Example 10 - Solving a Transient System with Adaptive Mesh Refinement

This example shows how a simple, linear transient system can be solved in parallel. The system is simple scalar convection-diffusion with a specified external velocity. The initial condition is given, and the solution is advanced in time with a standard Crank-Nicholson time-stepping strategy.

Also, we use this example to demonstrate writing out and reading in of solutions. We do 25 time steps, then save the solution and do another 25 time steps starting from the saved solution.

C++ include files that we need

        #include <iostream>
        #include <algorithm>
        #include <cmath>

Basic include file needed for the mesh functionality.

        #include "libmesh.h"
        #include "mesh.h"
        #include "mesh_refinement.h"
        #include "gmv_io.h"
        #include "equation_systems.h"
        #include "fe.h"
        #include "quadrature_gauss.h"
        #include "dof_map.h"
        #include "sparse_matrix.h"
        #include "numeric_vector.h"
        #include "dense_matrix.h"
        #include "dense_vector.h"
        
        #include "getpot.h"

Some (older) compilers do not offer full stream functionality, \p OStringStream works around this. Check example 9 for details.

        #include "o_string_stream.h"

This example will solve a linear transient system, so we need to include the \p TransientLinearImplicitSystem definition.

        #include "transient_system.h"
        #include "linear_implicit_system.h"
        #include "vector_value.h"

To refine the mesh we need an \p ErrorEstimator object to figure out which elements to refine.

        #include "error_vector.h"
        #include "kelly_error_estimator.h"

The definition of a geometric element

        #include "elem.h"

Function prototype. This function will assemble the system matrix and right-hand-side at each time step. Note that since the system is linear we technically do not need to assmeble the matrix at each time step, but we will anyway. In subsequent examples we will employ adaptive mesh refinement, and with a changing mesh it will be necessary to rebuild the system matrix.

        void assemble_cd (EquationSystems& es,
        		  const std::string& system_name);

Function prototype. This function will initialize the system. Initialization functions are optional for systems. They allow you to specify the initial values of the solution. If an initialization function is not provided then the default (0) solution is provided.

        void init_cd (EquationSystems& es,
        	      const std::string& system_name);

Exact solution function prototype. This gives the exact solution as a function of space and time. In this case the initial condition will be taken as the exact solution at time 0, as will the Dirichlet boundary conditions at time t.

        Real exact_solution (const Real x,
        		     const Real y,
        		     const Real t);
        
        Number exact_value (const Point& p,
        		    const Parameters& parameters,
        		    const std::string&,
        		    const std::string&)
        {
          return exact_solution(p(0), p(1), parameters.get<Real> ("time"));
        }

Begin the main program. Note that the first statement in the program throws an error if you are in complex number mode, since this example is only intended to work with real numbers.

        int main (int argc, char** argv)
        {

Initialize libMesh.

          libMesh::init (argc, argv);
        
        #ifndef ENABLE_AMR
          std::cerr << "ERROR: This example requires libMesh to be\n"
                    << "compiled with AMR support!"
                    << std::endl;
          return 0;
        #else
        
          {

Brief message to the user regarding the program name and command line arguments.

Use commandline parameter to specify if we are to read in an initial solution or generate it ourself

            std::cout << "Usage:\n"
              <<"\t " << argv[0] << " -init_timestep 0\n"
              << "OR\n"
              <<"\t " << argv[0] << " -read_solution -init_timestep 26\n"
              << std::endl;
        
            std::cout << "Running: " << argv[0];
        
            for (int i=1; i<argc; i++)
              std::cout << " " << argv[i];
        
            std::cout << std::endl << std::endl;

Create a GetPot object to parse the command line

            GetPot command_line (argc, argv);

This boolean value is obtained from the command line, it is true if the flag "-read_solution" is present, false otherwise. It indicates whether we are going to read in the mesh and solution files "saved_mesh.xda" and "saved_solution.xda" or whether we are going to start from scratch by just reading "mesh.xda"

            const bool read_solution   = command_line.search("-read_solution");

This value is also obtained from the commandline and it specifies the initial value for the t_step looping variable. We must distinguish between the two cases here, whether we read in the solution or we started from scratch, so that we do not overwrite the gmv output files.

            unsigned int init_timestep = 0;

Search the command line for the "init_timestep" flag and if it is present, set init_timestep accordingly.

            if(command_line.search("-init_timestep"))
              init_timestep = command_line.next(0);
            else
            {
              std::cerr << "ERROR: Initial timestep not specified\n" << std::endl;

This handy function will print the file name, line number, and then abort. Currrently the library does not use C++ exception handling.

              error();
            }

This value is also obtained from the command line, and specifies the number of time steps to take.

            unsigned int n_timesteps = 0;

Again do a search on the command line for the argument

            if(command_line.search("-n_timesteps"))
              n_timesteps = command_line.next(0);
            else
            {
              std::cerr << "ERROR: Number of timesteps not specified\n" << std::endl;
              error();
            }

Create a two-dimensional mesh.

            Mesh mesh (2);

Create an equation systems object.

            EquationSystems equation_systems (mesh);
            MeshRefinement mesh_refinement (mesh);

First we process the case where we do not read in the solution

            if(!read_solution)
            {

Read the mesh from file.

              mesh.read ("mesh.xda");

Uniformly refine the mesh 5 times

              if(!read_solution)
                mesh_refinement.uniformly_refine (5);

Print information about the mesh to the screen.

              mesh.print_info();

Declare the system and its variables. Begin by creating a transient system named "Convection-Diffusion".

              TransientLinearImplicitSystem & system = 
                equation_systems.add_system<TransientLinearImplicitSystem> 
                ("Convection-Diffusion");

Adds the variable "u" to "Convection-Diffusion". "u" will be approximated using first-order approximation.

              system.add_variable ("u", FIRST);

Give the system a pointer to the matrix assembly and initialization functions.

              system.attach_assemble_function (assemble_cd);
              system.attach_init_function (init_cd);

Initialize the data structures for the equation system.

              equation_systems.init ();
            }

Otherwise we read in the solution and mesh

            else 
            {

Read in the mesh stored in "saved_mesh.xda"

              mesh.read("saved_mesh.xda");

Print information about the mesh to the screen.

              mesh.print_info();

Read in the solution stored in "saved_solution.xda"

              equation_systems.read("saved_solution.xda", libMeshEnums::READ);

Get a reference to the system so that we can call update() on it

              TransientLinearImplicitSystem & system = 
                equation_systems.get_system<TransientLinearImplicitSystem> 
                ("Convection-Diffusion");

We need to call update to put system in a consistent state with the solution that was read in

              system.update();

Attach the same matrix assembly function as above. Note, we do not have to attach an init() function since we are initializing the system by reading in "saved_solution.xda"

              system.attach_assemble_function (assemble_cd);
            }

Prints information about the system to the screen.

            equation_systems.print_info();
        
            
            equation_systems.parameters.set<unsigned int>
              ("linear solver maximum iterations") = 250;
            equation_systems.parameters.set<Real>
              ("linear solver tolerance") = TOLERANCE;
              
            if(!read_solution)

Write out the initial condition

              GMVIO(mesh).write_equation_systems ("out.gmv.000",
        					  equation_systems);
            else

Write out the solution that was read in

              GMVIO(mesh).write_equation_systems ("solution_read_in.gmv",
        					  equation_systems);

The Convection-Diffusion system requires that we specify the flow velocity. We will specify it as a RealVectorValue data type and then use the Parameters object to pass it to the assemble function.

            equation_systems.parameters.set<RealVectorValue>("velocity") = 
              RealVectorValue (0.8, 0.8);

Solve the system "Convection-Diffusion". This will be done by looping over the specified time interval and calling the \p solve() member at each time step. This will assemble the system and call the linear solver.

            const Real dt = 0.025;
            Real time     = init_timestep*dt;

We do 25 timesteps both before and after writing out the intermediate solution

            for(unsigned int t_step=init_timestep; 
                             t_step<(init_timestep+n_timesteps); 
                             t_step++)
              {

Increment the time counter, set the time and the time step size as parameters in the EquationSystem.

                time += dt;
        
        	equation_systems.parameters.set<Real> ("time") = time;
        	equation_systems.parameters.set<Real> ("dt")   = dt;

A pretty update message

                std::cout << " Solving time step ";

As already seen in example 9, use a work-around for missing stream functionality (of older compilers). Add a set of scope braces to enforce data locality.

                {
        	  OStringStream out;
        
        	  OSSInt(out,2,t_step);
        	  out << ", time=";
        	  OSSRealzeroleft(out,6,3,time);
        	  out <<  "...";
        	  std::cout << out.str() << std::endl;
        	}

At this point we need to update the old solution vector. The old solution vector will be the current solution vector from the previous time step. We will do this by extracting the system from the \p EquationSystems object and using vector assignment. Since only \p TransientLinearImplicitSystems (and systems derived from them) contain old solutions we need to specify the system type when we ask for it.

                TransientLinearImplicitSystem &  system =
        	  equation_systems.get_system<TransientLinearImplicitSystem>("Convection-Diffusion");
        
        	*system.old_local_solution = *system.current_local_solution;

The number of refinement steps per time step.

                const unsigned int max_r_steps = 2;

A refinement loop.

                for (unsigned int r_step=0; r_step<max_r_steps; r_step++)
        	  {

Assemble & solve the linear system

                    system.solve();

Possibly refine the mesh

                    if (r_step+1 != max_r_steps)
        	      {
        		std::cout << "  Refining the mesh..." << std::endl;

The \p ErrorVector is a particular \p StatisticsVector for computing error information on a finite element mesh.

                        ErrorVector error;

The \p ErrorEstimator class interrogates a finite element solution and assigns to each element a positive error value. This value is used for deciding which elements to refine and which to coarsen. ErrorEstimator* error_estimator = new KellyErrorEstimator;

                        KellyErrorEstimator error_estimator;

Compute the error for each active element using the provided \p flux_jump indicator. Note in general you will need to provide an error estimator specifically designed for your application.

                        error_estimator.estimate_error (system,
        						error);

This takes the error in \p error and decides which elements will be coarsened or refined. Any element within 20% of the maximum error on any element will be refined, and any element within 7% of the minimum error on any element might be coarsened. Note that the elements flagged for refinement will be refined, but those flagged for coarsening _might_ be coarsened.

                        mesh_refinement.refine_fraction() = 0.80;
        		mesh_refinement.coarsen_fraction() = 0.07;
        		mesh_refinement.max_h_level() = 5;
        		mesh_refinement.flag_elements_by_error_fraction (error);

This call actually refines and coarsens the flagged elements.

                        mesh_refinement.refine_and_coarsen_elements();

This call reinitializes the \p EquationSystems object for the newly refined mesh. One of the steps in the reinitialization is projecting the \p solution, \p old_solution, etc... vectors from the old mesh to the current one.

                        equation_systems.reinit ();
        	      }	    
        	  }

Output evey 10 timesteps to file.

                if ( (t_step+1)%10 == 0)
        	  {
        	    OStringStream file_name;
        
        	    file_name << "out.gmv.";
        	    OSSRealzeroright(file_name,3,0,t_step+1);
        
        	    GMVIO(mesh).write_equation_systems (file_name.str(),
        						equation_systems);
        	  }
              }
        
              if(!read_solution)
              {
                mesh.write("saved_mesh.xda");
                equation_systems.write("saved_solution.xda", libMeshEnums::WRITE);
              }
          }
        #endif // #ifndef ENABLE_AMR

All done.

          return libMesh::close ();
        }

Here we define the initialization routine for the Convection-Diffusion system. This routine is responsible for applying the initial conditions to the system.

        void init_cd (EquationSystems& es,
        	      const std::string& system_name)
        {

It is a good idea to make sure we are initializing the proper system.

          assert (system_name == "Convection-Diffusion");

Get a reference to the Convection-Diffusion system object.

          TransientLinearImplicitSystem & system =
            es.get_system<TransientLinearImplicitSystem>("Convection-Diffusion");

Project initial conditions at time 0

          es.parameters.set<Real> ("time") = 0;
          
          system.project_solution(exact_value, NULL, es.parameters);
        }

This function defines the assembly routine which will be called at each time step. It is responsible for computing the proper matrix entries for the element stiffness matrices and right-hand sides.

        void assemble_cd (EquationSystems& es,
        		  const std::string& system_name)
        {
        #ifdef ENABLE_AMR

It is a good idea to make sure we are assembling the proper system.

          assert (system_name == "Convection-Diffusion");

Get a constant reference to the mesh object.

          const Mesh& mesh = es.get_mesh();

The dimension that we are running

          const unsigned int dim = mesh.mesh_dimension();

Get a reference to the Convection-Diffusion system object.

          TransientLinearImplicitSystem & system =
            es.get_system<TransientLinearImplicitSystem> ("Convection-Diffusion");

Get the Finite Element type for the first (and only) variable in the system.

          FEType fe_type = system.variable_type(0);

Build a Finite Element object of the specified type. Since the \p FEBase::build() member dynamically creates memory we will store the object as an \p AutoPtr. This can be thought of as a pointer that will clean up after itself.

          AutoPtr<FEBase> fe      (FEBase::build(dim, fe_type));
          AutoPtr<FEBase> fe_face (FEBase::build(dim, fe_type));

A Gauss quadrature rule for numerical integration. Let the \p FEType object decide what order rule is appropriate.

          QGauss qrule (dim,   fe_type.default_quadrature_order());
          QGauss qface (dim-1, fe_type.default_quadrature_order());

Tell the finite element object to use our quadrature rule.

          fe->attach_quadrature_rule      (&qrule);
          fe_face->attach_quadrature_rule (&qface);

Here we define some references to cell-specific data that will be used to assemble the linear system. We will start with the element Jacobian * quadrature weight at each integration point.

          const std::vector<Real>& JxW      = fe->get_JxW();
          const std::vector<Real>& JxW_face = fe_face->get_JxW();

The element shape functions evaluated at the quadrature points.

          const std::vector<std::vector<Real> >& phi = fe->get_phi();
          const std::vector<std::vector<Real> >& psi = fe_face->get_phi();

The element shape function gradients evaluated at the quadrature points.

          const std::vector<std::vector<RealGradient> >& dphi = fe->get_dphi();

The XY locations of the quadrature points used for face integration

          const std::vector<Point>& qface_points = fe_face->get_xyz();

A reference to the \p DofMap object for this system. The \p DofMap object handles the index translation from node and element numbers to degree of freedom numbers. We will talk more about the \p DofMap in future examples.

          const DofMap& dof_map = system.get_dof_map();

Define data structures to contain the element matrix and right-hand-side vector contribution. Following basic finite element terminology we will denote these "Ke" and "Fe".

          DenseMatrix<Number> Ke;
          DenseVector<Number> Fe;

This vector will hold the degree of freedom indices for the element. These define where in the global system the element degrees of freedom get mapped.

          std::vector<unsigned int> dof_indices;

Here we extract the velocity & parameters that we put in the EquationSystems object.

          const RealVectorValue velocity =
            es.parameters.get<RealVectorValue> ("velocity");
        
          const Real dt = es.parameters.get<Real>   ("dt");
          const Real time = es.parameters.get<Real> ("time");

Now we will loop over all the elements in the mesh that live on the local processor. We will compute the element matrix and right-hand-side contribution. Since the mesh will be refined we want to only consider the ACTIVE elements, hence we use a variant of the \p active_elem_iterator.

          MeshBase::const_element_iterator       el     = mesh.active_local_elements_begin();
          const MeshBase::const_element_iterator end_el = mesh.active_local_elements_end(); 
          
          for ( ; el != end_el; ++el)
            {

Store a pointer to the element we are currently working on. This allows for nicer syntax later.

              const Elem* elem = *el;

Get the degree of freedom indices for the current element. These define where in the global matrix and right-hand-side this element will contribute to.

              dof_map.dof_indices (elem, dof_indices);

Compute the element-specific data for the current element. This involves computing the location of the quadrature points (q_point) and the shape functions (phi, dphi) for the current element.

              fe->reinit (elem);

Zero the element matrix and right-hand side before summing them. We use the resize member here because the number of degrees of freedom might have changed from the last element. Note that this will be the case if the element type is different (i.e. the last element was a triangle, now we are on a quadrilateral).

              Ke.resize (dof_indices.size(),
        		 dof_indices.size());
        
              Fe.resize (dof_indices.size());

Now we will build the element matrix and right-hand-side. Constructing the RHS requires the solution and its gradient from the previous timestep. This myst be calculated at each quadrature point by summing the solution degree-of-freedom values by the appropriate weight functions.

              for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        	{

Values to hold the old solution & its gradient.

                  Number   u_old = 0.;
        	  Gradient grad_u_old;

Compute the old solution & its gradient.

                  for (unsigned int l=0; l<phi.size(); l++)
        	    {
        	      u_old      += phi[l][qp]*system.old_solution  (dof_indices[l]);

This will work, grad_u_old += dphi[l][qp]*system.old_solution (dof_indices[l]); but we can do it without creating a temporary like this:

                      grad_u_old.add_scaled (dphi[l][qp],system.old_solution (dof_indices[l]));
        	    }

Now compute the element matrix and RHS contributions.

                  for (unsigned int i=0; i<phi.size(); i++)
        	    {

The RHS contribution

                      Fe(i) += JxW[qp]*(

Mass matrix term

                                        u_old*phi[i][qp] + 
        				-.5*dt*(

Convection term (grad_u_old may be complex, so the order here is important!)

                                                (grad_u_old*velocity)*phi[i][qp] +

Diffusion term

                                                0.01*(grad_u_old*dphi[i][qp]))     
        				);
        	      
        	      for (unsigned int j=0; j<phi.size(); j++)
        		{

The matrix contribution

                          Ke(i,j) += JxW[qp]*(

Mass-matrix

                                              phi[i][qp]*phi[j][qp] + 
        				      .5*dt*(

Convection term

                                                     (velocity*dphi[j][qp])*phi[i][qp] +

Diffusion term

                                                     0.01*(dphi[i][qp]*dphi[j][qp]))      
        				      );
        		}
        	    } 
        	}

At this point the interior element integration has been completed. However, we have not yet addressed boundary conditions. For this example we will only consider simple Dirichlet boundary conditions imposed via the penalty method.

The following loops over the sides of the element. If the element has no neighbor on a side then that side MUST live on a boundary of the domain.

The penalty value.

                const Real penalty = 1.e10;

The following loops over the sides of the element. If the element has no neighbor on a side then that side MUST live on a boundary of the domain.

                for (unsigned int s=0; s<elem->n_sides(); s++)
        	  if (elem->neighbor(s) == NULL)
        	    {
        	      fe_face->reinit(elem,s);
        	      
        	      for (unsigned int qp=0; qp<qface.n_points(); qp++)
        		{
        		  const Number value = exact_solution (qface_points[qp](0),
        						       qface_points[qp](1),
        						       time);

RHS contribution

                          for (unsigned int i=0; i<psi.size(); i++)
        		    Fe(i) += penalty*JxW_face[qp]*value*psi[i][qp];

Matrix contribution

                          for (unsigned int i=0; i<psi.size(); i++)
        		    for (unsigned int j=0; j<psi.size(); j++)
        		      Ke(i,j) += penalty*JxW_face[qp]*psi[i][qp]*psi[j][qp];
        		}
        	    } 
              }

We have now built the element matrix and RHS vector in terms of the element degrees of freedom. However, it is possible that some of the element DOFs are constrained to enforce solution continuity, i.e. they are not really "free". We need to constrain those DOFs in terms of non-constrained DOFs to ensure a continuous solution. The \p DofMap::constrain_element_matrix_and_vector() method does just that.

              dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);

The element matrix and right-hand-side are now built for this element. Add them to the global matrix and right-hand-side vector. The \p PetscMatrix::add_matrix() and \p PetscVector::add_vector() members do this for us.

              system.matrix->add_matrix (Ke, dof_indices);
              system.rhs->add_vector    (Fe, dof_indices);
              
            }

Finished computing the sytem matrix and right-hand side.

        #endif // #ifdef ENABLE_AMR
        }

The program without comments:

 
   
  #include <iostream>
  #include <algorithm>
  #include <cmath>
  
  #include "libmesh.h"
  #include "mesh.h"
  #include "mesh_refinement.h"
  #include "gmv_io.h"
  #include "equation_systems.h"
  #include "fe.h"
  #include "quadrature_gauss.h"
  #include "dof_map.h"
  #include "sparse_matrix.h"
  #include "numeric_vector.h"
  #include "dense_matrix.h"
  #include "dense_vector.h"
  
  #include "getpot.h"
  
  #include "o_string_stream.h"
  
  #include "transient_system.h"
  #include "linear_implicit_system.h"
  #include "vector_value.h"
  
  #include "error_vector.h"
  #include "kelly_error_estimator.h"
  
  #include "elem.h"
  
  void assemble_cd (EquationSystems& es,
  		  const std::string& system_name);
  
  void init_cd (EquationSystems& es,
  	      const std::string& system_name);
  
  Real exact_solution (const Real x,
  		     const Real y,
  		     const Real t);
  
  Number exact_value (const Point& p,
  		    const Parameters& parameters,
  		    const std::string&,
  		    const std::string&)
  {
    return exact_solution(p(0), p(1), parameters.get<Real> ("time"));
  }
  
  
  
  int main (int argc, char** argv)
  {
    libMesh::init (argc, argv);
  
  #ifndef ENABLE_AMR
    std::cerr << "ERROR: This example requires libMesh to be\n"
              << "compiled with AMR support!"
              << std::endl;
    return 0;
  #else
  
    {    
  
  
      std::cout << "Usage:\n"
        <<"\t " << argv[0] << " -init_timestep 0\n"
        << "OR\n"
        <<"\t " << argv[0] << " -read_solution -init_timestep 26\n"
        << std::endl;
  
      std::cout << "Running: " << argv[0];
  
      for (int i=1; i<argc; i++)
        std::cout << " " << argv[i];
  
      std::cout << std::endl << std::endl;
  
      GetPot command_line (argc, argv);
  
  
      const bool read_solution   = command_line.search("-read_solution");
  
      unsigned int init_timestep = 0;
      
      if(command_line.search("-init_timestep"))
        init_timestep = command_line.next(0);
      else
      {
        std::cerr << "ERROR: Initial timestep not specified\n" << std::endl;
  
        error();
      }
  
  
  
      unsigned int n_timesteps = 0;
  
      if(command_line.search("-n_timesteps"))
        n_timesteps = command_line.next(0);
      else
      {
        std::cerr << "ERROR: Number of timesteps not specified\n" << std::endl;
        error();
      }
  
  
      Mesh mesh (2);
  
      EquationSystems equation_systems (mesh);
      MeshRefinement mesh_refinement (mesh);
  
      if(!read_solution)
      {
        mesh.read ("mesh.xda");
  
        if(!read_solution)
          mesh_refinement.uniformly_refine (5);
  
        mesh.print_info();
        
        
        TransientLinearImplicitSystem & system = 
          equation_systems.add_system<TransientLinearImplicitSystem> 
          ("Convection-Diffusion");
  
        system.add_variable ("u", FIRST);
  
        system.attach_assemble_function (assemble_cd);
        system.attach_init_function (init_cd);
  
        equation_systems.init ();
      }
  
      else 
      {
        mesh.read("saved_mesh.xda");
  
        mesh.print_info();
  
        equation_systems.read("saved_solution.xda", libMeshEnums::READ);
  
        TransientLinearImplicitSystem & system = 
          equation_systems.get_system<TransientLinearImplicitSystem> 
          ("Convection-Diffusion");
  
        system.update();
  
        system.attach_assemble_function (assemble_cd);
      }
  
      equation_systems.print_info();
  
      
      equation_systems.parameters.set<unsigned int>
        ("linear solver maximum iterations") = 250;
      equation_systems.parameters.set<Real>
        ("linear solver tolerance") = TOLERANCE;
        
      if(!read_solution)
        GMVIO(mesh).write_equation_systems ("out.gmv.000",
  					  equation_systems);
      else
        GMVIO(mesh).write_equation_systems ("solution_read_in.gmv",
  					  equation_systems);
  
        
      
      equation_systems.parameters.set<RealVectorValue>("velocity") = 
        RealVectorValue (0.8, 0.8);
      
      const Real dt = 0.025;
      Real time     = init_timestep*dt;
      
      for(unsigned int t_step=init_timestep; 
                       t_step<(init_timestep+n_timesteps); 
                       t_step++)
        {
  	time += dt;
  
  	equation_systems.parameters.set<Real> ("time") = time;
  	equation_systems.parameters.set<Real> ("dt")   = dt;
  
  	std::cout << " Solving time step ";
  	
  	{
  	  OStringStream out;
  
  	  OSSInt(out,2,t_step);
  	  out << ", time=";
  	  OSSRealzeroleft(out,6,3,time);
  	  out <<  "...";
  	  std::cout << out.str() << std::endl;
  	}
  	
  	TransientLinearImplicitSystem &  system =
  	  equation_systems.get_system<TransientLinearImplicitSystem>("Convection-Diffusion");
  
  	*system.old_local_solution = *system.current_local_solution;
  	
  	const unsigned int max_r_steps = 2;
  	
  	for (unsigned int r_step=0; r_step<max_r_steps; r_step++)
  	  {
  	    system.solve();
  	    
  	    if (r_step+1 != max_r_steps)
  	      {
  		std::cout << "  Refining the mesh..." << std::endl;
  
  		ErrorVector error;
  
  		KellyErrorEstimator error_estimator;
  		
  		error_estimator.estimate_error (system,
  						error);
  		
  		mesh_refinement.refine_fraction() = 0.80;
  		mesh_refinement.coarsen_fraction() = 0.07;
  		mesh_refinement.max_h_level() = 5;
  		mesh_refinement.flag_elements_by_error_fraction (error);
  		
  		mesh_refinement.refine_and_coarsen_elements();
  		
  		equation_systems.reinit ();
  	      }	    
  	  }
  	
  	if ( (t_step+1)%10 == 0)
  	  {
  	    OStringStream file_name;
  
  	    file_name << "out.gmv.";
  	    OSSRealzeroright(file_name,3,0,t_step+1);
  
  	    GMVIO(mesh).write_equation_systems (file_name.str(),
  						equation_systems);
  	  }
        }
  
        if(!read_solution)
        {
          mesh.write("saved_mesh.xda");
          equation_systems.write("saved_solution.xda", libMeshEnums::WRITE);
        }
    }
  #endif // #ifndef ENABLE_AMR
    
    return libMesh::close ();
  }
  
  void init_cd (EquationSystems& es,
  	      const std::string& system_name)
  {
    assert (system_name == "Convection-Diffusion");
  
    TransientLinearImplicitSystem & system =
      es.get_system<TransientLinearImplicitSystem>("Convection-Diffusion");
  
    es.parameters.set<Real> ("time") = 0;
    
    system.project_solution(exact_value, NULL, es.parameters);
  }
  
  
  
  void assemble_cd (EquationSystems& es,
  		  const std::string& system_name)
  {
  #ifdef ENABLE_AMR
    assert (system_name == "Convection-Diffusion");
    
    const Mesh& mesh = es.get_mesh();
    
    const unsigned int dim = mesh.mesh_dimension();
    
    TransientLinearImplicitSystem & system =
      es.get_system<TransientLinearImplicitSystem> ("Convection-Diffusion");
    
    FEType fe_type = system.variable_type(0);
    
    AutoPtr<FEBase> fe      (FEBase::build(dim, fe_type));
    AutoPtr<FEBase> fe_face (FEBase::build(dim, fe_type));
    
    QGauss qrule (dim,   fe_type.default_quadrature_order());
    QGauss qface (dim-1, fe_type.default_quadrature_order());
  
    fe->attach_quadrature_rule      (&qrule);
    fe_face->attach_quadrature_rule (&qface);
  
    const std::vector<Real>& JxW      = fe->get_JxW();
    const std::vector<Real>& JxW_face = fe_face->get_JxW();
    
    const std::vector<std::vector<Real> >& phi = fe->get_phi();
    const std::vector<std::vector<Real> >& psi = fe_face->get_phi();
  
    const std::vector<std::vector<RealGradient> >& dphi = fe->get_dphi();
  
    const std::vector<Point>& qface_points = fe_face->get_xyz();
      
    const DofMap& dof_map = system.get_dof_map();
    
    DenseMatrix<Number> Ke;
    DenseVector<Number> Fe;
    
    std::vector<unsigned int> dof_indices;
  
    const RealVectorValue velocity =
      es.parameters.get<RealVectorValue> ("velocity");
  
    const Real dt = es.parameters.get<Real>   ("dt");
    const Real time = es.parameters.get<Real> ("time");
  
    MeshBase::const_element_iterator       el     = mesh.active_local_elements_begin();
    const MeshBase::const_element_iterator end_el = mesh.active_local_elements_end(); 
    
    for ( ; el != end_el; ++el)
      {    
        const Elem* elem = *el;
        
        dof_map.dof_indices (elem, dof_indices);
  
        fe->reinit (elem);
        
        Ke.resize (dof_indices.size(),
  		 dof_indices.size());
  
        Fe.resize (dof_indices.size());
        
        for (unsigned int qp=0; qp<qrule.n_points(); qp++)
  	{
  	  Number   u_old = 0.;
  	  Gradient grad_u_old;
  	  
  	  for (unsigned int l=0; l<phi.size(); l++)
  	    {
  	      u_old      += phi[l][qp]*system.old_solution  (dof_indices[l]);
  	      
  	      grad_u_old.add_scaled (dphi[l][qp],system.old_solution (dof_indices[l]));
  	    }
  	  
  	  for (unsigned int i=0; i<phi.size(); i++)
  	    {
  	      Fe(i) += JxW[qp]*(
  				u_old*phi[i][qp] + 
  				-.5*dt*(
  					(grad_u_old*velocity)*phi[i][qp] +
  					
  					0.01*(grad_u_old*dphi[i][qp]))     
  				);
  	      
  	      for (unsigned int j=0; j<phi.size(); j++)
  		{
  		  Ke(i,j) += JxW[qp]*(
  				      phi[i][qp]*phi[j][qp] + 
  				      .5*dt*(
  					     (velocity*dphi[j][qp])*phi[i][qp] +
  					     0.01*(dphi[i][qp]*dphi[j][qp]))      
  				      );
  		}
  	    } 
  	} 
  
        {
  	const Real penalty = 1.e10;
  
  	for (unsigned int s=0; s<elem->n_sides(); s++)
  	  if (elem->neighbor(s) == NULL)
  	    {
  	      fe_face->reinit(elem,s);
  	      
  	      for (unsigned int qp=0; qp<qface.n_points(); qp++)
  		{
  		  const Number value = exact_solution (qface_points[qp](0),
  						       qface_points[qp](1),
  						       time);
  						       
  		  for (unsigned int i=0; i<psi.size(); i++)
  		    Fe(i) += penalty*JxW_face[qp]*value*psi[i][qp];
  
  		  for (unsigned int i=0; i<psi.size(); i++)
  		    for (unsigned int j=0; j<psi.size(); j++)
  		      Ke(i,j) += penalty*JxW_face[qp]*psi[i][qp]*psi[j][qp];
  		}
  	    } 
        } 
  
        
        dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);
        
        system.matrix->add_matrix (Ke, dof_indices);
        system.rhs->add_vector    (Fe, dof_indices);
        
      }
  #endif // #ifdef ENABLE_AMR
  }

The console output of the program:

***************************************************************
* Running Example  ./ex10-devel
***************************************************************
 
Usage:
	 ./ex10-devel -init_timestep 0
OR
	 ./ex10-devel -read_solution -init_timestep 26

Running: ./ex10-devel -n_timesteps 25 -init_timestep 0

 Mesh Information:
  mesh_dimension()=2
  spatial_dimension()=3
  n_nodes()=6273
  n_elem()=13650
   n_local_elem()=13650
   n_active_elem()=10240
  n_subdomains()=1
  n_processors()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System "Convection-Diffusion"
    Type "TransientLinearImplicit"
    Variables="u" 
    Finite Element Types="LAGRANGE" 
    Approximation Orders="FIRST" 
    n_dofs()=6273
    n_local_dofs()=6273
    n_constrained_dofs()=0
    n_vectors()=3

 Solving time step  0, time=0.0250...
  Refining the mesh...
 Solving time step  1, time=0.0500...
  Refining the mesh...
 Solving time step  2, time=0.0750...
  Refining the mesh...
 Solving time step  3, time=0.1000...
  Refining the mesh...
 Solving time step  4, time=0.1250...
  Refining the mesh...
 Solving time step  5, time=0.1500...
  Refining the mesh...
 Solving time step  6, time=0.1750...
  Refining the mesh...
 Solving time step  7, time=0.2000...
  Refining the mesh...
 Solving time step  8, time=0.2250...
  Refining the mesh...
 Solving time step  9, time=0.2500...
  Refining the mesh...
 Solving time step 10, time=0.2750...
  Refining the mesh...
 Solving time step 11, time=0.3000...
  Refining the mesh...
 Solving time step 12, time=0.3250...
  Refining the mesh...
 Solving time step 13, time=0.3500...
  Refining the mesh...
 Solving time step 14, time=0.3750...
  Refining the mesh...
 Solving time step 15, time=0.4000...
  Refining the mesh...
 Solving time step 16, time=0.4250...
  Refining the mesh...
 Solving time step 17, time=0.4500...
  Refining the mesh...
 Solving time step 18, time=0.4750...
  Refining the mesh...
 Solving time step 19, time=0.5000...
  Refining the mesh...
 Solving time step 20, time=0.5250...
  Refining the mesh...
 Solving time step 21, time=0.5500...
  Refining the mesh...
 Solving time step 22, time=0.5750...
  Refining the mesh...
 Solving time step 23, time=0.6000...
  Refining the mesh...
 Solving time step 24, time=0.6250...
  Refining the mesh...
 
***** Finished first 25 steps, now read in saved solution and continue *****
 
Usage:
	 ./ex10-devel -init_timestep 0
OR
	 ./ex10-devel -read_solution -init_timestep 26

Running: ./ex10-devel -read_solution -n_timesteps 25 -init_timestep 25

 Mesh Information:
  mesh_dimension()=2
  spatial_dimension()=3
  n_nodes()=713
  n_elem()=1018
   n_local_elem()=1018
   n_active_elem()=766
  n_subdomains()=1
  n_processors()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System "Convection-Diffusion"
    Type "TransientLinearImplicit"
    Variables="u" 
    Finite Element Types="LAGRANGE" 
    Approximation Orders="FIRST" 
    n_dofs()=713
    n_local_dofs()=713
    n_constrained_dofs()=122
    n_vectors()=3

 Solving time step 25, time=0.6500...
  Refining the mesh...
 Solving time step 26, time=0.6750...
  Refining the mesh...
 Solving time step 27, time=0.7000...
  Refining the mesh...
 Solving time step 28, time=0.7250...
  Refining the mesh...
 Solving time step 29, time=0.7500...
  Refining the mesh...
 Solving time step 30, time=0.7750...
  Refining the mesh...
 Solving time step 31, time=0.8000...
  Refining the mesh...
 Solving time step 32, time=0.8250...
  Refining the mesh...
 Solving time step 33, time=0.8500...
  Refining the mesh...
 Solving time step 34, time=0.8750...
  Refining the mesh...
 Solving time step 35, time=0.9000...
  Refining the mesh...
 Solving time step 36, time=0.9250...
  Refining the mesh...
 Solving time step 37, time=0.9500...
  Refining the mesh...
 Solving time step 38, time=0.9750...
  Refining the mesh...
 Solving time step 39, time=1.0000...
  Refining the mesh...
 Solving time step 40, time=1.0300...
  Refining the mesh...
 Solving time step 41, time=1.0500...
  Refining the mesh...
 Solving time step 42, time=1.0700...
  Refining the mesh...
 Solving time step 43, time=1.1000...
  Refining the mesh...
 Solving time step 44, time=1.1200...
  Refining the mesh...
 Solving time step 45, time=1.1500...
  Refining the mesh...
 Solving time step 46, time=1.1700...
  Refining the mesh...
 Solving time step 47, time=1.2000...
  Refining the mesh...
 Solving time step 48, time=1.2200...
  Refining the mesh...
 Solving time step 49, time=1.2500...
  Refining the mesh...
 
***************************************************************
* Done Running Example  ./ex10-devel
***************************************************************