From 28c50c4009dfe92b689f0d7b12ec6f2f5d284a74 Mon Sep 17 00:00:00 2001
From: Scott Miller <miller.scott.t@gmail.com>
Date: Thu, 8 Aug 2013 13:05:04 +0000
Subject: [PATCH] Merge step-51 botched commit.

git-svn-id: https://svn.dealii.org/trunk@30255 0785d39b-7218-0410-832d-ea1e28bc413d
---
 deal.II/examples/step-51/step-51.cc | 1481 ++++++++++++++-------------
 1 file changed, 755 insertions(+), 726 deletions(-)

diff --git a/deal.II/examples/step-51/step-51.cc b/deal.II/examples/step-51/step-51.cc
index ab4bf33dae..0657b04d70 100644
--- a/deal.II/examples/step-51/step-51.cc
+++ b/deal.II/examples/step-51/step-51.cc
@@ -32,855 +32,885 @@
 #include <deal.II/lac/solver_gmres.h>
 #include <deal.II/lac/precondition.h>
 #include <deal.II/grid/tria.h>
-#include <deal.II/dofs/dof_handler.h>
 #include <deal.II/grid/tria_accessor.h>
 #include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/grid_refinement.h>
 #include <deal.II/grid/tria_iterator.h>
 #include <deal.II/grid/tria_boundary_lib.h>
+#include <deal.II/dofs/dof_handler.h>
 #include <deal.II/dofs/dof_accessor.h>
 #include <deal.II/dofs/dof_renumbering.h>
 #include <deal.II/dofs/dof_tools.h>
 #include <deal.II/fe/fe_system.h>
 #include <deal.II/fe/fe_values.h>
 #include <deal.II/numerics/vector_tools.h>
+#include <deal.II/numerics/error_estimator.h>
 #include <deal.II/numerics/matrix_tools.h>
 #include <deal.II/numerics/data_out.h>
 
-//! New include:  output data on faces of a 
-//  triangulation
-#include <deal.II/numerics/data_out_faces.h>
-
-
 //! New include:  fe_face.h
 //  Explain that it implements fe on 
 //  codim=1 surfaces of a geometric discretization
 #include <deal.II/fe/fe_dgq.h>
 #include <deal.II/fe/fe_face.h>
 
+//! New include:  explain the chunk_xxx
 #include <deal.II/lac/chunk_sparse_matrix.h>
-#include <deal.II/numerics/data_out_faces.h>
-
-namespace Step51
-{
-  using namespace dealii;
-
-  // @sect3{Equation data}
-
-  // The structure of the analytic solution is the same as in step-7. There
-  // are two exceptions. Firstly, we also create a solution for the 3d case,
-  // and secondly, we take into account the convection velocity in the right
-  // hand side that is variable in this case.
-  template <int dim>
-  class SolutionBase
-  {
-  protected:
-    static const unsigned int  n_source_centers = 3;
-    static const Point<dim>    source_centers[n_source_centers];
-    static const double        width;
-  };
-
-
-  template <>
-  const Point<1>
-  SolutionBase<1>::source_centers[SolutionBase<1>::n_source_centers]
-    = { Point<1>(-1.0 / 3.0),
-        Point<1>(0.0),
-        Point<1>(+1.0 / 3.0)
-      };
-
-
-  template <>
-  const Point<2>
-  SolutionBase<2>::source_centers[SolutionBase<2>::n_source_centers]
-    = { Point<2>(-0.5, +0.5),
-        Point<2>(-0.5, -0.5),
-        Point<2>(+0.5, -0.5)
-      };
-
-  template <>
-  const Point<3>
-  SolutionBase<3>::source_centers[SolutionBase<3>::n_source_centers]
-  = { Point<3>(-0.5, +0.5, 0.25),
-      Point<3>(-0.6, -0.5, -0.125),
-      Point<3>(+0.5, -0.5, 0.5)   };
-
-  template <int dim>
-  const double SolutionBase<dim>::width = 1./5.;
-
 
+//! New include:  output data on faces of a 
+//  triangulation
+#include <deal.II/numerics/data_out_faces.h>
 
-  template <int dim>
-  class ConvectionVelocity : public TensorFunction<1,dim>
-  {
-  public:
-    ConvectionVelocity() : TensorFunction<1,dim>() {}
+using namespace dealii;
 
-    virtual Tensor<1,dim> value (const Point<dim> &p) const;
-  };
+// @sect3{Equation data}
 
+// The structure of the analytic solution is the same as in step-7. There
+// are two exceptions. Firstly, we also create a solution for the 3d case,
+// and secondly, we take into account the convection velocity in the right
+// hand side that is variable in this case.
+template <int dim>
+class SolutionBase
+{
+protected:
+  static const unsigned int  n_source_centers = 3;
+  static const Point<dim>    source_centers[n_source_centers];
+  static const double        width;
+};
 
 
-  template <int dim>
-  Tensor<1,dim>
-  ConvectionVelocity<dim>::value(const Point<dim> &p) const
-  {
-    Tensor<1,dim> convection;
-    switch (dim)
-      {
-      case 1:
-        convection[0] = 1;
-        break;
-      case 2:
-        convection[0] = p[1];
-        convection[1] = -p[0];
-        break;
-      case 3:
-        convection[0] = p[1];
-        convection[1] = -p[0];
-        convection[2] = 1;
-        break;
-      default:
-        Assert(false, ExcNotImplemented());
-      }
-    return convection;
-  }
+template <>
+const Point<1>
+SolutionBase<1>::source_centers[SolutionBase<1>::n_source_centers]
+= { Point<1>(-1.0 / 3.0),
+    Point<1>(0.0),
+    Point<1>(+1.0 / 3.0)
+};
 
 
-  template <int dim>
-  class Solution : public Function<dim>,
-                   protected SolutionBase<dim>
-  {
-  public:
-    Solution () : Function<dim>() {}
+template <>
+const Point<2>
+SolutionBase<2>::source_centers[SolutionBase<2>::n_source_centers]
+= { Point<2>(-0.5, +0.5),
+    Point<2>(-0.5, -0.5),
+    Point<2>(+0.5, -0.5)
+};
 
-    virtual double value (const Point<dim>   &p,
-			  const unsigned int  component = 0) const;
+template <>
+const Point<3>
+SolutionBase<3>::source_centers[SolutionBase<3>::n_source_centers]
+= { Point<3>(-0.5, +0.5, 0.25),
+    Point<3>(-0.6, -0.5, -0.125),
+    Point<3>(+0.5, -0.5, 0.5)   };
 
-    virtual Tensor<1,dim> gradient (const Point<dim>   &p,
-				    const unsigned int  component = 0) const;
-  };
+template <int dim>
+const double SolutionBase<dim>::width = 1./5.;
 
 
 
-  template <int dim>
-  double Solution<dim>::value (const Point<dim>   &p,
-                               const unsigned int) const
-  {
-    double return_value = 0;
-    for (unsigned int i=0; i<this->n_source_centers; ++i)
-      {
-        const Point<dim> x_minus_xi = p - this->source_centers[i];
-        return_value += std::exp(-x_minus_xi.square() /
-                                 (this->width * this->width));
-      }
+template <int dim>
+class ConvectionVelocity : public TensorFunction<1,dim>
+{
+public:
+  ConvectionVelocity() : TensorFunction<1,dim>() {}
 
-    return return_value /
-      Utilities::fixed_power<dim>(std::sqrt(2. * numbers::PI) * this->width);
-  }
+  virtual Tensor<1,dim> value (const Point<dim> &p) const;
+};
 
 
 
-  template <int dim>
-  Tensor<1,dim> Solution<dim>::gradient (const Point<dim>   &p,
-                                         const unsigned int) const
-  {
-    Tensor<1,dim> return_value;
+template <int dim>
+Tensor<1,dim>
+ConvectionVelocity<dim>::value(const Point<dim> &p) const
+{
+  Tensor<1,dim> convection;
+  switch (dim)
+    {
+    case 1:
+      convection[0] = 1;
+      break;
+    case 2:
+      convection[0] = p[1];
+      convection[1] = -p[0];
+      break;
+    case 3:
+      convection[0] = p[1];
+      convection[1] = -p[0];
+      convection[2] = 1;
+      break;
+    default:
+      Assert(false, ExcNotImplemented());
+    }
+  return convection;
+}
 
-    for (unsigned int i=0; i<this->n_source_centers; ++i)
-      {
-        const Point<dim> x_minus_xi = p - this->source_centers[i];
 
-        return_value += (-2 / (this->width * this->width) *
-                         std::exp(-x_minus_xi.square() /
-                                  (this->width * this->width)) *
-                         x_minus_xi);
-      }
+template <int dim>
+class Solution : public Function<dim>,
+                 protected SolutionBase<dim>
+{
+public:
+  Solution () : Function<dim>() {}
 
-    return return_value / Utilities::fixed_power<dim>(std::sqrt(2 * numbers::PI) *
-                                                      this->width);
-  }
+  virtual double value (const Point<dim>   &p,
+                        const unsigned int  component = 0) const;
 
+  virtual Tensor<1,dim> gradient (const Point<dim>   &p,
+                                  const unsigned int  component = 0) const;
+};
 
 
-  template <int dim>
-  class SolutionAndGradient : public Function<dim>,
-                              protected SolutionBase<dim>
-  {
-  public:
-    SolutionAndGradient () : Function<dim>(dim) {}
 
-    virtual void vector_value (const Point<dim>   &p,
-                               Vector<double>     &v) const
+template <int dim>
+double Solution<dim>::value (const Point<dim>   &p,
+                             const unsigned int) const
+{
+  double return_value = 0;
+  for (unsigned int i=0; i<this->n_source_centers; ++i)
     {
-      AssertDimension(v.size(), dim+1);
-      Solution<dim> solution;
-      Tensor<1,dim> grad = solution.gradient(p);
-      for (unsigned int d=0; d<dim; ++d)
-        v[d] = -grad[d];
-      v[dim] = solution.value(p);
+      const Point<dim> x_minus_xi = p - this->source_centers[i];
+      return_value += std::exp(-x_minus_xi.square() /
+                               (this->width * this->width));
     }
-  };
 
+  return return_value /
+    Utilities::fixed_power<dim>(std::sqrt(2. * numbers::PI) * this->width);
+}
 
 
-  template <int dim>
-  class RightHandSide : public Function<dim>,
-                        protected SolutionBase<dim>
-  {
-  public:
-    RightHandSide () : Function<dim>() {}
 
-    virtual double value (const Point<dim>   &p,
-			  const unsigned int  component = 0) const;
+template <int dim>
+Tensor<1,dim> Solution<dim>::gradient (const Point<dim>   &p,
+                                       const unsigned int) const
+{
+  Tensor<1,dim> return_value;
 
-  private:
-    const ConvectionVelocity<dim> convection_velocity;
-  };
+  for (unsigned int i=0; i<this->n_source_centers; ++i)
+    {
+      const Point<dim> x_minus_xi = p - this->source_centers[i];
 
+      return_value += (-2 / (this->width * this->width) *
+                       std::exp(-x_minus_xi.square() /
+                                (this->width * this->width)) *
+                       x_minus_xi);
+    }
 
-  template <int dim>
-  double RightHandSide<dim>::value (const Point<dim>   &p,
-                                    const unsigned int) const
-  {
-    Tensor<1,dim> convection = convection_velocity.value(p);
-    double return_value = 0;
-    for (unsigned int i=0; i<this->n_source_centers; ++i)
-      {
-        const Point<dim> x_minus_xi = p - this->source_centers[i];
-
-        return_value +=
-          ((2*dim - 2*convection*x_minus_xi - 4*x_minus_xi.square()/
-            (this->width * this->width)) /
-           (this->width * this->width) *
-           std::exp(-x_minus_xi.square() /
-                    (this->width * this->width)));
-      }
+  return return_value / Utilities::fixed_power<dim>(std::sqrt(2 * numbers::PI) *
+                                                    this->width);
+}
 
-    return return_value / Utilities::fixed_power<dim>(std::sqrt(2 * numbers::PI)
-                                                      * this->width);
-  }
 
 
+template <int dim>
+class SolutionAndGradient : public Function<dim>,
+                            protected SolutionBase<dim>
+{
+public:
+  SolutionAndGradient () : Function<dim>(dim) {}
 
-  template <int dim>
-  class HDG
+  virtual void vector_value (const Point<dim>   &p,
+                             Vector<double>     &v) const
   {
-  public:
-    enum RefinementMode
-    {
-      global_refinement, adaptive_refinement
-    };
-
-    HDG (const unsigned int degree,
-            const RefinementMode refinement_mode);
-    void run ();
-
-  private:
-    void setup_system ();
-    void assemble_system (const bool reconstruct_trace = false);
-    void solve ();
-    void postprocess ();
-    void refine_mesh ();
-    void output_results (const unsigned int cycle);
-
-    Triangulation<dim>   triangulation;
-
-    const MappingQ<dim>  mapping;
+    AssertDimension(v.size(), dim+1);
+    Solution<dim> solution;
+    Tensor<1,dim> grad = solution.gradient(p);
+    for (unsigned int d=0; d<dim; ++d)
+      v[d] = -grad[d];
+    v[dim] = solution.value(p);
+  }
+};
 
-	// local (element interior) solutions
-    FESystem<dim>        fe_local;
-    DoFHandler<dim>      dof_handler_local;
 
-	// global (trace/skeleton) solution
-	// Note that FE_FaceQ<dim> represents
-	// finite element data on the faces/edges
-	// of our triangulation
-    FE_FaceQ<dim>        fe;
-    DoFHandler<dim>      dof_handler;
 
-	// post-processed solution
-    FE_DGQ<dim>          fe_u_post;
-    DoFHandler<dim>      dof_handler_u_post;
-    
-    // Dirichlet BCs are strongly enforced
-    // on the "skeleton" solution
-    ConstraintMatrix     constraints;
-    
-    
-    // Comment on "chunk" here.
-    // First, set up objects for the global
-    // solution
-    ChunkSparsityPattern sparsity_pattern;
-    ChunkSparseMatrix<double> system_matrix;
+template <int dim>
+class RightHandSide : public Function<dim>,
+                      protected SolutionBase<dim>
+{
+public:
+  RightHandSide () : Function<dim>() {}
 
-    Vector<double>       solution;
-    Vector<double>       system_rhs;
+  virtual double value (const Point<dim>   &p,
+                        const unsigned int  component = 0) const;
 
-	// Local solution values
-    Vector<double>       solution_local;
-    
-    // HDG solutions can be post-processed
-    // to gain one order of accuracy.
-    // <code>solution_u_post</code> will be
-    // our post-processed DG solution on the
-    // interior of cells represented by a 
-    // DG solution of order (degree+1)
-    Vector<double>       solution_u_post;
+private:
+  const ConvectionVelocity<dim> convection_velocity;
+};
 
-	// Same as step-7:
-    const RefinementMode refinement_mode;
 
-    ConvergenceTable     convergence_table;
-  };
+template <int dim>
+double RightHandSide<dim>::value (const Point<dim>   &p,
+                                  const unsigned int) const
+{
+  Tensor<1,dim> convection = convection_velocity.value(p);
+  double return_value = 0;
+  for (unsigned int i=0; i<this->n_source_centers; ++i)
+    {
+      const Point<dim> x_minus_xi = p - this->source_centers[i];
+
+      return_value +=
+        ((2*dim - 2*convection*x_minus_xi - 4*x_minus_xi.square()/
+          (this->width * this->width)) /
+         (this->width * this->width) *
+         std::exp(-x_minus_xi.square() /
+                  (this->width * this->width)));
+    }
 
+  return return_value / Utilities::fixed_power<dim>(std::sqrt(2 * numbers::PI)
+                                                    * this->width);
+}
 
 
-  template <int dim>
-  HDG<dim>::HDG (const unsigned int degree,
-                       const RefinementMode refinement_mode) :
-    mapping  (1),
-    fe_local (FE_DGQ<dim>(degree), dim,
-              FE_DGQ<dim>(degree), 1),
-    dof_handler_local (triangulation),
-    fe (degree),
-    dof_handler (triangulation),
-    fe_u_post (degree+1),
-    dof_handler_u_post (triangulation),
-    refinement_mode (refinement_mode)
-  {}
 
+template <int dim>
+class Step51
+{
+public:
+  enum RefinementMode
+    {
+      global_refinement, adaptive_refinement
+    };
 
+  Step51 (const unsigned int degree,
+          const RefinementMode refinement_mode);
+  void run ();
+
+private:
+  void setup_system ();
+  void assemble_system (const bool reconstruct_trace = false);
+  void solve ();
+  void postprocess ();
+  void refine_grid (const unsigned int cylce);
+  void output_results (const unsigned int cycle);
+
+  Triangulation<dim>   triangulation;
+
+  const MappingQ<dim>  mapping;
+  
+  // local (element interior) solutions
+  FESystem<dim>        fe_local;
+  DoFHandler<dim>      dof_handler_local;
+
+  // global (trace/skeleton) solution
+  // Note that FE_FaceQ<dim> represents
+  // finite element data on the faces/edges
+  // of our triangulation
+  FE_FaceQ<dim>        fe;
+  DoFHandler<dim>      dof_handler;
+
+  // post-processed solution
+  FE_DGQ<dim>          fe_u_post;
+  DoFHandler<dim>      dof_handler_u_post;
+
+  // Dirichlet BCs are strongly enforced
+  // on the "skeleton" solution
+  ConstraintMatrix     constraints;
+  
+  // Comment on chunk.
+  ChunkSparsityPattern sparsity_pattern;
+  ChunkSparseMatrix<double> system_matrix;
+
+  // Global/skeleton solution/rhs
+  Vector<double>       solution;
+  Vector<double>       system_rhs;
+
+  // Local elementwise solution
+  Vector<double>       solution_local;
+  
+  // HDG solutions can be post-processed
+  // to gain one order of accuracy.
+  // <code>solution_u_post</code> will be
+  // our post-processed DG solution on the
+  // interior of cells represented by a 
+  // DG solution of order (degree+1)
+  Vector<double>       solution_u_post;
+
+  // Same as step-7:
+  const RefinementMode refinement_mode;
+
+  ConvergenceTable     convergence_table;
+};
+
+
+
+template <int dim>
+Step51<dim>::Step51 (const unsigned int degree,
+                     const RefinementMode refinement_mode) :
+  mapping  (3),
+  fe_local (FE_DGQ<dim>(degree), dim,
+            FE_DGQ<dim>(degree), 1),
+  dof_handler_local (triangulation),
+  fe (degree),
+  dof_handler (triangulation),
+  fe_u_post (degree+1),
+  dof_handler_u_post (triangulation),
+  refinement_mode (refinement_mode)
+{}
+
+
+
+template <int dim>
+void
+Step51<dim>::setup_system ()
+{
+  dof_handler_local.distribute_dofs(fe_local);
+  dof_handler.distribute_dofs(fe);
+  dof_handler_u_post.distribute_dofs(fe_u_post);
+
+  std::cout << "   Number of degrees of freedom: "
+            << dof_handler.n_dofs()
+            << std::endl;
+
+  solution.reinit (dof_handler.n_dofs());
+  system_rhs.reinit (dof_handler.n_dofs());
+
+  solution_local.reinit (dof_handler_local.n_dofs());
+  solution_u_post.reinit (dof_handler_u_post.n_dofs());
+
+  constraints.clear ();
+  DoFTools::make_hanging_node_constraints (dof_handler, constraints);
+  typename FunctionMap<dim>::type boundary_functions;
+  Solution<dim> solution;
+  boundary_functions[0] = &solution;
+  VectorTools::project_boundary_values (mapping, dof_handler,
+                                        boundary_functions,
+                                        QGauss<dim-1>(fe.degree+1),
+                                        constraints);
+  constraints.close ();
 
-  template <int dim>
-  void
-  HDG<dim>::setup_system ()
   {
-    dof_handler_local.distribute_dofs(fe_local);
-    dof_handler.distribute_dofs(fe);
-    dof_handler_u_post.distribute_dofs(fe_u_post);
+    CompressedSimpleSparsityPattern csp (dof_handler.n_dofs());
+    DoFTools::make_sparsity_pattern (dof_handler, csp,
+                                     constraints, false);
+    sparsity_pattern.copy_from(csp, fe.dofs_per_face);
+  }
+  system_matrix.reinit (sparsity_pattern);
+}
 
-    std::cout << "   Number of degrees of freedom: "
-              << dof_handler.n_dofs()
-              << std::endl;
 
-    solution.reinit (dof_handler.n_dofs());
-    system_rhs.reinit (dof_handler.n_dofs());
 
-    solution_local.reinit (dof_handler_local.n_dofs());
-    solution_u_post.reinit (dof_handler_u_post.n_dofs());
+template <int dim>
+void
+Step51<dim>::assemble_system (const bool trace_reconstruct)
+{
+  QGauss<dim>   quadrature_formula(fe.degree+1);
+  QGauss<dim-1> face_quadrature_formula(fe.degree+1);
+
+  FEValues<dim> fe_values_local (mapping, fe_local, quadrature_formula,
+                                 update_values | update_gradients |
+                                 update_JxW_values | update_quadrature_points);
+  FEFaceValues<dim> fe_face_values (mapping, fe, face_quadrature_formula,
+                                    update_values | update_normal_vectors |
+                                    update_quadrature_points |
+                                    update_JxW_values);
+  FEFaceValues<dim> fe_face_values_local (mapping, fe_local,
+                                          face_quadrature_formula,
+                                          update_values);
+
+  const unsigned int n_q_points    = quadrature_formula.size();
+  const unsigned int n_face_q_points = face_quadrature_formula.size();
+
+  const unsigned int dofs_per_cell = fe.dofs_per_cell;
+  const unsigned int loc_dofs_per_cell = fe_local.dofs_per_cell;
+
+  FullMatrix<double> ll_matrix (loc_dofs_per_cell, loc_dofs_per_cell);
+  FullMatrix<double> lf_matrix (loc_dofs_per_cell, dofs_per_cell);
+  FullMatrix<double> fl_matrix (dofs_per_cell, loc_dofs_per_cell);
+  FullMatrix<double> tmp_matrix (dofs_per_cell, loc_dofs_per_cell);
+  FullMatrix<double> ff_matrix (dofs_per_cell, dofs_per_cell);
+  Vector<double>     l_rhs (loc_dofs_per_cell);
+  Vector<double>     f_rhs (dofs_per_cell);
+  Vector<double>     tmp_rhs (loc_dofs_per_cell);
+
+  std::vector<types::global_dof_index> dof_indices (dofs_per_cell);
+  std::vector<types::global_dof_index> loc_dof_indices (loc_dofs_per_cell);
+
+  std::vector<Tensor<1,dim> > q_phi (loc_dofs_per_cell);
+  std::vector<double>         q_phi_div (loc_dofs_per_cell);
+  std::vector<double>         u_phi (loc_dofs_per_cell);
+  std::vector<Tensor<1,dim> > u_phi_grad (loc_dofs_per_cell);
+  std::vector<double>         tr_phi (dofs_per_cell);
+
+  std::vector<double> trace_values(n_face_q_points);
+
+  // Choose stabilization parameter to be 5 * diffusion = 5
+  const double tau_stab_diffusion = 5.;
+
+  ConvectionVelocity<dim> convection_velocity;
+  RightHandSide<dim> right_hand_side;
+  const Solution<dim> exact_solution;
+
+  const FEValuesExtractors::Vector fluxes (0);
+  const FEValuesExtractors::Scalar scalar (dim);
+
+  std::vector<std::vector<unsigned int> >
+    fe_local_support_on_face(GeometryInfo<dim>::faces_per_cell);
+  for (unsigned int face=0; face<GeometryInfo<dim>::faces_per_cell; ++face)
+    for (unsigned int i=0; i<loc_dofs_per_cell; ++i)
+      if (fe_local.has_support_on_face(i,face))
+        fe_local_support_on_face[face].push_back(i);
+  std::vector<std::vector<unsigned int> >
+    fe_support_on_face(GeometryInfo<dim>::faces_per_cell);
+  for (unsigned int face=0; face<GeometryInfo<dim>::faces_per_cell; ++face)
+    for (unsigned int i=0; i<dofs_per_cell; ++i)
+      if (fe.has_support_on_face(i,face))
+        fe_support_on_face[face].push_back(i);
+
+  typename DoFHandler<dim>::active_cell_iterator
+    cell = dof_handler.begin_active(),
+    loc_cell = dof_handler_local.begin_active(),
+    endc = dof_handler.end();
+  for (; cell!=endc; ++cell, ++loc_cell)
+    {
+      ll_matrix = 0;
+      l_rhs = 0;
+      if (!trace_reconstruct)
+        {
+          lf_matrix = 0;
+          fl_matrix = 0;
+          ff_matrix = 0;
+          f_rhs = 0;
+        }
+      fe_values_local.reinit (loc_cell);
 
-    constraints.clear ();
-    DoFTools::make_hanging_node_constraints (dof_handler, constraints);
-    std::map<unsigned int,double> boundary_values;
-    typename FunctionMap<dim>::type boundary_functions;
-    Solution<dim> solution;
-    boundary_functions[0] = &solution;
-    VectorTools::project_boundary_values (mapping, dof_handler,
-                                          boundary_functions,
-                                          QGauss<dim-1>(fe.degree+1),
-                                          boundary_values);
-    for (std::map<unsigned int,double>::iterator it = boundary_values.begin();
-         it != boundary_values.end(); ++it)
-      if (constraints.is_constrained(it->first) == false)
+      for (unsigned int q=0; q<n_q_points; ++q)
         {
-          constraints.add_line(it->first);
-          constraints.set_inhomogeneity(it->first, it->second);
+          const double rhs_value
+            = right_hand_side.value(fe_values_local.quadrature_point(q));
+          const Tensor<1,dim> convection
+            = convection_velocity.value(fe_values_local.quadrature_point(q));
+          const double JxW = fe_values_local.JxW(q);
+          for (unsigned int k=0; k<loc_dofs_per_cell; ++k)
+            {
+              q_phi[k] = fe_values_local[fluxes].value(k,q);
+              q_phi_div[k] = fe_values_local[fluxes].divergence(k,q);
+              u_phi[k] = fe_values_local[scalar].value(k,q);
+              u_phi_grad[k] = fe_values_local[scalar].gradient(k,q);
+            }
+          for (unsigned int i=0; i<loc_dofs_per_cell; ++i)
+            {
+              for (unsigned int j=0; j<loc_dofs_per_cell; ++j)
+                ll_matrix(i,j) += (
+                                   q_phi[i] * q_phi[j]
+                                   -
+                                   q_phi_div[i] * u_phi[j]
+                                   +
+                                   u_phi[i] * q_phi_div[j]
+                                   -
+                                   (u_phi_grad[i] * convection) * u_phi[j]
+                                   ) * JxW;
+              l_rhs(i) += u_phi[i] * rhs_value * JxW;
+            }
         }
-    constraints.close ();
 
-    {
-      CompressedSimpleSparsityPattern csp (dof_handler.n_dofs());
-      DoFTools::make_sparsity_pattern (dof_handler, csp,
-                                       constraints, false);
-      sparsity_pattern.copy_from(csp, fe.dofs_per_face);
-    }
-    system_matrix.reinit (sparsity_pattern);
-  }
+      for (unsigned int face=0; face<GeometryInfo<dim>::faces_per_cell; ++face)
+        {
+          fe_face_values_local.reinit(loc_cell, face);
+          fe_face_values.reinit(cell, face);
+          if (trace_reconstruct)
+            fe_face_values.get_function_values (solution, trace_values);
 
+          for (unsigned int q=0; q<n_face_q_points; ++q)
+            {
+              const double JxW = fe_face_values.JxW(q);
+              const Point<dim> normal = fe_face_values.normal_vector(q);
+              const Tensor<1,dim> convection
+                = convection_velocity.value(fe_face_values.quadrature_point(q));
+              const double tau_stab = (tau_stab_diffusion +
+                                       std::abs(convection * normal));
+
+              for (unsigned int k=0; k<fe_local_support_on_face[face].size(); ++k)
+                {
+                  const unsigned int kk=fe_local_support_on_face[face][k];
+                  q_phi[k] = fe_face_values_local[fluxes].value(kk,q);
+                  u_phi[k] = fe_face_values_local[scalar].value(kk,q);
+                }
+              if (!trace_reconstruct)
+                {
+                  for (unsigned int k=0; k<fe_support_on_face[face].size(); ++k)
+                    tr_phi[k] =
+                      fe_face_values.shape_value(fe_support_on_face[face][k],q);
+                  for (unsigned int i=0; i<fe_local_support_on_face[face].size(); ++i)
+                    for (unsigned int j=0; j<fe_support_on_face[face].size(); ++j)
+                      {
+                        const unsigned int ii=fe_local_support_on_face[face][i];
+                        const unsigned int jj=fe_support_on_face[face][j];
+                        lf_matrix(ii,jj) += (
+                                             (q_phi[i] * normal
+                                              +
+                                              (convection * normal -
+                                               tau_stab) * u_phi[i])
+                                             * tr_phi[j]
+                                             ) * JxW;
+                        fl_matrix(jj,ii) -= (
+                                             (q_phi[i] * normal
+                                              +
+                                              tau_stab * u_phi[i])
+                                             * tr_phi[j]
+                                             ) * JxW;
+                      }
 
+                  for (unsigned int i=0; i<fe_support_on_face[face].size(); ++i)
+                    for (unsigned int j=0; j<fe_support_on_face[face].size(); ++j)
+                      {
+                        const unsigned int ii=fe_support_on_face[face][i];
+                        const unsigned int jj=fe_support_on_face[face][j];
+                        ff_matrix(ii,jj) += (
+                                             (convection * normal - tau_stab) *
+                                             tr_phi[i] * tr_phi[j]
+                                             ) * JxW;
+                      }
 
-  template <int dim>
-  void
-  HDG<dim>::assemble_system (const bool trace_reconstruct)
-  {
-    QGauss<dim>   quadrature_formula(fe.degree+1);
-    QGauss<dim-1> face_quadrature_formula(fe.degree+1);
-
-    FEValues<dim> fe_values_local (mapping, fe_local, quadrature_formula,
-                                   update_values | update_gradients |
-                                   update_JxW_values | update_quadrature_points);
-    FEFaceValues<dim> fe_face_values (mapping, fe, face_quadrature_formula,
-                                      update_values | update_normal_vectors |
-                                      update_quadrature_points |
-                                      update_JxW_values);
-    FEFaceValues<dim> fe_face_values_local (mapping, fe_local,
-                                            face_quadrature_formula,
-                                            update_values);
-
-    const unsigned int n_q_points    = quadrature_formula.size();
-    const unsigned int n_face_q_points = face_quadrature_formula.size();
-
-    const unsigned int dofs_per_cell = fe.dofs_per_cell;
-    const unsigned int loc_dofs_per_cell = fe_local.dofs_per_cell;
-
-    FullMatrix<double> ll_matrix (loc_dofs_per_cell, loc_dofs_per_cell);
-    FullMatrix<double> lf_matrix (loc_dofs_per_cell, dofs_per_cell);
-    FullMatrix<double> fl_matrix (dofs_per_cell, loc_dofs_per_cell);
-    FullMatrix<double> tmp_matrix (dofs_per_cell, loc_dofs_per_cell);
-    FullMatrix<double> ff_matrix (dofs_per_cell, dofs_per_cell);
-    Vector<double>     l_rhs (loc_dofs_per_cell);
-    Vector<double>     f_rhs (dofs_per_cell);
-    Vector<double>     tmp_rhs (loc_dofs_per_cell);
-
-    std::vector<types::global_dof_index> dof_indices (dofs_per_cell);
-    std::vector<types::global_dof_index> loc_dof_indices (loc_dofs_per_cell);
-
-    ConvectionVelocity<dim> convection;
-    std::vector<Tensor<1,dim> > convection_values (n_q_points);
-    std::vector<Tensor<1,dim> > convection_values_face (n_face_q_points);
-
-    std::vector<double> trace_values(n_face_q_points);
-
-    // Choose stabilization parameter to be 5 * diffusion = 5
-    const double tau_stab_diffusion = 5.;
-    std::vector<double> tau_stab (n_q_points);
-
-    RightHandSide<dim> right_hand_side;
-    std::vector<double> rhs_values (n_q_points);
-
-    const Solution<dim> exact_solution;
-    std::vector<double> neumann_values (n_face_q_points);
-
-    const FEValuesExtractors::Vector gradients (0);
-    const FEValuesExtractors::Scalar values (dim);
-
-    typename DoFHandler<dim>::active_cell_iterator
-    cell = dof_handler.begin_active(),
-    loc_cell = dof_handler_local.begin_active(),
-    endc = dof_handler.end();
-    for (; cell!=endc; ++cell, ++loc_cell)
-      {
-        if (!trace_reconstruct)
-          {
-            lf_matrix = 0;
-            fl_matrix = 0;
-            ff_matrix = 0;
-            f_rhs = 0;
-          }
-        fe_values_local.reinit (loc_cell);
-        right_hand_side.value_list (fe_values_local.get_quadrature_points(),
-                                    rhs_values);
-        convection.value_list(fe_values_local.get_quadrature_points(),
-                              convection_values);
-
-        for (unsigned int i=0; i<loc_dofs_per_cell; ++i)
-          for (unsigned int j=0; j<loc_dofs_per_cell; ++j)
-            {
-              double sum = 0;
-              for (unsigned int q=0; q<n_q_points; ++q)
-                sum += (fe_values_local[gradients].value(i,q) *
-                        fe_values_local[gradients].value(j,q)
-                        -
-                        fe_values_local[gradients].divergence(i,q) *
-                        fe_values_local[values].value(j,q)
-                        +
-                        fe_values_local[values].value(i,q) *
-                        fe_values_local[gradients].divergence(j,q)
-                        -
-                        fe_values_local[values].value(j,q) *
-                        (fe_values_local[values].gradient(i,q) *
-                         convection_values[q])
-                        ) * fe_values_local.JxW(q);
-              ll_matrix(i,j) = sum;
-            }
-        for (unsigned int i=0; i<loc_dofs_per_cell; ++i)
-          {
-            double sum = 0.;
-            for (unsigned int q=0; q<n_q_points; ++q)
-              sum += rhs_values[q] * fe_values_local.JxW(q) *
-                fe_values_local[values].value(i,q);
-            l_rhs(i) = sum;
-          }
-
-        for (unsigned int face=0; face<GeometryInfo<dim>::faces_per_cell; ++face)
-          {
-            fe_face_values_local.reinit(loc_cell, face);
-            fe_face_values.reinit(cell, face);
-            const std::vector<double> &JxW = fe_face_values.get_JxW_values();
-            const std::vector<Point<dim> > &normals =
-              fe_face_values.get_normal_vectors();
-            convection.value_list(fe_face_values.get_quadrature_points(),
-                                  convection_values_face);
-            for (unsigned int q=0; q<n_face_q_points; ++q)
-              tau_stab[q] = (tau_stab_diffusion +
-                             std::abs(convection_values_face[q] * normals[q]));
-            if (!trace_reconstruct)
-              {
-                for (unsigned int i=0; i<dofs_per_cell; ++i)
-                  for (unsigned int j=0; j<loc_dofs_per_cell; ++j)
+                  if (cell->face(face)->at_boundary()
+                      &&
+                      (cell->face(face)->boundary_indicator() == 1))
                     {
-                      double sum_lf = 0., sum_fl = 0.;
-                      for (unsigned int q=0; q<n_face_q_points; ++q)
+                      const double neumann_value =
+                        exact_solution.value(fe_face_values.quadrature_point(q));
+                      for (unsigned int i=0; i<fe_support_on_face[face].size(); ++i)
                         {
-                          sum_lf += (fe_face_values.shape_value(i,q) *
-                                     (fe_face_values_local[gradients].value(j,q) *
-                                      normals[q]
-                                      +
-                                      (convection_values_face[q] *
-                                       normals[q]
-                                       -
-                                       tau_stab[q]) *
-                                      fe_face_values_local[values].value(j,q))
-                                     ) * JxW[q];
-                          sum_fl += (fe_face_values.shape_value(i,q) *
-                                     (fe_face_values_local[gradients].value(j,q) *
-                                      normals[q]
-                                      +
-                                      tau_stab[q] *
-                                      fe_face_values_local[values].value(j,q))
-                                     ) * JxW[q];
+                          const unsigned int ii=fe_support_on_face[face][i];
+                          f_rhs(ii) -= tr_phi[i] * neumann_value * JxW;
                         }
-                      lf_matrix(j,i) += sum_lf;
-                      fl_matrix(i,j) -= sum_fl;
                     }
-                for (unsigned int i=0; i<dofs_per_cell; ++i)
-                  for (unsigned int j=0; j<dofs_per_cell; ++j)
-                    {
-                      double sum = 0;
-                      for (unsigned int q=0; q<n_face_q_points; ++q)
-                        sum += ((convection_values_face[q] * normals[q]
-                                 -
-                                 tau_stab[q]
-                                 ) *
-                                fe_face_values.shape_value(i,q) *
-                                fe_face_values.shape_value(j,q)
-                                ) * JxW[q];
-                      ff_matrix(i,j) += sum;
-                    }
-                if (cell->face(face)->at_boundary()
-                    &&
-                    (cell->face(face)->boundary_indicator() == 1))
+                }
+
+              for (unsigned int i=0; i<fe_local_support_on_face[face].size(); ++i)
+                for (unsigned int j=0; j<fe_local_support_on_face[face].size(); ++j)
                   {
-                    exact_solution.value_list(fe_face_values.get_quadrature_points(),
-                                              neumann_values);
-                    for (unsigned int i=0; i<dofs_per_cell; ++i)
-                      {
-                        double sum = 0;
-                        for (unsigned int q=0; q<n_face_q_points; ++q)
-                          sum -= (fe_face_values.shape_value(i,q) *
-                                  neumann_values[q]) * JxW[q];
-                        f_rhs(i) += sum;
-                      }
+                    const unsigned int ii=fe_local_support_on_face[face][i];
+                    const unsigned int jj=fe_local_support_on_face[face][j];
+                    ll_matrix(ii,jj) += tau_stab * u_phi[i] * u_phi[j] * JxW;
                   }
-              }
-            for (unsigned int i=0; i<loc_dofs_per_cell; ++i)
-              for (unsigned int j=0; j<loc_dofs_per_cell; ++j)
-                {
-                  double sum = 0;
-                  for (unsigned int q=0; q<n_face_q_points; ++q)
-                    sum += (tau_stab[q] *
-                            fe_face_values_local[values].value(i,q) *
-                            fe_face_values_local[values].value(j,q)) * JxW[q];
-                  ll_matrix(i,j) += sum;
-                }
 
-            // compute the local right hand side contributions from trace
-            if (trace_reconstruct)
-              {
-                fe_face_values.get_function_values (solution, trace_values);
-                for (unsigned int i=0; i<loc_dofs_per_cell; ++i)
+              // compute the local right hand side contributions from trace
+              if (trace_reconstruct)
+                for (unsigned int i=0; i<fe_local_support_on_face[face].size(); ++i)
                   {
-                    double sum = 0;
-                    for (unsigned int q=0; q<n_face_q_points; ++q)
-                      sum += ((fe_face_values_local[gradients].value(i,q) *
-                               normals[q]) *
-                              trace_values[q]
-                              +
-                              fe_face_values_local[values].value(i,q) *
-                              (convection_values_face[q] * normals[q]
-                               -
-                               tau_stab[q]) * trace_values[q]) * JxW[q];
-                    l_rhs(i) -= sum;
+                    const unsigned int ii=fe_local_support_on_face[face][i];
+                    l_rhs(ii) -= (q_phi[i] * normal
+                                  +
+                                  u_phi[i] * (convection * normal - tau_stab)
+                                  ) * trace_values[q] * JxW;
                   }
-              }
-          }
-
-		// invert ll_matrix and overwrite
-        ll_matrix.gauss_jordan();
-        if (!trace_reconstruct)
-          {
-            // tmp_matrix = fl_matrix * ll_matrix
-            fl_matrix.mmult(tmp_matrix, ll_matrix);
-            
-            // f_rhs = tmp_matrix * l_rhs
-            tmp_matrix.vmult_add(f_rhs, l_rhs);
-            
-            // ff_matrix = ff_matrix + tmp_matrix * lf_matrix
-            tmp_matrix.mmult(ff_matrix, lf_matrix, true);
-            cell->get_dof_indices(dof_indices);
-            constraints.distribute_local_to_global (ff_matrix, f_rhs,
-                                                    dof_indices,
-                                                    system_matrix, system_rhs);
-          }
-        else
-          {
-            ll_matrix.vmult(tmp_rhs, l_rhs);
-            loc_cell->set_dof_values(tmp_rhs, solution_local);
-          }
-      }
-  }
+            }
+        }
 
+      ll_matrix.gauss_jordan();
+      if (trace_reconstruct == false)
+        {
+          fl_matrix.mmult(tmp_matrix, ll_matrix);
+          tmp_matrix.vmult_add(f_rhs, l_rhs);
+          tmp_matrix.mmult(ff_matrix, lf_matrix, true);
+          cell->get_dof_indices(dof_indices);
+          constraints.distribute_local_to_global (ff_matrix, f_rhs,
+                                                  dof_indices,
+                                                  system_matrix, system_rhs);
+        }
+      else
+        {
+          ll_matrix.vmult(tmp_rhs, l_rhs);
+          loc_cell->set_dof_values(tmp_rhs, solution_local);
+        }
+    }
+}
 
 
-  template <int dim>
-  void HDG<dim>::solve ()
-  {
-    SolverControl solver_control (system_matrix.m()*10,
-                                  1e-10*system_rhs.l2_norm());
-    SolverGMRES<> solver (solver_control, 50);
-    solver.solve (system_matrix, solution, system_rhs,
-                  PreconditionIdentity());
 
-    std::cout << "   Number of GMRES iterations: " << solver_control.last_step()
-              << std::endl;
+template <int dim>
+void Step51<dim>::solve ()
+{
+  SolverControl solver_control (system_matrix.m()*10,
+                                1e-10*system_rhs.l2_norm());
+  SolverGMRES<> solver (solver_control, 50);
+  solver.solve (system_matrix, solution, system_rhs,
+                PreconditionIdentity());
 
-    system_matrix.clear();
-    sparsity_pattern.reinit(0,0,0,1);
-    constraints.distribute(solution);
+  std::cout << "   Number of GMRES iterations: " << solver_control.last_step()
+            << std::endl;
 
-    // update local values
-    assemble_system(true);
-  }
+  system_matrix.clear();
+  sparsity_pattern.reinit(0,0,0,1);
+  constraints.distribute(solution);
+
+  // update local values
+  assemble_system(true);
+}
 
 
 
-  template <int dim>
-  void
-  HDG<dim>::postprocess()
-  {
-    const unsigned int n_active_cells=triangulation.n_active_cells();
-    Vector<float> difference_per_cell (triangulation.n_active_cells());
-
-    ComponentSelectFunction<dim> value_select (dim, dim+1);
-    VectorTools::integrate_difference (mapping, dof_handler_local,
-                                       solution_local,
-                                       SolutionAndGradient<dim>(),
-                                       difference_per_cell,
-                                       QGauss<dim>(fe.degree+2),
-                                       VectorTools::L2_norm,
-                                       &value_select);
-    const double L2_error = difference_per_cell.l2_norm();
-
-    ComponentSelectFunction<dim> gradient_select (std::pair<unsigned int,unsigned int>(0, dim),
-                                                  dim+1);
-    VectorTools::integrate_difference (mapping, dof_handler_local,
-                                       solution_local,
-                                       SolutionAndGradient<dim>(),
-                                       difference_per_cell,
-                                       QGauss<dim>(fe.degree+2),
-                                       VectorTools::L2_norm,
-                                       &gradient_select);
-    const double grad_error = difference_per_cell.l2_norm();
-
-    convergence_table.add_value("cells", n_active_cells);
-    convergence_table.add_value("dofs", dof_handler.n_dofs());
-    convergence_table.add_value("val L2", L2_error);
-    convergence_table.add_value("grad L2", grad_error);
-
-    // construct post-processed solution with (hopefully) higher order of
-    // accuracy
-    QGauss<dim> quadrature(fe_u_post.degree+1);
-    FEValues<dim> fe_values(mapping, fe_u_post, quadrature,
-                            update_values | update_JxW_values |
-                            update_gradients);
-
-    const unsigned int n_q_points = quadrature.size();
-    std::vector<double> u_values(n_q_points);
-    std::vector<Tensor<1,dim> > u_gradients(n_q_points);
-    FEValuesExtractors::Vector gradients(0);
-    FEValuesExtractors::Scalar values(dim);
-    FEValues<dim> fe_values_local(mapping, fe_local, quadrature, update_values);
-    FullMatrix<double> cell_matrix(fe_u_post.dofs_per_cell,
-                                   fe_u_post.dofs_per_cell);
-    Vector<double> cell_rhs(fe_u_post.dofs_per_cell);
-    Vector<double> cell_sol(fe_u_post.dofs_per_cell);
-
-    typename DoFHandler<dim>::active_cell_iterator
-      cell_loc = dof_handler_local.begin_active(),
-      cell = dof_handler_u_post.begin_active(),
-      endc = dof_handler_u_post.end();
-    for ( ; cell != endc; ++cell, ++cell_loc)
-      {
-        fe_values.reinit(cell);
-        fe_values_local.reinit(cell_loc);
-
-        fe_values_local[values].get_function_values(solution_local, u_values);
-        fe_values_local[gradients].get_function_values(solution_local, u_gradients);
-        for (unsigned int i=1; i<fe_u_post.dofs_per_cell; ++i)
-          {
-            for (unsigned int j=0; j<fe_u_post.dofs_per_cell; ++j)
-              {
-                double sum = 0;
-                for (unsigned int q=0; q<quadrature.size(); ++q)
-                  sum += (fe_values.shape_grad(i,q) *
-                          fe_values.shape_grad(j,q)
-                          ) * fe_values.JxW(q);
-                cell_matrix(i,j) = sum;
-              }
-            double sum = 0;
-            for (unsigned int q=0; q<quadrature.size(); ++q)
-              sum -= (fe_values.shape_grad(i,q) * u_gradients[q]
-                      ) * fe_values.JxW(q);
-            cell_rhs(i) = sum;
-          }
-        for (unsigned int j=0; j<fe_u_post.dofs_per_cell; ++j)
-          {
-            double sum = 0;
-            for (unsigned int q=0; q<quadrature.size(); ++q)
-              sum += fe_values.shape_value(j,q) * fe_values.JxW(q);
-            cell_matrix(0,j) = sum;
-          }
+template <int dim>
+void
+Step51<dim>::postprocess()
+{
+  const unsigned int n_active_cells=triangulation.n_active_cells();
+  Vector<float> difference_per_cell (triangulation.n_active_cells());
+
+  ComponentSelectFunction<dim> value_select (dim, dim+1);
+  VectorTools::integrate_difference (mapping, dof_handler_local,
+                                     solution_local,
+                                     SolutionAndGradient<dim>(),
+                                     difference_per_cell,
+                                     QGauss<dim>(fe.degree+2),
+                                     VectorTools::L2_norm,
+                                     &value_select);
+  const double L2_error = difference_per_cell.l2_norm();
+
+  ComponentSelectFunction<dim> gradient_select (std::pair<unsigned int,unsigned int>(0, dim),
+                                                dim+1);
+  VectorTools::integrate_difference (mapping, dof_handler_local,
+                                     solution_local,
+                                     SolutionAndGradient<dim>(),
+                                     difference_per_cell,
+                                     QGauss<dim>(fe.degree+2),
+                                     VectorTools::L2_norm,
+                                     &gradient_select);
+  const double grad_error = difference_per_cell.l2_norm();
+
+  convergence_table.add_value("cells", n_active_cells);
+  convergence_table.add_value("dofs", dof_handler.n_dofs());
+  convergence_table.add_value("val L2", L2_error);
+  convergence_table.add_value("grad L2", grad_error);
+
+  // construct post-processed solution with (hopefully) higher order of
+  // accuracy
+  QGauss<dim> quadrature(fe_u_post.degree+1);
+  FEValues<dim> fe_values(mapping, fe_u_post, quadrature,
+                          update_values | update_JxW_values |
+                          update_gradients);
+
+  const unsigned int n_q_points = quadrature.size();
+  std::vector<double> u_values(n_q_points);
+  std::vector<Tensor<1,dim> > u_gradients(n_q_points);
+  FEValuesExtractors::Vector fluxes(0);
+  FEValuesExtractors::Scalar scalar(dim);
+  FEValues<dim> fe_values_local(mapping, fe_local, quadrature, update_values);
+  FullMatrix<double> cell_matrix(fe_u_post.dofs_per_cell,
+                                 fe_u_post.dofs_per_cell);
+  Vector<double> cell_rhs(fe_u_post.dofs_per_cell);
+  Vector<double> cell_sol(fe_u_post.dofs_per_cell);
+
+  typename DoFHandler<dim>::active_cell_iterator
+    cell_loc = dof_handler_local.begin_active(),
+    cell = dof_handler_u_post.begin_active(),
+    endc = dof_handler_u_post.end();
+  for ( ; cell != endc; ++cell, ++cell_loc)
+    {
+      fe_values.reinit(cell);
+      fe_values_local.reinit(cell_loc);
+
+      fe_values_local[scalar].get_function_values(solution_local, u_values);
+      fe_values_local[fluxes].get_function_values(solution_local, u_gradients);
+      for (unsigned int i=1; i<fe_u_post.dofs_per_cell; ++i)
         {
+          for (unsigned int j=0; j<fe_u_post.dofs_per_cell; ++j)
+            {
+              double sum = 0;
+              for (unsigned int q=0; q<quadrature.size(); ++q)
+                sum += (fe_values.shape_grad(i,q) *
+                        fe_values.shape_grad(j,q)
+                        ) * fe_values.JxW(q);
+              cell_matrix(i,j) = sum;
+            }
           double sum = 0;
           for (unsigned int q=0; q<quadrature.size(); ++q)
-            sum += u_values[q] * fe_values.JxW(q);
-          cell_rhs(0) = sum;
+            sum -= (fe_values.shape_grad(i,q) * u_gradients[q]
+                    ) * fe_values.JxW(q);
+          cell_rhs(i) = sum;
         }
-
-        cell_matrix.gauss_jordan();
-        cell_matrix.vmult(cell_sol, cell_rhs);
-        cell->distribute_local_to_global(cell_sol, solution_u_post);
+      for (unsigned int j=0; j<fe_u_post.dofs_per_cell; ++j)
+        {
+          double sum = 0;
+          for (unsigned int q=0; q<quadrature.size(); ++q)
+            sum += fe_values.shape_value(j,q) * fe_values.JxW(q);
+          cell_matrix(0,j) = sum;
+        }
+      {
+        double sum = 0;
+        for (unsigned int q=0; q<quadrature.size(); ++q)
+          sum += u_values[q] * fe_values.JxW(q);
+        cell_rhs(0) = sum;
       }
 
-    VectorTools::integrate_difference (mapping, dof_handler_u_post,
-                                       solution_u_post,
-                                       Solution<dim>(),
-                                       difference_per_cell,
-                                       QGauss<dim>(fe.degree+3),
-                                       VectorTools::L2_norm);
-    double post_error = difference_per_cell.l2_norm();
-    convergence_table.add_value("val L2-post", post_error);
-  }
+      cell_matrix.gauss_jordan();
+      cell_matrix.vmult(cell_sol, cell_rhs);
+      cell->distribute_local_to_global(cell_sol, solution_u_post);
+    }
 
+  VectorTools::integrate_difference (mapping, dof_handler_u_post,
+                                     solution_u_post,
+                                     Solution<dim>(),
+                                     difference_per_cell,
+                                     QGauss<dim>(fe.degree+3),
+                                     VectorTools::L2_norm);
+  double post_error = difference_per_cell.l2_norm();
+  convergence_table.add_value("val L2-post", post_error);
+}
 
 
-  template <int dim>
-  void HDG<dim>::output_results (const unsigned int cycle)
-  {
-    std::string filename;
+
+template <int dim>
+void Step51<dim>::output_results (const unsigned int cycle)
+{
+  std::string filename;
+  switch (refinement_mode)
+    {
+    case global_refinement:
+      filename = "solution-global";
+      break;
+    case adaptive_refinement:
+      filename = "solution-adaptive";
+      break;
+    default:
+      Assert (false, ExcNotImplemented());
+    }
+    
+  std::string face_out(filename);
+  face_out += "-face";
+
+  filename += "-q" + Utilities::int_to_string(fe.degree,1);
+  filename += "-" + Utilities::int_to_string(cycle,2);
+  filename += ".vtk";
+  std::ofstream output (filename.c_str());
+
+  DataOut<dim> data_out;
+  std::vector<std::string> names (dim, "gradient");
+  names.push_back ("solution");
+  std::vector<DataComponentInterpretation::DataComponentInterpretation>
+    component_interpretation
+    (dim+1, DataComponentInterpretation::component_is_part_of_vector);
+  component_interpretation[dim]
+    = DataComponentInterpretation::component_is_scalar;
+  data_out.add_data_vector (dof_handler_local, solution_local,
+                            names, component_interpretation);
+                            
+  // Post-processed solution:  can now add more than 1 dof_handler to 
+  // the DataOut object!
+  std::vector<std::string> post_name(1,"u_post");
+  std::vector<DataComponentInterpretation::DataComponentInterpretation> 
+			 post_comp_type(1, DataComponentInterpretation::component_is_scalar);
+  data_out.add_data_vector (dof_handler_u_post, solution_u_post,
+						  post_name, post_comp_type);
+
+  data_out.build_patches (fe.degree);
+  data_out.write_vtk (output);
+    
+  face_out += "-q" + Utilities::int_to_string(fe.degree,1);
+  face_out += "-" + Utilities::int_to_string(cycle,2);
+  face_out += ".vtk";
+  std::ofstream face_output (face_out.c_str());
+
+  DataOutFaces<dim> data_out_face(false);
+  std::vector<std::string> face_name(1,"lambda");
+  std::vector<DataComponentInterpretation::DataComponentInterpretation> 
+			 face_component_type(1, DataComponentInterpretation::component_is_scalar);
+
+  data_out_face.add_data_vector (dof_handler, 
+							  	solution, 
+								face_name,
+								face_component_type);
+								
+  data_out_face.build_patches (fe.degree);
+  data_out_face.write_vtk (face_output);
+}
+
+
+
+template <int dim>
+void Step51<dim>::refine_grid (const unsigned int cycle)
+{
+  const bool do_cube = true;
+  if (cycle == 0)
+    {
+      if (!do_cube)
+        {
+          GridGenerator::hyper_ball (triangulation);
+          static const HyperBallBoundary<dim> boundary;
+          triangulation.set_boundary(0, boundary);
+          triangulation.refine_global(6-2*dim);
+        }
+      else
+        GridGenerator::subdivided_hyper_cube (triangulation, 2, -1, 1);
+    }
+  else
     switch (refinement_mode)
       {
       case global_refinement:
-        filename = "solution-global";
-        break;
+        {
+          if (do_cube)
+            {
+              triangulation.clear();
+              GridGenerator::subdivided_hyper_cube (triangulation, 2+(cycle%2), -1, 1);
+              triangulation.refine_global(3-dim+cycle/2);
+            }
+          else
+            triangulation.refine_global (1);
+          break;
+        }
+
       case adaptive_refinement:
-        filename = "solution-adaptive";
+      {
+        Vector<float> estimated_error_per_cell (triangulation.n_active_cells());
+
+        FEValuesExtractors::Scalar scalar(dim);
+        typename FunctionMap<dim>::type neumann_boundary;
+        KellyErrorEstimator<dim>::estimate (dof_handler_local,
+                                            QGauss<dim-1>(3),
+                                            neumann_boundary,
+                                            solution_local,
+                                            estimated_error_per_cell,
+                                            fe_local.component_mask(scalar));
+
+        GridRefinement::refine_and_coarsen_fixed_number (triangulation,
+                                                         estimated_error_per_cell,
+                                                         0.3, 0.);
+
+        triangulation.execute_coarsening_and_refinement ();
+
         break;
+      }
+
       default:
+      {
         Assert (false, ExcNotImplemented());
       }
-      
-    std::string face_out(filename);
-    face_out += "-face";
-
-    filename += "-q" + Utilities::int_to_string(fe.degree,1);
-    filename += "-" + Utilities::int_to_string(cycle,2);
-    filename += ".vtk";
-    std::ofstream output (filename.c_str());
-
-    DataOut<dim> data_out;
-    std::vector<std::string> names (dim, "gradient");
-    names.push_back ("solution");
-    std::vector<DataComponentInterpretation::DataComponentInterpretation>
-      component_interpretation
-      (dim+1, DataComponentInterpretation::component_is_part_of_vector);
-    component_interpretation[dim]
-      = DataComponentInterpretation::component_is_scalar;
-    data_out.add_data_vector (dof_handler_local, solution_local,
-                              names, component_interpretation);
-
-	// Post-processed solution
-	std::vector<std::string> post_name(1,"u_post");
-    std::vector<DataComponentInterpretation::DataComponentInterpretation> 
-    			 post_comp_type(1, DataComponentInterpretation::component_is_scalar);
-    data_out.add_data_vector (dof_handler_u_post, solution_u_post,
-                              post_name, post_comp_type);
-                              
-    // build patches based on the highest degree, i.e. the post-proc'd soln                          
-    data_out.build_patches (fe_u_post.degree);
-    data_out.write_vtk (output);
-    
-    face_out += "-q" + Utilities::int_to_string(fe.degree,1);
-    face_out += "-" + Utilities::int_to_string(cycle,2);
-    face_out += ".vtk";
-    std::ofstream face_output (face_out.c_str());
-    
-    DataOutFaces<dim> data_out_face(false);
-    std::vector<std::string> face_name(1,"lambda");
-    std::vector<DataComponentInterpretation::DataComponentInterpretation> 
-    			 face_component_type(1, DataComponentInterpretation::component_is_scalar);
-    
-    data_out_face.add_data_vector (dof_handler, 
-    								solution, 
-    								face_name,
-    								face_component_type);
-    								
-    data_out_face.build_patches (fe.degree);
-    data_out_face.write_vtk (face_output);
-    
+      }
   }
 
 
 
 
-  template <int dim>
-  void HDG<dim>::run ()
-  {
-    const bool do_cube = true;
-    if (!do_cube)
-      {
-        GridGenerator::hyper_ball (triangulation);
-        static const HyperBallBoundary<dim> boundary;
-        triangulation.set_boundary(0, boundary);
-        triangulation.refine_global(6-2*dim);
-      }
 
-    for (unsigned int cycle=0; cycle<10; ++cycle)
-      {
-        std::cout << "Cycle " << cycle << ':' << std::endl;
-
-        if (do_cube)
-          {
-            triangulation.clear();
-            GridGenerator::subdivided_hyper_cube (triangulation, 2+(cycle%2), -1, 1);
-            triangulation.refine_global(3-dim+cycle/2);
-          }
-        else triangulation.refine_global(1);
-
-        setup_system ();
-        assemble_system (false);
-        solve ();
-        postprocess();
-        output_results (cycle);
-      }
+template <int dim>
+void Step51<dim>::run ()
+{
+  for (unsigned int cycle=0; cycle<10; ++cycle)
+    {
+      std::cout << "Cycle " << cycle << ':' << std::endl;
+      
+      refine_grid (cycle);
+      setup_system ();
+      assemble_system (false);
+      solve ();
+      postprocess();
+      output_results (cycle);
+    }
 
 
 
-    convergence_table.set_precision("val L2", 3);
-    convergence_table.set_scientific("val L2", true);
-    convergence_table.set_precision("grad L2", 3);
-    convergence_table.set_scientific("grad L2", true);
-    convergence_table.set_precision("val L2-post", 3);
-    convergence_table.set_scientific("val L2-post", true);
+  convergence_table.set_precision("val L2", 3);
+  convergence_table.set_scientific("val L2", true);
+  convergence_table.set_precision("grad L2", 3);
+  convergence_table.set_scientific("grad L2", true);
+  convergence_table.set_precision("val L2-post", 3);
+  convergence_table.set_scientific("val L2-post", true);
 
-    convergence_table
-      .evaluate_convergence_rates("val L2", "cells", ConvergenceTable::reduction_rate_log2, dim);
-    convergence_table
-      .evaluate_convergence_rates("grad L2", "cells", ConvergenceTable::reduction_rate_log2, dim);
-    convergence_table
-      .evaluate_convergence_rates("val L2-post", "cells", ConvergenceTable::reduction_rate_log2, dim);
-    convergence_table.write_text(std::cout);
-  }
+  convergence_table
+    .evaluate_convergence_rates("val L2", "cells", ConvergenceTable::reduction_rate_log2, dim);
+  convergence_table
+    .evaluate_convergence_rates("grad L2", "cells", ConvergenceTable::reduction_rate_log2, dim);
+  convergence_table
+    .evaluate_convergence_rates("val L2-post", "cells", ConvergenceTable::reduction_rate_log2, dim);
+  convergence_table.write_text(std::cout);
 }
 
 
@@ -891,7 +921,6 @@ int main (int argc, char** argv)
   try
     {
       using namespace dealii;
-      using namespace Step51;
 
       deallog.depth_console (0);
 
@@ -902,7 +931,7 @@ int main (int argc, char** argv)
                   << "=============================================" << std::endl
                   << std::endl;
 
-        HDG<dim> hdg_problem (1, HDG<dim>::adaptive_refinement);
+        Step51<dim> hdg_problem (1, Step51<dim>::adaptive_refinement);
         hdg_problem.run ();
 
         std::cout << std::endl;
@@ -913,7 +942,7 @@ int main (int argc, char** argv)
                   << "===========================================" << std::endl
                   << std::endl;
 
-        HDG<dim> hdg_problem (1, HDG<dim>::global_refinement);
+        Step51<dim> hdg_problem (1, Step51<dim>::global_refinement);
         hdg_problem.run ();
 
         std::cout << std::endl;
@@ -924,7 +953,7 @@ int main (int argc, char** argv)
                   << "===========================================" << std::endl
                   << std::endl;
 
-        HDG<dim> hdg_problem (3, HDG<dim>::global_refinement);
+        Step51<dim> hdg_problem (3, Step51<dim>::global_refinement);
         hdg_problem.run ();
 
         std::cout << std::endl;
-- 
2.39.5