From: Scott Miller <miller.scott.t@gmail.com>
Date: Thu, 8 Aug 2013 02:07:07 +0000 (+0000)
Subject: Add output for step-51
X-Git-Tag: v8.1.0~1118
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=4c443c0feb1c68650f6f9820bf372ff0fb64477b;p=dealii.git

Add output for step-51

git-svn-id: https://svn.dealii.org/trunk@30254 0785d39b-7218-0410-832d-ea1e28bc413d
---

diff --git a/deal.II/examples/step-51/step-51.cc b/deal.II/examples/step-51/step-51.cc
index d67bbfb248..ab4bf33dae 100644
--- a/deal.II/examples/step-51/step-51.cc
+++ b/deal.II/examples/step-51/step-51.cc
@@ -16,7 +16,7 @@
 
  *
  * Author: Martin Kronbichler, TU Muenchen,
- *         Scott T. Miller, xxx, 2013
+ *         Scott T. Miller, The Pennsylvania State University, 2013
  */
 
 
@@ -32,829 +32,855 @@
 #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>
 
 #include <deal.II/lac/chunk_sparse_matrix.h>
 #include <deal.II/numerics/data_out_faces.h>
 
-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
+namespace Step51
 {
-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)   };
+  using namespace dealii;
 
-template <int dim>
-const double SolutionBase<dim>::width = 1./5.;
+  // @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>
-class ConvectionVelocity : public TensorFunction<1,dim>
-{
-public:
-  ConvectionVelocity() : TensorFunction<1,dim>() {}
+  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)
+      };
 
-  virtual Tensor<1,dim> value (const Point<dim> &p) const;
-};
 
+  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>
-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 <int dim>
+  const double SolutionBase<dim>::width = 1./5.;
 
 
-template <int dim>
-class Solution : public Function<dim>,
-                 protected SolutionBase<dim>
-{
-public:
-  Solution () : Function<dim>() {}
 
-  virtual double value (const Point<dim>   &p,
-                        const unsigned int  component = 0) const;
+  template <int dim>
+  class ConvectionVelocity : public TensorFunction<1,dim>
+  {
+  public:
+    ConvectionVelocity() : TensorFunction<1,dim>() {}
 
-  virtual Tensor<1,dim> gradient (const Point<dim>   &p,
-                                  const unsigned int  component = 0) const;
-};
+    virtual Tensor<1,dim> value (const Point<dim> &p) 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)
-    {
-      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>
+  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;
+  }
 
-  return return_value /
-    Utilities::fixed_power<dim>(std::sqrt(2. * numbers::PI) * this->width);
-}
 
+  template <int dim>
+  class Solution : public Function<dim>,
+                   protected SolutionBase<dim>
+  {
+  public:
+    Solution () : 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;
+    virtual Tensor<1,dim> gradient (const Point<dim>   &p,
+				    const unsigned int  component = 0) const;
+  };
 
-  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);
-    }
 
-  return return_value / Utilities::fixed_power<dim>(std::sqrt(2 * numbers::PI) *
-                                                    this->width);
-}
+  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));
+      }
 
+    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) {}
 
-  virtual void vector_value (const Point<dim>   &p,
-                             Vector<double>     &v) const
+  template <int dim>
+  Tensor<1,dim> Solution<dim>::gradient (const Point<dim>   &p,
+                                         const unsigned int) const
   {
-    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);
-  }
-};
+    Tensor<1,dim> return_value;
 
+    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 RightHandSide : public Function<dim>,
-                      protected SolutionBase<dim>
-{
-public:
-  RightHandSide () : 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;
 
-private:
-  const ConvectionVelocity<dim> convection_velocity;
-};
 
+  template <int dim>
+  class SolutionAndGradient : public Function<dim>,
+                              protected SolutionBase<dim>
+  {
+  public:
+    SolutionAndGradient () : Function<dim>(dim) {}
 
-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)
+    virtual void vector_value (const Point<dim>   &p,
+                               Vector<double>     &v) const
     {
-      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)));
+      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);
     }
+  };
 
-  return return_value / Utilities::fixed_power<dim>(std::sqrt(2 * numbers::PI)
-                                                    * this->width);
-}
-
-
-
-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;
-
-  FESystem<dim>        fe_local;
-  DoFHandler<dim>      dof_handler_local;
+  template <int dim>
+  class RightHandSide : public Function<dim>,
+                        protected SolutionBase<dim>
+  {
+  public:
+    RightHandSide () : Function<dim>() {}
 
-  FE_FaceQ<dim>        fe;
-  DoFHandler<dim>      dof_handler;
+    virtual double value (const Point<dim>   &p,
+			  const unsigned int  component = 0) const;
 
-  FE_DGQ<dim>          fe_u_post;
-  DoFHandler<dim>      dof_handler_u_post;
+  private:
+    const ConvectionVelocity<dim> convection_velocity;
+  };
 
-  ConstraintMatrix     constraints;
-  ChunkSparsityPattern sparsity_pattern;
-  ChunkSparseMatrix<double> system_matrix;
 
-  Vector<double>       solution;
-  Vector<double>       system_rhs;
+  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)));
+      }
 
-  Vector<double>       solution_local;
-  Vector<double>       solution_u_post;
+    return return_value / Utilities::fixed_power<dim>(std::sqrt(2 * numbers::PI)
+                                                      * this->width);
+  }
 
-  const RefinementMode refinement_mode;
 
-  ConvergenceTable     convergence_table;
-};
 
+  template <int dim>
+  class HDG
+  {
+  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;
+
+	// 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;
+
+    Vector<double>       solution;
+    Vector<double>       system_rhs;
+
+	// 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;
+
+	// Same as step-7:
+    const RefinementMode refinement_mode;
+
+    ConvergenceTable     convergence_table;
+  };
+
+
+
+  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>
+  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);
 
-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)
-{}
+    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>::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 ();
+    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)
+        {
+          constraints.add_line(it->first);
+          constraints.set_inhomogeneity(it->first, it->second);
+        }
+    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);
+    {
+      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);
   }
-  system_matrix.reinit (sparsity_pattern);
-}
 
 
 
-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
+  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)
-    {
-      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);
-
-      for (unsigned int q=0; q<n_q_points; ++q)
-        {
-          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 (; 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)
             {
-              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;
+              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 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;
-                      }
-
-                  if (cell->face(face)->at_boundary()
-                      &&
-                      (cell->face(face)->boundary_indicator() == 1))
+        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)
                     {
-                      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)
+                      double sum_lf = 0., sum_fl = 0.;
+                      for (unsigned int q=0; q<n_face_q_points; ++q)
                         {
-                          const unsigned int ii=fe_support_on_face[face][i];
-                          f_rhs(ii) -= tr_phi[i] * neumann_value * JxW;
+                          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];
                         }
+                      lf_matrix(j,i) += sum_lf;
+                      fl_matrix(i,j) -= sum_fl;
                     }
-                }
-
-              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)
+                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))
                   {
-                    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;
+                    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;
+                      }
                   }
+              }
+            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)
-                for (unsigned int i=0; i<fe_local_support_on_face[face].size(); ++i)
+            // 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)
                   {
-                    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;
+                    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;
                   }
-            }
-        }
-
-      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);
-        }
-    }
-}
+              }
+          }
+
+		// 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);
+          }
+      }
+  }
 
 
 
-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());
+  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;
+    std::cout << "   Number of GMRES iterations: " << solver_control.last_step()
+              << std::endl;
 
-  system_matrix.clear();
-  sparsity_pattern.reinit(0,0,0,1);
-  constraints.distribute(solution);
+    system_matrix.clear();
+    sparsity_pattern.reinit(0,0,0,1);
+    constraints.distribute(solution);
 
-  // update local values
-  assemble_system(true);
-}
+    // update local values
+    assemble_system(true);
+  }
 
 
 
-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 -= (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)
+  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;
+          }
         {
           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;
+            sum += u_values[q] * fe_values.JxW(q);
+          cell_rhs(0) = 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;
-      }
-
-      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);
-}
 
+        cell_matrix.gauss_jordan();
+        cell_matrix.vmult(cell_sol, cell_rhs);
+        cell->distribute_local_to_global(cell_sol, solution_u_post);
+      }
 
-
-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());
-    }
-
-  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);
-
-  data_out.build_patches (fe.degree);
-  data_out.write_vtk (output);
-}
+    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 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
+  template <int dim>
+  void HDG<dim>::output_results (const unsigned int cycle)
+  {
+    std::string filename;
     switch (refinement_mode)
       {
       case global_refinement:
-        {
-          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;
-        }
-
+        filename = "solution-global";
+        break;
       case adaptive_refinement:
-      {
-        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 ();
-
+        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
+	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);
+      }
 
-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);
-    }
+    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);
+      }
 
 
 
-  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);
+  }
 }
 
 
@@ -865,6 +891,7 @@ int main (int argc, char** argv)
   try
     {
       using namespace dealii;
+      using namespace Step51;
 
       deallog.depth_console (0);
 
@@ -875,7 +902,7 @@ int main (int argc, char** argv)
                   << "=============================================" << std::endl
                   << std::endl;
 
-        Step51<dim> hdg_problem (1, Step51<dim>::adaptive_refinement);
+        HDG<dim> hdg_problem (1, HDG<dim>::adaptive_refinement);
         hdg_problem.run ();
 
         std::cout << std::endl;
@@ -886,7 +913,7 @@ int main (int argc, char** argv)
                   << "===========================================" << std::endl
                   << std::endl;
 
-        Step51<dim> hdg_problem (1, Step51<dim>::global_refinement);
+        HDG<dim> hdg_problem (1, HDG<dim>::global_refinement);
         hdg_problem.run ();
 
         std::cout << std::endl;
@@ -897,7 +924,7 @@ int main (int argc, char** argv)
                   << "===========================================" << std::endl
                   << std::endl;
 
-        Step51<dim> hdg_problem (3, Step51<dim>::global_refinement);
+        HDG<dim> hdg_problem (3, HDG<dim>::global_refinement);
         hdg_problem.run ();
 
         std::cout << std::endl;