+
+                                    // We define the quadrature
+                                    // formulae for the cell and the
+                                    // face terms of the
+                                    // discretization.
+    QGauss4<dim>   quadrature;
+    QGauss4<dim-1> face_quadrature;
+    
+                                    // And there are two solution
+                                    // vectors, that store the
+                                    // solutions to the problems
+                                    // corresponding to the two
+                                    // different assembling routines
+                                    // ``assemble_system1'' and
+                                    // ``assemble_system2'';
+    Vector<double>       solution1;
+    Vector<double>       solution2;
+    Vector<double>       right_hand_side;
+    
+                                    // Finally this class includes an
+                                    // object of the
+                                    // DGTransportEquations class
+                                    // described above.
+    DGTransportEquation<dim>     dg;
+};
+
+
+
+                                // Now for the implementation of the
+                                // main class. Constructor and
+                                // destructor follow the same
+                                // pattern that was used previously,
+                                // so we need not comment on these
+                                // two functions:  
 template <int dim>
-TransportProblem<dim>::TransportProblem () :
+DGMethod<dim>::DGMethod () :
                 fe (1),
                dof_handler (triangulation)
 {}
 
 
 template <int dim>
-TransportProblem<dim>::~TransportProblem () 
+DGMethod<dim>::~DGMethod () 
 {
   dof_handler.clear ();
 };
@@ -280,172 +565,460 @@ TransportProblem<dim>::~TransportProblem ()
 
 
 template <int dim>
-void TransportProblem<dim>::setup_system ()
+void DGMethod<dim>::setup_system ()
 {
-                                  // To distribute degrees of
-                                  // freedom, the ``dof_handler''
-                                  // variable takes only the finite
-                                  // element object. In this case, it
-                                  // will distribute four degrees of
-                                  // freedom per cell.
+                                  // First we need to distribute the
+                                  // DoFs.
   dof_handler.distribute_dofs (fe);
 
+                                  // The DoFs of a cell are coupled
+                                  // with all DoFs of all neighboring
+                                  // cells.  Therefore the maximum
+                                  // number of matrix entries is
+                                  // needed when all neighbors of a
+                                  // cell are once more refined than
+                                  // the cell under consideration.
   sparsity_pattern.reinit (dof_handler.n_dofs(),
                           dof_handler.n_dofs(),
-                          dof_handler.max_couplings_between_dofs());
+                          (GeometryInfo<dim>::faces_per_cell
+                           *GeometryInfo<dim>::subfaces_per_face+1)*fe.dofs_per_cell);
+  
+                                  // For DG discretizations we call
+                                  // the function analogue to
+                                  // DoFTools::make_sparsity_pattern.
   DoFTools::make_flux_sparsity_pattern (dof_handler, sparsity_pattern);
+  
+                                  // All following function calls are
+                                  // already known.
   sparsity_pattern.compress();
   
   system_matrix.reinit (sparsity_pattern);
 
-  solution.reinit (dof_handler.n_dofs());
+  solution1.reinit (dof_handler.n_dofs());
+  solution2.reinit (dof_handler.n_dofs());
   right_hand_side.reinit (dof_handler.n_dofs());
 };
 
 
-
+                                // We proceed with the
+                                // ``assemble_system1'' function that
+                                // implements the DG discretization
+                                // in its first version. This
+                                // function repeatedly calls the
+                                // ``assemble_cell_term'' and
+                                // ``assemble_face_term1'' functions
+                                // of the DGTransportEquation object.
+                                // The ``assemble_face_term1''
+                                // function takes two
+                                // FEFaceValuesBase objects; one for
+                                // the shape functions on the current
+                                // cell and the other for shape
+                                // functions on the neighboring cell
+                                // under consideration. Both objects
+                                // are either of class FEFaceValues
+                                // or of class FESubfaceValues (both
+                                // derived from FEFaceValuesBase)
+                                // according to following cases
+                                // already mentioned in the
+                                // introduction:
+                                //
+                                // 1. face is at boundary (current
+                                // cell: FEFaceValues, neighboring
+                                // cell does not exist);
+                                //
+                                // 2. neighboring cell is finer
+                                // (current cell: FESubfaceValues,
+                                // neighboring cell: FEFaceValues);
+                                //
+                                // 3. neighboring cell is of the same
+                                // refinement level (both, current
+                                // and neighboring cell:
+                                // FEFaceValues);
+                                //
+                                // 4. neighboring cell is coarser
+                                // (current cell: FEFaceValues,
+                                // neighboring cell:
+                                // FESubfaceValues).
+                                //
+                                // If we considered globally refined
+                                // meshes then only cases 1 and 3
+                                // would occur. But as we consider
+                                // also locally refined meshes we
+                                // need to distinguish all four cases
+                                // making the following assembling
+                                // function a bit longish.
 template <int dim>
-void TransportProblem<dim>::assemble_system () 
+void DGMethod<dim>::assemble_system1 () 
 {
-                                  // See Cockburn paper for the proper quadrature.
-  QGauss4<dim>  quadrature;
-  QGauss4<dim-1>  face_quadrature;
-  
   const unsigned int dofs_per_cell = dof_handler.get_fe().dofs_per_cell;
   vector<unsigned int> dofs (dofs_per_cell);
   vector<unsigned int> dofs_neighbor (dofs_per_cell);
 
+                                  // First we create the Update flags
+                                  // for the FEValues and the
+                                  // FEFaceValues objects.
   UpdateFlags update_flags = UpdateFlags(update_values
                                         | update_gradients
                                         | update_q_points
                                         | update_JxW_values);
   
+                                  // Note, that on faces we do not
+                                  // need gradients but we need
+                                  // normal vectors.
   UpdateFlags face_update_flags = UpdateFlags(update_values
                                              | update_q_points
                                              | update_JxW_values
                                              | update_normal_vectors);
+  
+                                  // On the neighboring cell we only
+                                  // need the shape values. Given a
+                                  // specific face, the quadrature
+                                  // points and `JxW values' are the
+                                  // same as for the current cells,
+                                  // the normal vectors are known to
+                                  // be the negative of the normal
+                                  // vectors of the current cell.
+  UpdateFlags neighbor_face_update_flags = UpdateFlags(update_values);
    
-
+                                  // Then we create the FEValues
+                                  // object. Note, that since version
+                                  // 3.2.0 the constructor of this
+                                  // class takes a Mapping object as
+                                  // first argument. Although the
+                                  // constructor without Mapping
+                                  // argument is still supported it
+                                  // is recommended to use the new
+                                  // constructor. This reduces the
+                                  // effect of `hidden magic' (the
+                                  // old constructor implicitely
+                                  // assumes a MappingQ1 mapping) and
+                                  // makes it easier to change the
+                                  // Mapping object later.
   FEValues<dim> fe_v (
     mapping, fe, quadrature, update_flags);
+  
+                                  // Similarly we create the
+                                  // FEFaceValues and FESubfaceValues
+                                  // objects for both, the current
+                                  // and the neighboring cell. Within
+                                  // the following nested loop over
+                                  // all cells and all faces of the
+                                  // cell they will be reinited to
+                                  // the current cell and the face
+                                  // (and subface) number.
   FEFaceValues<dim> fe_v_face (
     mapping, fe, face_quadrature, face_update_flags);
   FESubfaceValues<dim> fe_v_subface (
     mapping, fe, face_quadrature, face_update_flags);
   FEFaceValues<dim> fe_v_face_neighbor (
-    mapping, fe, face_quadrature, UpdateFlags(update_values | update_default));
+    mapping, fe, face_quadrature, neighbor_face_update_flags);
   FESubfaceValues<dim> fe_v_subface_neighbor (
-    mapping, fe, face_quadrature, UpdateFlags(update_values | update_default));
-
-                                  // includes the u and v terms
-  FullMatrix<double> cell_matrix (dofs_per_cell, dofs_per_cell);
-                                  // includes u_hat and v terms
-  FullMatrix<double> cell_inflow_matrix (dofs_per_cell, dofs_per_cell);
+    mapping, fe, face_quadrature, neighbor_face_update_flags);
+
+                                  // Now we create the cell matrices
+                                  // and vectors. Here we need two
+                                  // cell matrices, both for face
+                                  // terms that include test
+                                  // functions ``v'' (shape functions
+                                  // of the current cell). To be more
+                                  // precise, the first matrix will
+                                  // include the `u and v terms' and
+                                  // the second that will include the
+                                  // `un and v terms'. Here we recall
+                                  // our the convention that `un' is
+                                  // the short cut for `u_neighbor'
+                                  // and represents the $u_hat$, see
+                                  // introduction.
+  FullMatrix<double> u_v_matrix (dofs_per_cell, dofs_per_cell);
+  FullMatrix<double> un_v_matrix (dofs_per_cell, dofs_per_cell);
 
   Vector<double>  cell_vector (dofs_per_cell);
 
+                                  // Furthermore we need some cell
+                                  // and face iterators
   DoFHandler<dim>::active_cell_iterator cell = dof_handler.begin_active(),
                                        endc = dof_handler.end();
   DoFHandler<dim>::face_iterator face;
   DoFHandler<dim>::cell_iterator neighbor;
-  DoFHandler<dim>::cell_iterator neighbor_child;
+  DoFHandler<dim>::active_cell_iterator neighbor_child;
 
+                                  // Now we start the loop over all
+                                  // active cells
   for (;cell!=endc; ++cell) 
     {
-                                      // re-init fe values for this cell
+                                      // and reinit the FEValues
+                                      // object for the current cell,
       fe_v.reinit (cell);
 
-      cell_matrix.clear ();
-      cell_vector.clear ();
-
-      dg_assembler.assemble_cell_term(fe_v,
-                                     cell_matrix,
-                                     cell_vector);
-      
+                                      // Call the function that
+                                      // assembles the cell
+                                      // terms. The first argument is
+                                      // the FEValues that was
+                                      // already reinited on current
+                                      // the cell.
+      dg.assemble_cell_term(fe_v,
+                           u_v_matrix,
+                           cell_vector);
+
+                                      // As in previous example steps
+                                      // the vector `dofs' includes
+                                      // the dof_indices.
       cell->get_dof_indices (dofs);
 
+                                      // This is the start of the
+                                      // nested loop over all faces.
       for (unsigned int face_no=0; face_no<GeometryInfo<dim>::faces_per_cell; ++face_no)
        {
+                                          // First we set the face
+                                          // iterator.
          face = cell->face(face_no);
-         
-         cell_inflow_matrix.clear();
 
+                                          // Now we distinguish the
+                                          // four different cases in
+                                          // the ordering mentioned
+                                          // above. We start with
+                                          // faces belonging to the
+                                          // boundary of the domain.
          if (face->at_boundary())
            {
+                                              // We reinit the
+                                              // FEFaceValues object
+                                              // to the current face
              fe_v_face.reinit (cell, face_no);
 
-             dg_assembler.assemble_face_term(fe_v_face,
-                                             fe_v_face,
-                                             cell_matrix,
-                                             cell_inflow_matrix,
-                                             cell_vector);
+                                              // and assemble the
+                                              // corresponding face
+                                              // terms. Here, the
+                                              // second and fourth
+                                              // arguments are only
+                                              // dummy arguments. On
+                                              // the boundary of the
+                                              // domain the
+                                              // ``assemble_face_term1''
+                                              // function will not
+                                              // access to shape
+                                              // values on the
+                                              // non-existent
+                                              // neighboring
+                                              // cell. Also,
+                                              // ``un_v_matrix'' will
+                                              // be unchanged.
+             dg.assemble_face_term1(fe_v_face,
+                                    fe_v_face,
+                                    u_v_matrix,
+                                    un_v_matrix,
+                                    cell_vector);
            }
-         else // if (!face->at_boundary())
+         else
            {
-             Assert (cell->neighbor(face_no).state() == valid, ExcInternalError());
+                                              // When we are not on the
+                                              // boundary of the
+                                              // domain then there
+                                              // must exist a
+                                              // neighboring cell.
              neighbor = cell->neighbor(face_no);
-             
-             if (face->has_children())  // i.e. neighbor is one level more refined than cell
+
+                                              // We proceed with the
+                                              // second and most
+                                              // complicated case:
+                                              // the neighboring cell
+                                              // is more refined than
+                                              // the current cell. As
+                                              // in deal.II
+                                              // neighboring cells
+                                              // are restricted to
+                                              // have a level
+                                              // difference of not
+                                              // more than one, the
+                                              // neighboring cell is
+                                              // known to be only
+                                              // ONCE more refined
+                                              // than the current
+                                              // cell. Furthermore
+                                              // also the face is
+                                              // once more refined,
+                                              // i.e. it has
+                                              // children.
+             if (face->has_children())
                {
-                                                  // store which number #cell# has in the
-                                                  // list of neighbors of #neighbor#
-                 const unsigned int neighbor2=cell->neighbor_of_neighbor(face_no);
+                                                  // first we store
+                                                  // which number the
+                                                  // current cell has
+                                                  // in the list of
+                                                  // neighbors of the
+                                                  // neighboring
+                                                  // cell. Hence,
+                                                  // neighbor->neighbor(neighbor2)
+                                                  // equals the
+                                                  // current cell
+                                                  // ``cell''.
+                 const unsigned int neighbor2=
+                   cell->neighbor_of_neighbor(face_no);
                  
                  
-                                                  // loop over all subfaces
-                 for (unsigned int subface_no=0; subface_no<GeometryInfo<dim>::subfaces_per_face;
+                                                  // We loop over
+                                                  // subfaces
+                 for (unsigned int subface_no=0;
+                      subface_no<GeometryInfo<dim>::subfaces_per_face;
                       ++subface_no)
                    {
-                                                      // get an iterator pointing to the
-                                                      // cell behind the present subface
+                                                      // and set the
+                                                      // cell
+                                                      // iterator
+                                                      // ``neighbor_child''
+                                                      // to the cell
+                                                      // placed
+                                                      // `behind' the
+                                                      // current
+                                                      // subface.
                      neighbor_child = neighbor->child(GeometryInfo<dim>::
                                                       child_cell_on_face(neighbor2,subface_no));
+
+                                                      // As these are
+                                                      // quite
+                                                      // complicated
+                                                      // indirections
+                                                      // we check for
+                                                      // the internal
+                                                      // consistency.
                      Assert (neighbor_child->face(neighbor2) == face->child(subface_no),
                              ExcInternalError());
                      Assert (!neighbor_child->has_children(), ExcInternalError());
 
+                                                      // As already
+                                                      // mentioned
+                                                      // above for
+                                                      // this case
+                                                      // (case 2) we
+                                                      // employ the
+                                                      // FESubfaceValues
+                                                      // of the
+                                                      // current
+                                                      // cell, here
+                                                      // reinited for
+                                                      // the current
+                                                      // cell, face
+                                                      // and subface,
+                                                      // and we
+                                                      // employ the
+                                                      // FEFaceValues
+                                                      // of the
+                                                      // neighboring
+                                                      // child cell.
                      fe_v_subface.reinit (cell, face_no, subface_no);
                      fe_v_face_neighbor.reinit (neighbor_child, neighbor2);
-       
-                     cell_inflow_matrix.clear();
-           
-                     dg_assembler.assemble_face_term(fe_v_subface,
-                                                     fe_v_face_neighbor,
-                                                     cell_matrix,
-                                                     cell_inflow_matrix,
-                                                     cell_vector);
-                 
-                                                      // get indices of dofs of neighbor_child cell
+
+                     dg.assemble_face_term1(fe_v_subface,
+                                            fe_v_face_neighbor,
+                                            u_v_matrix,
+                                            un_v_matrix,
+                                            cell_vector);
+                     
+                                                      // get dof
+                                                      // indices of
+                                                      // the
+                                                      // neighbor_child
+                                                      // cell
                      neighbor_child->get_dof_indices (dofs_neighbor);
                                                                
-                                                      // distribute cell matrix
+                                                      // distribute
+                                                      // cell matrix
+                                                      // to the
+                                                      // system_matrix
                      for (unsigned int i=0; i<dofs_per_cell; ++i)
                        for (unsigned int k=0; k<dofs_per_cell; ++k)
                          system_matrix.add(dofs[i], dofs_neighbor[k],
-                                           cell_inflow_matrix(i,k));
+                                           un_v_matrix(i,k));
+
+                                                      // In the
+                                                      // ``assemble_face_term1''
+                                                      // function contributions to
+                                                      // the cell matrices and the
+                                                      // cell vector are only
+                                                      // ADDED. Therefore on each
+                                                      // subface we need to reset the
+                                                      // un_v_matrix
+                                                      // to zero, before assembling
+                                                      // the face terms corresponding
+                                                      // to the following neighbor_child cell.
+                     un_v_matrix.clear();
                    }
                }
-             else // if (!face->has_children())
+                                              // End of ``if
+                                              // (face->has_children())''
+             else
                {
+                                                  // We proceed with
+                                                  // case 3,
+                                                  // i.e. neighboring
+                                                  // cell is of the
+                                                  // same refinement
+                                                  // level as the
+                                                  // current cell.
                  if (neighbor->level() == cell->level()) 
                    {
-                                                      // store which number #cell# has in the
-                                                      // list of neighbors of #neighbor#
+                                                  // Like before we
+                                                  // store which
+                                                  // number the
+                                                  // current cell has
+                                                  // in the list of
+                                                  // neighbors of the
+                                                  // neighboring
+                                                  // cell.
                      const unsigned int neighbor2=cell->neighbor_of_neighbor(face_no);
 
+                                                      // We reinit
+                                                      // the
+                                                      // FEFaceValues
+                                                      // of the
+                                                      // current and
+                                                      // neighboring
+                                                      // cell to the
+                                                      // current face
+                                                      // and assemble
+                                                      // the
+                                                      // corresponding
+                                                      // face terms.
                      fe_v_face.reinit (cell, face_no);
                      fe_v_face_neighbor.reinit (neighbor, neighbor2);
                      
-                     dg_assembler.assemble_face_term(fe_v_face,
-                                                     fe_v_face_neighbor,
-                                                     cell_matrix,
-                                                     cell_inflow_matrix,
-                                                     cell_vector);
+                     dg.assemble_face_term1(fe_v_face,
+                                            fe_v_face_neighbor,
+                                            u_v_matrix,
+                                            un_v_matrix,
+                                            cell_vector);
+                                                      // End of ``if
+                                                      // (neighbor->level()
+                                                      // ==
+                                                      // cell->level())''
                    }
-                 else // if (neighbor->level() < cell->level()) i.e. neighbor is one level coarser than cell
+                 else
                    {
+                                                      // Finally we
+                                                      // consider
+                                                      // case 4. When
+                                                      // the
+                                                      // neighboring
+                                                      // cell is not
+                                                      // finer and
+                                                      // not of the
+                                                      // same
+                                                      // refinement
+                                                      // level as the
+                                                      // current cell
+                                                      // it must be
+                                                      // coarser.
                      Assert(neighbor->level() < cell->level(), ExcInternalError());
 
+                                                      // Find out the
+                                                      // how many'th
+                                                      // face_no and
+                                                      // subface_no
+                                                      // the current
+                                                      // face is
+                                                      // w.r.t. the
+                                                      // neighboring
+                                                      // cell.
                      const std::pair<unsigned int, unsigned int> faceno_subfaceno=
                        cell->neighbor_of_coarser_neighbor(face_no);
                      const unsigned int neighbor_face_no=faceno_subfaceno.first,
@@ -454,127 +1027,460 @@ void TransportProblem<dim>::assemble_system ()
                      Assert (neighbor->neighbor(neighbor_face_no)
                              ->child(GeometryInfo<dim>::child_cell_on_face(
                                face_no,neighbor_subface_no)) == cell, ExcInternalError());
-                       
-                                                      // now 'neighbor_face_no' stores the number
-                                                      // of a face in the list of faces of 'neighbor'.
-                                                      // This face has got a subface that is 
-                                                      // between 'cell' and 'neighbor'.
-                                                      // 'neighbor_subface_no' stores the number
-                                                      // of this subface in the list of subfaces of this
-                                                      // face 'neighbor->face(neighbor_face_no)'
-                                                      // that is between 'cell' and 'neighbor'
+
+                                                      // Reinit the
+                                                      // appropriate
+                                                      // FEFaceValues
+                                                      // and assemble
+                                                      // the face
+                                                      // terms.
                      fe_v_face.reinit (cell, face_no);
                      fe_v_subface_neighbor.reinit (neighbor, neighbor_face_no,
                                                    neighbor_subface_no);
                      
-                     dg_assembler.assemble_face_term(fe_v_face,
-                                                     fe_v_subface_neighbor,
-                                                     cell_matrix,
-                                                     cell_inflow_matrix,
-                                                     cell_vector);
-                   } // else // if (neighbor->level() < cell->level())
-
-                                                  // get indices of dofs of neighbor_child cell
+                     dg.assemble_face_term1(fe_v_face,
+                                            fe_v_subface_neighbor,
+                                            u_v_matrix,
+                                            un_v_matrix,
+                                            cell_vector);
+                   }
+
+                                                  // Get dof indices
+                                                  // of the
+                                                  // neighbor_child
+                                                  // cell,
                  neighbor->get_dof_indices (dofs_neighbor);
                                                                
-                                                  // distribute cell_inflow_matrix
+                                                  // distribute the
+                                                  // un_v_matrix,
                  for (unsigned int i=0; i<dofs_per_cell; ++i)
                    for (unsigned int k=0; k<dofs_per_cell; ++k)
                      system_matrix.add(dofs[i], dofs_neighbor[k],
-                                       cell_inflow_matrix(i,k));
-               } // else // if (!face->has_children())
-           }  // else // if (!face->at_boundary())
-       } //for (face_no...)
+                                       un_v_matrix(i,k));
+                 
+                                                  // and clear the
+                                                  // ``un_v_matrix''
+                                                  // on each face.
+                 un_v_matrix.clear();
+               }
+                                              // End of ``face not at boundary'':
+           }
+                                          // End of loop over all faces:
+       }
       
-                                      // distribute cell matrix
+                                      // Finally we distribute the
+                                      // u_v_matrix,
       for (unsigned int i=0; i<dofs_per_cell; ++i)
        for (unsigned int j=0; j<dofs_per_cell; ++j)
-         system_matrix.add(dofs[i], dofs[j], cell_matrix(i,j));
+         system_matrix.add(dofs[i], dofs[j], u_v_matrix(i,j));
       
-                                      // distribute cell vector
+                                      // the cell vector
       for (unsigned int i=0; i<dofs_per_cell; ++i)
        right_hand_side(dofs[i]) += cell_vector(i);
-    }  // for (cell...)
+
+                                      // and clear them both.
+      u_v_matrix.clear ();
+      cell_vector.clear ();
+    }
 };
 
 
 
+                                // We proceed with the
+                                // ``assemble_system2'' function that
+                                // implements the DG discretization
+                                // in its second version. This
+                                // function is very similar to the
+                                // ``assemble_system1''
+                                // function. Therefore, here we only
+                                // discuss the differences between
+                                // the two functions. This function
+                                // repeatedly calls the
+                                // ``assemble_face_term2'' function
+                                // of the DGTransportEquation object,
+                                // that assembles the face terms
+                                // written as a sum of integrals over
+                                // all faces. Therefore, we need to
+                                // make sure that each face is
+                                // treated only once. This is achieved
+                                // by introducing the rule:
+                                // 
+                                // a) If the current and the
+                                // neighboring cells are of the same
+                                // refinement level we access and
+                                // treat the face from the cell with
+                                // lower index.
+                                //
+                                // b) If the two cells are of
+                                // different refinement levels we
+                                // access and treat the face from the
+                                // coarser cell.
+                                //
+                                // Due to rule b) we do not need to
+                                // consider case 4 (neighboring cell
+                                // is coarser) any more.
+
 template <int dim>
-void TransportProblem<dim>::solve () 
-{  
-  SolverControl           solver_control (1000, 1e-12);
+void DGMethod<dim>::assemble_system2 () 
+{
+  const unsigned int dofs_per_cell = dof_handler.get_fe().dofs_per_cell;
+  vector<unsigned int> dofs (dofs_per_cell);
+  vector<unsigned int> dofs_neighbor (dofs_per_cell);
+
+  UpdateFlags update_flags = UpdateFlags(update_values
+                                        | update_gradients
+                                        | update_q_points
+                                        | update_JxW_values);
+  
+  UpdateFlags face_update_flags = UpdateFlags(update_values
+                                             | update_q_points
+                                             | update_JxW_values
+                                             | update_normal_vectors);
+   
+  UpdateFlags neighbor_face_update_flags = UpdateFlags(update_values);
+
+                                  // Here we do not need
+                                  // ``fe_v_face_neighbor'' as case 4
+                                  // does not occur.
+  FEValues<dim> fe_v (
+    mapping, fe, quadrature, update_flags);
+  FEFaceValues<dim> fe_v_face (
+    mapping, fe, face_quadrature, face_update_flags);
+  FESubfaceValues<dim> fe_v_subface (
+    mapping, fe, face_quadrature, face_update_flags);
+  FEFaceValues<dim> fe_v_face_neighbor (
+    mapping, fe, face_quadrature, neighbor_face_update_flags);
+
+
+  FullMatrix<double> u_v_matrix (dofs_per_cell, dofs_per_cell);
+  FullMatrix<double> un_v_matrix (dofs_per_cell, dofs_per_cell);
+  
+                                  // Additionally we need following
+                                  // two cell matrices, both for face
+                                  // term that include test function
+                                  // ``vn'' (shape functions of the
+                                  // neighboring cell). To be more
+                                  // precise, the first matrix will
+                                  // include the `u and vn terms' and
+                                  // the second that will include the
+                                  // `un and vn terms'.
+  FullMatrix<double> u_vn_matrix (dofs_per_cell, dofs_per_cell);
+  FullMatrix<double> un_vn_matrix (dofs_per_cell, dofs_per_cell);
+  
+  Vector<double>  cell_vector (dofs_per_cell);
+
+                                  // Furthermore, here we define a
+                                  // dummy matrix and rhs to
+                                  // emphasize when arguments of the
+                                  // ``assemble_face_term2''
+                                  // functions will not be access.
+  FullMatrix<double> dummy_matrix;
+  Vector<double>     dummy_rhs;
+
+                                  // The following lines are roughly
+                                  // the same as in the previous
+                                  // function.
+  DoFHandler<dim>::active_cell_iterator cell = dof_handler.begin_active(),
+                                       endc = dof_handler.end();
+  DoFHandler<dim>::face_iterator face;
+  DoFHandler<dim>::cell_iterator neighbor;
+  DoFHandler<dim>::cell_iterator neighbor_child;
+
+  for (;cell!=endc; ++cell) 
+    {
+      fe_v.reinit (cell);
+
+      dg.assemble_cell_term(fe_v,
+                           u_v_matrix,
+                           cell_vector);
+      
+      cell->get_dof_indices (dofs);
+
+      for (unsigned int face_no=0; face_no<GeometryInfo<dim>::faces_per_cell; ++face_no)
+       {
+         face = cell->face(face_no);
+
+                                          // Case 1:
+         if (face->at_boundary())
+           {
+             fe_v_face.reinit (cell, face_no);
+
+             dg.assemble_face_term2(fe_v_face,
+                                    fe_v_face,
+                                    u_v_matrix,
+                                    dummy_matrix,
+                                    dummy_matrix,
+                                    dummy_matrix,
+                                    cell_vector);
+           }
+         else
+           {
+             Assert (cell->neighbor(face_no).state() == valid, ExcInternalError());
+             neighbor = cell->neighbor(face_no);
+
+                                              // Case 2:
+             if (face->has_children())
+               {
+                 const unsigned int neighbor2=
+                   cell->neighbor_of_neighbor(face_no);
+                 
+                 for (unsigned int subface_no=0;
+                      subface_no<GeometryInfo<dim>::subfaces_per_face;
+                      ++subface_no)
+                   {
+                     neighbor_child = neighbor->child(
+                       GeometryInfo<dim>::child_cell_on_face(neighbor2,subface_no));
+                     Assert (neighbor_child->face(neighbor2) == face->child(subface_no),
+                             ExcInternalError());
+                     Assert (!neighbor_child->has_children(), ExcInternalError());
+
+                     fe_v_subface.reinit (cell, face_no, subface_no);
+                     fe_v_face_neighbor.reinit (neighbor_child, neighbor2);
+
+                     dg.assemble_face_term2(fe_v_subface,
+                                            fe_v_face_neighbor,
+                                            u_v_matrix,
+                                            un_v_matrix,
+                                            u_vn_matrix,
+                                            un_vn_matrix,
+                                            dummy_rhs);
+                 
+                     neighbor_child->get_dof_indices (dofs_neighbor);
+                                                               
+                     for (unsigned int i=0; i<dofs_per_cell; ++i)
+                       for (unsigned int j=0; j<dofs_per_cell; ++j)
+                         {
+                           system_matrix.add(dofs[i], dofs_neighbor[j],
+                                             un_v_matrix(i,j));
+                           system_matrix.add(dofs_neighbor[i], dofs[j],
+                                             u_vn_matrix(i,j));
+                           system_matrix.add(dofs_neighbor[i], dofs_neighbor[j],
+                                             un_vn_matrix(i,j));
+                         }
+                     
+                     un_v_matrix.clear();
+                     u_vn_matrix.clear();
+                     un_vn_matrix.clear();
+                   }
+               }
+             else
+               {
+                                                  // Case 3, with the
+                                                  // additional rule
+                                                  // a)
+                 if (neighbor->level() == cell->level() &&
+                     neighbor->index() > cell->index()) 
+                   {
+                     const unsigned int neighbor2=cell->neighbor_of_neighbor(face_no);
+
+                     fe_v_face.reinit (cell, face_no);
+                     fe_v_face_neighbor.reinit (neighbor, neighbor2);
+                     
+                     dg.assemble_face_term2(fe_v_face,
+                                            fe_v_face_neighbor,
+                                            u_v_matrix,
+                                            un_v_matrix,
+                                            u_vn_matrix,
+                                            un_vn_matrix,
+                                            dummy_rhs);
+
+                     neighbor->get_dof_indices (dofs_neighbor);
+
+                     for (unsigned int i=0; i<dofs_per_cell; ++i)
+                       for (unsigned int j=0; j<dofs_per_cell; ++j)
+                         {
+                           system_matrix.add(dofs[i], dofs_neighbor[j],
+                                             un_v_matrix(i,j));
+                           system_matrix.add(dofs_neighbor[i], dofs[j],
+                                             u_vn_matrix(i,j));
+                           system_matrix.add(dofs_neighbor[i], dofs_neighbor[j],
+                                             un_vn_matrix(i,j));
+                         }
+                     
+                     un_v_matrix.clear();
+                     u_vn_matrix.clear();
+                     un_vn_matrix.clear();
+                   }
+
+                                                  // Due to rule b)
+                                                  // we do not need
+                                                  // to consider case
+                                                  // 4.
+               }
+           }
+       }
+      
+      for (unsigned int i=0; i<dofs_per_cell; ++i)
+       for (unsigned int j=0; j<dofs_per_cell; ++j)
+         system_matrix.add(dofs[i], dofs[j], u_v_matrix(i,j));
+      
+      for (unsigned int i=0; i<dofs_per_cell; ++i)
+       right_hand_side(dofs[i]) += cell_vector(i);
+      
+      u_v_matrix.clear ();
+      cell_vector.clear ();
+    }
+};
+
+                                // For this simple solver we use the
+                                // simplest possible solver, called
+                                // richardson iteration, that
+                                // represents a simple defect
+                                // correction. This, in combination
+                                // with a block SSOR preconditioner,
+                                // that uses the special block matrix
+                                // structur of system matrices
+                                // arising from DG
+                                // discretizations. The size of these
+                                // blocks are the number of DoFs
+                                // per cell. Here, we use a SSOR
+                                // preconditioning as we have not
+                                // renumbered the DoFs according to
+                                // the flow field. If the DoFs are
+                                // renumbered downstream the flow,
+                                // then a block Gauss-Seidel
+                                // preconditioner (see the
+                                // PreconditionBlockSOR class with
+                                // relaxation=1) makes a much better
+                                // job.
+template <int dim>
+void DGMethod<dim>::solve (Vector<double> &solution) 
+{
+  SolverControl           solver_control (1000, 1e-12, false, false);
   PrimitiveVectorMemory<> vector_memory;
   SolverRichardson<>      solver (solver_control, vector_memory);
 
+                                  // Here we create the
+                                  // preconditioner,
   PreconditionBlockSSOR<double> preconditioner;
+
+                                  // we asigned the matrix to it and
+                                  // set the right block size.
   preconditioner.initialize(system_matrix, fe.dofs_per_cell);
+
+                                  // As the inverses of the diagonal
+                                  // blocks are needed in each
+                                  // preconditioner step, it is wise
+                                  // to invert the diagonal blocks of
+                                  // the matrix before starting the
+                                  // solver. Otherwise, the diagonal
+                                  // blocks are inverted in each
+                                  // preconditioner step,
+                                  // significantly slowing down the
+                                  // linear solving process.
   preconditioner.invert_diagblocks();
-  
+
+                                  // After these preparations we are
+                                  // ready to start the linear solver.
   solver.solve (system_matrix, solution, right_hand_side,
                preconditioner);
 };
 
 
+                                // We refine the grid according to a
+                                // very simple refinement criterion,
+                                // namely the gradients of the
+                                // solution. As here we consider the
+                                // DG(1) method (i.e. we use
+                                // piecewise bilinear shape
+                                // functions) we could simply compute
+                                // the gradients on each cell. But we
+                                // do not want to base our refinement
+                                // indicator on the gradients on each
+                                // cell only, but want to base them
+                                // also on jumps of the discontinuous
+                                // solution function over faces
+                                // between neighboring cells. The
+                                // simpliest way of doing that is to
+                                // compute approximative gradients by
+                                // difference quotients including the
+                                // cell under consideration and its
+                                // neighbors. This is done by the
+                                // DerivativeApproximation class that
+                                // computes the approximate
+                                // gradients in a way similar to the
+                                // GradientEstimation described in
+                                // Step 9 of this tutorial. According
+                                // to the argumentation in Step 9,
+                                // here we consider
+                                // $h^{1+d/2}|\nabla_h
+                                // u_h|$. Futhermore we note that we
+                                // do not consider approximate
+                                // second derivatives because
+                                // solutions to the linear advection
+                                // equation are in general not in H^2
+                                // but in H^1 (to be more precise, in
+                                // H^1_\beta) only.
 template <int dim>
-void TransportProblem<dim>::refine_grid ()
+void DGMethod<dim>::refine_grid ()
 {
-  Vector<float> estimated_error_per_cell (triangulation.n_active_cells());
-
-  FunctionMap<dim>::type neumann_boundary;
-
-  KellyErrorEstimator<dim>::estimate (dof_handler,
-                                     QGauss3<dim-1>(),
-                                     neumann_boundary,
-                                     solution,
-                                     estimated_error_per_cell);
+                                  // The DerivativeApproximation
+                                  // class computes the gradients to
+                                  // float precision. This is
+                                  // sufficient as they are
+                                  // approximate and serve as
+                                  // refinement indicators only.
+  Vector<float> gradient_indicator (triangulation.n_active_cells());
+
+                                  // Now the approximate gradients
+                                  // are computed
+  DerivativeApproximation::approximate_gradient (mapping,
+                                                dof_handler,
+                                                solution2,
+                                                gradient_indicator);
+
+                                  // and they are cell-wise scaled by
+                                  // the factor $h^{1+d/2}$
+  DoFHandler<dim>::active_cell_iterator cell = dof_handler.begin_active(),
+                                       endc = dof_handler.end();
+  for (unsigned int cell_no=0; cell!=endc; ++cell, ++cell_no)
+    gradient_indicator(cell_no)*=std::pow(cell->diameter(), 1+1.0*dim/2);
 
+                                  // Finally they serve as refinement
+                                  // indicator.
   GridRefinement::refine_and_coarsen_fixed_number (triangulation,
-                                                  estimated_error_per_cell,
-                                                  0.3, 0.03);
+                                                  gradient_indicator,
+                                                  0.3, 0.1);
 
   triangulation.execute_coarsening_and_refinement ();
-};
-
+}
 
 
+                                // The output of this program
+                                // consists of eps-files of the
+                                // adaptively refined grids and the
+                                // numerical solutions given in
+                                // gnuplot format. This was covered
+                                // in previous examples and will not
+                                // be further commented on.
 template <int dim>
-void TransportProblem<dim>::output_results (const unsigned int cycle) const
+void DGMethod<dim>::output_results (const unsigned int cycle) const
 {
-                                  // We want to write the grid in
-                                  // each cycle. Here is another way
-                                  // to quickly produce a filename
-                                  // based on the cycle number. It
-                                  // assumes that the numbers `0'
-                                  // through `9' are represented
-                                  // consecutively in the character
-                                  // set (which is the case in all
-                                  // known character sets). However,
-                                  // this will only work if the cycle
-                                  // number is less than ten, which
-                                  // we check by an assertion.
+                                  // Write the grid in eps format.
   std::string filename = "grid-";
   filename += ('0' + cycle);
   Assert (cycle < 10, ExcInternalError());
   
   filename += ".eps";
+  cout << "Writing grid to <" << filename << ">..." << endl;
   std::ofstream eps_output (filename.c_str());
 
-                                  // Using this filename, we write
-                                  // each grid as a postscript file.
   GridOut grid_out;
   grid_out.write_eps (triangulation, eps_output);
-
-                                  // output of the solution
+  
+                                  // Output of the solution in
+                                  // gnuplot format.
   filename = "sol-";
   filename += ('0' + cycle);
   Assert (cycle < 10, ExcInternalError());
   
   filename += ".gnuplot";
+  cout << "Writing solution to <" << filename << ">..." << endl;
   std::ofstream gnuplot_output (filename.c_str());
   
   DataOut<dim> data_out;
   data_out.attach_dof_handler (dof_handler);
-  data_out.add_data_vector (solution, "u");
+  data_out.add_data_vector (solution2, "u");
 
   data_out.build_patches ();
   
@@ -582,11 +1488,23 @@ void TransportProblem<dim>::output_results (const unsigned int cycle) const
 };
 
 
-
+                                // The following ``run'' function is
+                                // similar to previous examples. The
+                                // only difference is that the
+                                // problem is assembled and solved
+                                // twice on each refinement step;
+                                // first by ``assemble_system1'' that
+                                // implements the first version and
+                                // then by ``assemble_system2'' that
+                                // implements the second version of
+                                // writing the DG
+                                // discretization. Furthermore the
+                                // time needed by each of the two
+                                // assembling routines is measured.
 template <int dim>
-void TransportProblem<dim>::run () 
+void DGMethod<dim>::run () 
 {
-  for (unsigned int cycle=0; cycle<3; ++cycle)
+  for (unsigned int cycle=0; cycle<6; ++cycle)
     {
       std::cout << "Cycle " << cycle << ':' << std::endl;
 
@@ -597,18 +1515,7 @@ void TransportProblem<dim>::run ()
          triangulation.refine_global (3);
        }
       else
-                                        // In case this is not the
-                                        // first cycle, we want to
-                                        // refine the grid. Unlike
-                                        // the global refinement
-                                        // employed in the last
-                                        // example, we now use the
-                                        // adaptive procedure
-                                        // described in the function
-                                        // which we now call:
-       {
-         refine_grid ();
-       };
+       refine_grid ();
       
 
       std::cout << "   Number of active cells:       "
@@ -620,97 +1527,62 @@ void TransportProblem<dim>::run ()
       std::cout << "   Number of degrees of freedom: "
                << dof_handler.n_dofs()
                << std::endl;
-      
-      assemble_system ();
-      solve ();
+
+                                      // The constructor of the Timer
+                                      // class automatically starts
+                                      // the time measurement.
+      Timer assemble_timer;
+                                      // First assembling routine.
+      assemble_system1 ();
+                                      // The operator () accesses the
+                                      // current time without
+                                      // disturbing the time
+                                      // measurement.
+      cout << "Time of assemble_system1: " << assemble_timer() << endl;
+      solve (solution1);
+
+                                      // As preparation for the
+                                      // second assembling routine we
+                                      // reinit the system matrix, the
+                                      // right hand side vector and
+                                      // the Timer object.
+      system_matrix.reinit();
+      right_hand_side.clear();
+      assemble_timer.reset();
+
+                                      // We start the Timer,
+      assemble_timer.start();
+                                      // call the second assembling routine
+      assemble_system2 ();
+                                      // and access the current time.
+      cout << "Time of assemble_system2: " << assemble_timer() << endl;
+      solve (solution2);
+
+                                      // To make sure that both
+                                      // versions of the DG method
+                                      // yield the same
+                                      // discretization and hence the
+                                      // same solution we check the
+                                      // two solutions for equality.
+      solution1-=solution2;
+      const double difference=solution1.linfty_norm();
+      if (difference<1e-13)
+       cout << "solution1 and solution2 do not differ." << endl;
+
+                                      // Finally we perform the
+                                      // output.
       output_results (cycle);
     }
 }
 
+
+
 int main () 
 {
+  DGMethod<2> dgmethod_2d;
+  dgmethod_2d.run ();
 
-                                  // The general idea behind the
-                                  // layout of this function is as
-                                  // follows: let's try to run the
-                                  // program as we did before...
-  try
-    {
-      TransportProblem<2> Transport_problem_2d;
-      Transport_problem_2d.run ();
-    }
-                                  // ...and if this should fail, try
-                                  // to gather as much information as
-                                  // possible. Specifically, if the
-                                  // exception that was thrown is an
-                                  // object of a class that is
-                                  // derived from the C++ standard
-                                  // class ``exception'', then we can
-                                  // use the ``what'' member function
-                                  // to get a string which describes
-                                  // the reason why the exception was
-                                  // thrown. 
-                                  //
-                                  // The deal.II exception classes
-                                  // are all derived from the
-                                  // standard class, and in
-                                  // particular, the ``exc.what()''
-                                  // function will return
-                                  // approximately the same string as
-                                  // would be generated if the
-                                  // exception was thrown using the
-                                  // ``Assert'' macro. You have seen
-                                  // the output of such an exception
-                                  // in the previous example, and you
-                                  // then know that it contains the
-                                  // file and line number of where
-                                  // the exception occured, and some
-                                  // other information. This is also
-                                  // what would be printed in the
-                                  // following.
-  catch (std::exception &exc)
-    {
-      std::cerr << std::endl << std::endl
-               << "----------------------------------------------------"
-               << std::endl;
-      std::cerr << "Exception on processing: " << std::endl
-               << exc.what() << std::endl
-               << "Aborting!" << std::endl
-               << "----------------------------------------------------"
-               << std::endl;
-                                      // We can't do much more than
-                                      // printing as much information
-                                      // as we can get to, so abort
-                                      // with error:
-      return 1;
-    }
-                                  // If the exception that was thrown
-                                  // somewhere was not an object of a
-                                  // class derived from the standard
-                                  // ``exception'' class, then we
-                                  // can't do anything at all. We
-                                  // then simply print an error
-                                  // message and exit.
-  catch (...) 
-    {
-      std::cerr << std::endl << std::endl
-               << "----------------------------------------------------"
-               << std::endl;
-      std::cerr << "Unknown exception!" << std::endl
-               << "Aborting!" << std::endl
-               << "----------------------------------------------------"
-               << std::endl;
-      return 1;
-    };
-
-                                  // If we got to this point, there
-                                  // was no exception which
-                                  // propagated up to the main
-                                  // function (maybe there were some,
-                                  // but they were caught somewhere
-                                  // in the program or the
-                                  // library). Therefore, the program
-                                  // performed as was expected and we
-                                  // can return without error.
   return 0;
 };
+
+

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