step-77.h

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) const
 *   {
 *     return std::sin(2 * numbers::PI * (p[0] + p[1]));
 *   }
 * 
 * 
 * @endcode
 * 
 * 
 * <a name="ThecodeMinimalSurfaceProblemcodeclassimplementation"></a> 
 * <h3>The <code>MinimalSurfaceProblem</code> class implementation</h3>
 * 
 
 * 
 * 
 * <a name="Constructorandsetupfunctions"></a> 
 * <h4>Constructor and set up functions</h4>
 * 
 
 * 
 * The following few functions are also essentially copies of what
 * @ref step_15 "step-15" already does, and so there is little to discuss.
 * 
 * @code
 *   template <int dim>
 *   MinimalSurfaceProblem<dim>::MinimalSurfaceProblem()
 *     : dof_handler(triangulation)
 *     , fe(1)
 *     , computing_timer(std::cout, TimerOutput::never, TimerOutput::wall_times)
 *   {}
 * 
 * 
 * 
 *   template <int dim>
 *   void MinimalSurfaceProblem<dim>::setup_system(const bool initial_step)
 *   {
 *     TimerOutput::Scope t(computing_timer, "set up");
 * 
 *     if (initial_step)
 *       {
 *         dof_handler.distribute_dofs(fe);
 *         current_solution.reinit(dof_handler.n_dofs());
 * 
 *         hanging_node_constraints.clear();
 *         DoFTools::make_hanging_node_constraints(dof_handler,
 *                                                 hanging_node_constraints);
 *         hanging_node_constraints.close();
 *       }
 * 
 *     DynamicSparsityPattern dsp(dof_handler.n_dofs());
 *     DoFTools::make_sparsity_pattern(dof_handler, dsp);
 * 
 *     hanging_node_constraints.condense(dsp);
 * 
 *     sparsity_pattern.copy_from(dsp);
 *     jacobian_matrix.reinit(sparsity_pattern);
 *     jacobian_matrix_factorization.reset();
 *   }
 * 
 * 
 * 
 * @endcode
 * 
 * 
 * <a name="AssemblingandfactorizingtheJacobianmatrix"></a> 
 * <h4>Assembling and factorizing the Jacobian matrix</h4>
 * 
 
 * 
 * The following function is then responsible for assembling and factorizing
 * the Jacobian matrix. The first half of the function is in essence the
 * `assemble_system()` function of @ref step_15 "step-15", except that it does not deal with
 * also forming a right hand side vector (i.e., the residual) since we do not
 * always have to do these operations at the same time.
 *   
 
 * 
 * We put the whole assembly functionality into a code block enclosed by curly
 * braces so that we can use a TimerOutput::Scope variable to measure how much
 * time is spent in this code block, excluding everything that happens in this
 * function after the matching closing brace `}`.
 * 
 * @code
 *   template <int dim>
 *   void MinimalSurfaceProblem<dim>::compute_and_factorize_jacobian(
 *     const Vector<double> &evaluation_point)
 *   {
 *     {
 *       TimerOutput::Scope t(computing_timer, "assembling the Jacobian");
 * 
 *       std::cout << "  Computing Jacobian matrix" << std::endl;
 * 
 *       const QGauss<dim> quadrature_formula(fe.degree + 1);
 * 
 *       jacobian_matrix = 0;
 * 
 *       FEValues<dim> fe_values(fe,
 *                               quadrature_formula,
 *                               update_gradients | update_quadrature_points |
 *                                 update_JxW_values);
 * 
 *       const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
 *       const unsigned int n_q_points    = quadrature_formula.size();
 * 
 *       FullMatrix<double> cell_matrix(dofs_per_cell, dofs_per_cell);
 * 
 *       std::vector<Tensor<1, dim>> evaluation_point_gradients(n_q_points);
 * 
 *       std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
 * 
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 *         {
 *           cell_matrix = 0;
 * 
 *           fe_values.reinit(cell);
 * 
 *           fe_values.get_function_gradients(evaluation_point,
 *                                            evaluation_point_gradients);
 * 
 *           for (unsigned int q = 0; q < n_q_points; ++q)
 *             {
 *               const double coeff =
 *                 1.0 / std::sqrt(1 + evaluation_point_gradients[q] *
 *                                       evaluation_point_gradients[q]);
 * 
 *               for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *                 {
 *                   for (unsigned int j = 0; j < dofs_per_cell; ++j)
 *                     cell_matrix(i, j) +=
 *                       (((fe_values.shape_grad(i, q)    // ((\nabla \phi_i
 *                          * coeff                       //   * a_n
 *                          * fe_values.shape_grad(j, q)) //   * \nabla \phi_j)
 *                         -                              //  -
 *                         (fe_values.shape_grad(i, q)    //  (\nabla \phi_i
 *                          * coeff * coeff * coeff       //   * a_n^3
 *                          *
 *                          (fe_values.shape_grad(j, q)       //   * (\nabla \phi_j
 *                           * evaluation_point_gradients[q]) //      * \nabla u_n)
 *                          * evaluation_point_gradients[q])) //   * \nabla u_n)))
 *                        * fe_values.JxW(q));                // * dx
 *                 }
 *             }
 * 
 *           cell->get_dof_indices(local_dof_indices);
 *           hanging_node_constraints.distribute_local_to_global(cell_matrix,
 *                                                               local_dof_indices,
 *                                                               jacobian_matrix);
 *         }
 * 
 *       std::map<types::global_dof_index, double> boundary_values;
 *       VectorTools::interpolate_boundary_values(dof_handler,
 *                                                0,
 *                                                Functions::ZeroFunction<dim>(),
 *                                                boundary_values);
 *       Vector<double> dummy_solution(dof_handler.n_dofs());
 *       Vector<double> dummy_rhs(dof_handler.n_dofs());
 *       MatrixTools::apply_boundary_values(boundary_values,
 *                                          jacobian_matrix,
 *                                          dummy_solution,
 *                                          dummy_rhs);
 *     }
 * 
 * @endcode
 * 
 * The second half of the function then deals with factorizing the
 * so-computed matrix. To do this, we first create a new SparseDirectUMFPACK
 * object and by assigning it to the member variable
 * `jacobian_matrix_factorization`, we also destroy whatever object that
 * pointer previously pointed to (if any). Then we tell the object to
 * factorize the Jacobian.
 *     
 
 * 
 * As above, we enclose this block of code into curly braces and use a timer
 * to assess how long this part of the program takes.
 *     
 
 * 
 * (Strictly speaking, we don't actually need the matrix any more after we
 * are done here, and could throw the matrix object away. A code intended to
 * be memory efficient would do this, and only create the matrix object in
 * this function, rather than as a member variable of the surrounding class.
 * We omit this step here because using the same coding style as in previous
 * tutorial programs breeds familiarity with the common style and helps make
 * these tutorial programs easier to read.)
 * 
 * @code
 *     {
 *       TimerOutput::Scope t(computing_timer, "factorizing the Jacobian");
 * 
 *       std::cout << "  Factorizing Jacobian matrix" << std::endl;
 * 
 *       jacobian_matrix_factorization = std::make_unique<SparseDirectUMFPACK>();
 *       jacobian_matrix_factorization->factorize(jacobian_matrix);
 *     }
 *   }
 * 
 * 
 * 
 * @endcode
 * 
 * 
 * <a name="Computingtheresidualvector"></a> 
 * <h4>Computing the residual vector</h4>
 * 
 
 * 
 * The second part of what `assemble_system()` used to do in @ref step_15 "step-15" is
 * computing the residual vector, i.e., the right hand side vector of the
 * Newton linear systems. We have broken this out of the previous function,
 * but the following function will be easy to understand if you understood
 * what `assemble_system()` in @ref step_15 "step-15" did. Importantly, however, we need to
 * compute the residual not linearized around the current solution vector, but
 * whatever we get from KINSOL. This is necessary for operations such as line
 * search where we want to know what the residual @f$F(U^k + \alpha_k \delta
 * U^K)@f$ is for different values of @f$\alpha_k@f$; KINSOL in those cases simply
 * gives us the argument to the function @f$F@f$ and we then compute the residual
 * @f$F(\cdot)@f$ at this point.
 *   
 
 * 
 * The function prints the norm of the so-computed residual at the end as a
 * way for us to follow along the progress of the program.
 * 
 * @code
 *   template <int dim>
 *   void MinimalSurfaceProblem<dim>::compute_residual(
 *     const Vector<double> &evaluation_point,
 *     Vector<double> &      residual)
 *   {
 *     TimerOutput::Scope t(computing_timer, "assembling the residual");
 * 
 *     std::cout << "  Computing residual vector..." << std::flush;
 * 
 *     const QGauss<dim> quadrature_formula(fe.degree + 1);
 *     FEValues<dim>     fe_values(fe,
 *                             quadrature_formula,
 *                             update_gradients | update_quadrature_points |
 *                               update_JxW_values);
 * 
 *     const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
 *     const unsigned int n_q_points    = quadrature_formula.size();
 * 
 *     Vector<double>              cell_residual(dofs_per_cell);
 *     std::vector<Tensor<1, dim>> evaluation_point_gradients(n_q_points);
 * 
 *     std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
 * 
 *     for (const auto &cell : dof_handler.active_cell_iterators())
 *       {
 *         cell_residual = 0;
 *         fe_values.reinit(cell);
 * 
 *         fe_values.get_function_gradients(evaluation_point,
 *                                          evaluation_point_gradients);
 * 
 * 
 *         for (unsigned int q = 0; q < n_q_points; ++q)
 *           {
 *             const double coeff =
 *               1.0 / std::sqrt(1 + evaluation_point_gradients[q] *
 *                                     evaluation_point_gradients[q]);
 * 
 *             for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *               cell_residual(i) =
 *                 (fe_values.shape_grad(i, q)      // \nabla \phi_i
 *                  * coeff                         // * a_n
 *                  * evaluation_point_gradients[q] // * \nabla u_n
 *                  * fe_values.JxW(q));            // * dx
 *           }
 * 
 *         cell->get_dof_indices(local_dof_indices);
 *         for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *           residual(local_dof_indices[i]) += cell_residual(i);
 *       }
 * 
 *     hanging_node_constraints.condense(residual);
 * 
 *     for (const types::global_dof_index i :
 *          DoFTools::extract_boundary_dofs(dof_handler))
 *       residual(i) = 0;
 * 
 *     for (const types::global_dof_index i :
 *          DoFTools::extract_hanging_node_dofs(dof_handler))
 *       residual(i) = 0;
 * 
 *     std::cout << " norm=" << residual.l2_norm() << std::endl;
 *   }
 * 
 * 
 * 
 * @endcode
 * 
 * 
 * <a name="SolvinglinearsystemswiththeJacobianmatrix"></a> 
 * <h4>Solving linear systems with the Jacobian matrix</h4>
 * 
 
 * 
 * Next up is the function that implements the solution of a linear system
 * with the Jacobian matrix. Since we have already factored the matrix when we
 * built the matrix, solving a linear system comes down to applying the
 * inverse matrix to the given right hand side vector: This is what the
 * SparseDirectUMFPACK::vmult() function does that we use here. Following
 * this, we have to make sure that we also address the values of hanging nodes
 * in the solution vector, and this is done using
 * AffineConstraints::distribute().
 *   
 
 * 
 * The function takes an additional, but unused, argument `tolerance` that
 * indicates how accurately we have to solve the linear system. The meaning of
 * this argument is discussed in the introduction in the context of the
 * "Eisenstat Walker trick", but since we are using a direct rather than an
 * iterative solver, we are not using this opportunity to solve linear systems
 * only inexactly.
 * 
 * @code
 *   template <int dim>
 *   void MinimalSurfaceProblem<dim>::solve(const Vector<double> &rhs,
 *                                          Vector<double> &      solution,
 *                                          const double /*tolerance*/)
 *   {
 *     TimerOutput::Scope t(computing_timer, "linear system solve");
 * 
 *     std::cout << "  Solving linear system" << std::endl;
 * 
 *     jacobian_matrix_factorization->vmult(solution, rhs);
 * 
 *     hanging_node_constraints.distribute(solution);
 *   }
 * 
 * 
 * 
 * @endcode
 * 
 * 
 * <a name="Refiningthemeshsettingboundaryvaluesandgeneratinggraphicaloutput"></a> 
 * <h4>Refining the mesh, setting boundary values, and generating graphical output</h4>
 * 
 
 * 
 * The following three functions are again simply copies of the ones in
 * @ref step_15 "step-15":
 * 
 * @code
 *   template <int dim>
 *   void MinimalSurfaceProblem<dim>::refine_mesh()
 *   {
 *     Vector<float> estimated_error_per_cell(triangulation.n_active_cells());
 * 
 *     KellyErrorEstimator<dim>::estimate(
 *       dof_handler,
 *       QGauss<dim - 1>(fe.degree + 1),
 *       std::map<types::boundary_id, const Function<dim> *>(),
 *       current_solution,
 *       estimated_error_per_cell);
 * 
 *     GridRefinement::refine_and_coarsen_fixed_number(triangulation,
 *                                                     estimated_error_per_cell,
 *                                                     0.3,
 *                                                     0.03);
 * 
 *     triangulation.prepare_coarsening_and_refinement();
 * 
 *     SolutionTransfer<dim> solution_transfer(dof_handler);
 *     solution_transfer.prepare_for_coarsening_and_refinement(current_solution);
 * 
 *     triangulation.execute_coarsening_and_refinement();
 * 
 *     dof_handler.distribute_dofs(fe);
 * 
 *     Vector<double> tmp(dof_handler.n_dofs());
 *     solution_transfer.interpolate(current_solution, tmp);
 *     current_solution = std::move(tmp);
 * 
 *     hanging_node_constraints.clear();
 * 
 *     DoFTools::make_hanging_node_constraints(dof_handler,
 *                                             hanging_node_constraints);
 *     hanging_node_constraints.close();
 * 
 *     hanging_node_constraints.distribute(current_solution);
 * 
 *     set_boundary_values();
 * 
 *     setup_system(/*initial_step=*/false);
 *   }
 * 
 * 
 * 
 *   template <int dim>
 *   void MinimalSurfaceProblem<dim>::set_boundary_values()
 *   {
 *     std::map<types::global_dof_index, double> boundary_values;
 *     VectorTools::interpolate_boundary_values(dof_handler,
 *                                              0,
 *                                              BoundaryValues<dim>(),
 *                                              boundary_values);
 *     for (const auto &boundary_value : boundary_values)
 *       current_solution(boundary_value.first) = boundary_value.second;
 * 
 *     hanging_node_constraints.distribute(current_solution);
 *   }
 * 
 * 
 * 
 *   template <int dim>
 *   void MinimalSurfaceProblem<dim>::output_results(
 *     const unsigned int refinement_cycle)
 *   {
 *     TimerOutput::Scope t(computing_timer, "graphical output");
 * 
 *     DataOut<dim> data_out;
 * 
 *     data_out.attach_dof_handler(dof_handler);
 *     data_out.add_data_vector(current_solution, "solution");
 *     data_out.build_patches();
 * 
 *     const std::string filename =
 *       "solution-" + Utilities::int_to_string(refinement_cycle, 2) + ".vtu";
 *     std::ofstream output(filename);
 *     data_out.write_vtu(output);
 *   }
 * 
 * 
 * 
 * @endcode
 * 
 * 
 * <a name="Therunfunctionandtheoveralllogicoftheprogram"></a> 
 * <h4>The run() function and the overall logic of the program</h4>
 * 
 
 * 
 * The only function that *really* is interesting in this program is the one
 * that drives the overall algorithm of starting on a coarse mesh, doing some
 * mesh refinement cycles, and on each mesh using KINSOL to find the solution
 * of the nonlinear algebraic equation we obtain from discretization on this
 * mesh. The `refine_mesh()` function above makes sure that the solution on
 * one mesh is used as the starting guess on the next mesh. We also use a
 * TimerOutput object to measure how much time every operation on each mesh
 * costs, and reset the timer at the beginning of each cycle.
 *   
 
 * 
 * As discussed in the introduction, it is not necessary to solve problems on
 * coarse meshes particularly accurately since these will only solve as
 * starting guesses for the next mesh. As a consequence, we will use a target
 * tolerance of
 * @f$\tau=10^{-3} \frac{1}{10^k}@f$ for the @f$k@f$th mesh refinement cycle.
 *   
 
 * 
 * All of this is encoded in the first part of this function:
 * 
 * @code
 *   template <int dim>
 *   void MinimalSurfaceProblem<dim>::run()
 *   {
 *     GridGenerator::hyper_ball(triangulation);
 *     triangulation.refine_global(2);
 * 
 *     setup_system(/*initial_step=*/true);
 *     set_boundary_values();
 * 
 *     for (unsigned int refinement_cycle = 0; refinement_cycle < 6;
 *          ++refinement_cycle)
 *       {
 *         computing_timer.reset();
 *         std::cout << "Mesh refinement step " << refinement_cycle << std::endl;
 * 
 *         if (refinement_cycle != 0)
 *           refine_mesh();
 * 
 *         const double target_tolerance = 1e-3 * std::pow(0.1, refinement_cycle);
 *         std::cout << "  Target_tolerance: " << target_tolerance << std::endl
 *                   << std::endl;
 * 
 * @endcode
 * 
 * This is where the fun starts. At the top we create the KINSOL solver
 * object and feed it with an object that encodes a number of additional
 * specifics (of which we only change the nonlinear tolerance we want to
 * reach; but you might want to look into what other members the
 * SUNDIALS::KINSOL::AdditionalData class has and play with them).
 * 
 * @code
 *         {
 *           typename SUNDIALS::KINSOL<Vector<double>>::AdditionalData
 *             additional_data;
 *           additional_data.function_tolerance = target_tolerance;
 * 
 *           SUNDIALS::KINSOL<Vector<double>> nonlinear_solver(additional_data);
 * 
 * @endcode
 * 
 * Then we have to describe the operations that were already mentioned
 * in the introduction. In essence, we have to teach KINSOL how to (i)
 * resize a vector to the correct size, (ii) compute the residual
 * vector, (iii) compute the Jacobian matrix (during which we also
 * compute its factorization), and (iv) solve a linear system with the
 * Jacobian.
 *           
 
 * 
 * All four of these operations are represented by member variables of
 * the SUNDIALS::KINSOL class that are of type `std::function`, i.e.,
 * they are objects to which we can assign a pointer to a function or,
 * as we do here, a "lambda function" that takes the appropriate
 * arguments and returns the appropriate information. By convention,
 * KINSOL wants that functions doing something nontrivial return an
 * integer where zero indicates success. It turns out that we can do
 * all of this in just 25 lines of code.
 *           
 
 * 
 * (If you're not familiar what "lambda functions" are, take
 * a look at @ref step_12 "step-12" or at the
 * [wikipedia page](https://en.wikipedia.org/wiki/Anonymous_function)
 * on the subject. The idea of lambda functions is that one
 * wants to define a function with a certain set of
 * arguments, but (i) not make it a named functions because,
 * typically, the function is used in only one place and it
 * seems unnecessary to give it a global name; and (ii) that
 * the function has access to some of the variables that
 * exist at the place where it is defined, including member
 * variables. The syntax of lambda functions is awkward, but
 * ultimately quite useful.)
 *           
 
 * 
 * At the very end of the code block we then tell KINSOL to go to work
 * and solve our problem. The member functions called from the
 * 'residual', 'setup_jacobian', and 'solve_jacobian_system' functions
 * will then print output to screen that allows us to follow along
 * with the progress of the program.
 * 
 * @code
 *           nonlinear_solver.reinit_vector = [&](Vector<double> &x) {
 *             x.reinit(dof_handler.n_dofs());
 *           };
 * 
 *           nonlinear_solver.residual =
 *             [&](const Vector<double> &evaluation_point,
 *                 Vector<double> &      residual) {
 *               compute_residual(evaluation_point, residual);
 * 
 *               return 0;
 *             };
 * 
 *           nonlinear_solver.setup_jacobian =
 *             [&](const Vector<double> &current_u,
 *                 const Vector<double> & /*current_f*/) {
 *               compute_and_factorize_jacobian(current_u);
 * 
 *               return 0;
 *             };
 * 
 *           nonlinear_solver.solve_with_jacobian = [&](const Vector<double> &rhs,
 *                                                      Vector<double> &      dst,
 *                                                      const double tolerance) {
 *             this->solve(rhs, dst, tolerance);
 * 
 *             return 0;
 *           };
 * 
 *           nonlinear_solver.solve(current_solution);
 *         }
 * 
 * @endcode
 * 
 * The rest is then just house-keeping: Writing data to a file for
 * visualizing, and showing a summary of the timing collected so that we
 * can interpret how long each operation has taken, how often it was
 * executed, etc:
 * 
 * @code
 *         output_results(refinement_cycle);
 * 
 *         computing_timer.print_summary();
 * 
 *         std::cout << std::endl;
 *       }
 *   }
 * } // namespace Step77
 * 
 * 
 * int main()
 * {
 *   try
 *     {
 *       using namespace Step77;
 * 
 *       MinimalSurfaceProblem<2> laplace_problem_2d;
 *       laplace_problem_2d.run();
 *     }
 *   catch (std::exception &exc)
 *     {
 *       std::cerr << std::endl
 *                 << std::endl
 *                 << "----------------------------------------------------"
 *                 << std::endl;
 *       std::cerr << "Exception on processing: " << std::endl
 *                 << exc.what() << std::endl
 *                 << "Aborting!" << std::endl
 *                 << "----------------------------------------------------"
 *                 << std::endl;
 * 
 *       return 1;
 *     }
 *   catch (...)
 *     {
 *       std::cerr << std::endl
 *                 << std::endl
 *                 << "----------------------------------------------------"
 *                 << std::endl;
 *       std::cerr << "Unknown exception!" << std::endl
 *                 << "Aborting!" << std::endl
 *                 << "----------------------------------------------------"
 *                 << std::endl;
 *       return 1;
 *     }
 *   return 0;
 * }
 * @endcode
<a name="Results"></a><h1>Results</h1>
 
 
When running the program, you get output that looks like this:
@code
Mesh refinement step 0
  Target_tolerance: 0.001
 
  Computing residual vector... norm=0.231202
  Computing Jacobian matrix
  Factorizing Jacobian matrix
  Solving linear system
  Computing residual vector... norm=0.231202
  Computing residual vector... norm=0.171585
  Solving linear system
  Computing residual vector... norm=0.171585
  Computing residual vector... norm=0.127245
  Computing residual vector... norm=0.0796471
  Solving linear system
  Computing residual vector... norm=0.0796471
  Computing residual vector... norm=0.0625301
  Solving linear system
  Computing residual vector... norm=0.0625301
  Computing residual vector... norm=0.0498864
  Solving linear system
  Computing residual vector... norm=0.0498864
  Computing residual vector... norm=0.0407765
  Solving linear system
  Computing residual vector... norm=0.0407765
  Computing residual vector... norm=0.0341589
  Solving linear system
  Computing residual vector... norm=0.0341589
  Computing residual vector... norm=0.0292867
  Solving linear system
  Computing residual vector... norm=0.0292867
  Computing residual vector... norm=0.0256309
  Computing residual vector... norm=0.0223448
  Solving linear system
  Computing residual vector... norm=0.0223448
  Computing residual vector... norm=0.0202797
  Computing residual vector... norm=0.0183817
  Solving linear system
  Computing residual vector... norm=0.0183817
  Computing residual vector... norm=0.0170464
  Computing residual vector... norm=0.0157967
  Computing Jacobian matrix
  Factorizing Jacobian matrix
  Solving linear system
  Computing residual vector... norm=0.0157967
  Computing residual vector... norm=0.0141572
  Computing residual vector... norm=0.012657
  Solving linear system
  Computing residual vector... norm=0.012657
  Computing residual vector... norm=0.0116863
  Computing residual vector... norm=0.0107696
  Solving linear system
  Computing residual vector... norm=0.0107696
  Computing residual vector... norm=0.0100986
  Computing residual vector... norm=0.00944829
  Computing residual vector... norm=0.00822576
  Solving linear system
  Computing residual vector... norm=0.00822576
  Computing residual vector... norm=0.00781983
  Computing residual vector... norm=0.00741619
  Computing residual vector... norm=0.00661792
  Solving linear system
  Computing residual vector... norm=0.00661792
  Computing residual vector... norm=0.00630571
  Computing residual vector... norm=0.00599457
  Computing residual vector... norm=0.00537663
  Solving linear system
  Computing residual vector... norm=0.00537663
  Computing residual vector... norm=0.00512813
  Computing residual vector... norm=0.00488033
  Computing residual vector... norm=0.00438751
  Computing residual vector... norm=0.00342052
  Solving linear system
  Computing residual vector... norm=0.00342052
  Computing residual vector... norm=0.00326581
  Computing residual vector... norm=0.00311176
  Computing residual vector... norm=0.00280617
  Computing residual vector... norm=0.00220992
  Solving linear system
  Computing residual vector... norm=0.00220992
  Computing residual vector... norm=0.00209976
  Computing residual vector... norm=0.00199943
  Solving linear system
  Computing residual vector... norm=0.00199942
  Computing residual vector... norm=0.00190953
  Computing residual vector... norm=0.00182005
  Computing residual vector... norm=0.00164259
  Computing residual vector... norm=0.00129652
 
 
+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |     0.192s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| assembling the Jacobian         |         2 |    0.0141s |       7.4% |
| assembling the residual         |        61 |     0.168s |        88% |
| factorizing the Jacobian        |         2 |    0.0016s |      0.83% |
| graphical output                |         1 |   0.00385s |         2% |
| linear system solve             |        19 |    0.0013s |      0.68% |
+---------------------------------+-----------+------------+------------+
 
 
Mesh refinement step 1
  Target_tolerance: 0.0001
 
  Computing residual vector... norm=0.0883422
  Computing Jacobian matrix
  Factorizing Jacobian matrix
  Solving linear system
  Computing residual vector... norm=0.0883422
  Computing residual vector... norm=0.0607066
  Solving linear system
  Computing residual vector... norm=0.0607066
  Computing residual vector... norm=0.0437266
  Solving linear system
  Computing residual vector... norm=0.0437266
  Computing residual vector... norm=0.0327999
  Solving linear system
  Computing residual vector... norm=0.0327999
  Computing residual vector... norm=0.0255418
  Solving linear system
  Computing residual vector... norm=0.0255417
  Computing residual vector... norm=0.0206042
  Solving linear system
  Computing residual vector... norm=0.0206042
  Computing residual vector... norm=0.0171602
  Solving linear system
  Computing residual vector... norm=0.0171602
  Computing residual vector... norm=0.014689
  Solving linear system
 
[...]
@endcode
 
The way this should be interpreted is most easily explained by looking at
the first few lines of the output on the first mesh:
@code
Mesh refinement step 0
Mesh refinement step 0
  Target_tolerance: 0.001
 
  Computing residual vector... norm=0.231202
  Computing Jacobian matrix
  Factorizing Jacobian matrix
  Solving linear system
  Computing residual vector... norm=0.231202
  Computing residual vector... norm=0.171585
  Solving linear system
  Computing residual vector... norm=0.171585
  Computing residual vector... norm=0.127245
  Computing residual vector... norm=0.0796471
  Solving linear system
  Computing residual vector... norm=0.0796471
  ...
@endcode
What is happening is this:
- In the first residual computation, KINSOL computes the residual to see whether
  the desired tolerance has been reached. The answer is no, so it requests the
  user program to compute the Jacobian matrix (and the function then also
  factorizes the matrix via SparseDirectUMFPACK).
- KINSOL then instructs us to solve a linear system of the form
  @f$J_k \, \delta U_k = -F_k@f$ with this matrix and the previously computed
  residual vector.
- It is then time to determine how far we want to go in this direction,
  i.e., do line search. To this end, KINSOL requires us to compute the
  residual vector @f$F(U_k + \alpha_k \delta U_k)@f$ for different step lengths
  @f$\alpha_k@f$. For the first step above, it finds an acceptable @f$\alpha_k@f$
  after two tries, the second time around it takes three tries.
- Having found a suitable updated solution @f$U_{k+1}@f$, the process is
  repeated except now KINSOL is happy with the current Jacobian matrix
  and does not instruct us to re-build the matrix and its factorization,
  and instead asks us to solve a linear system with that same matrix.
 
The program also writes the solution to a VTU file at the end
of each mesh refinement cycle, and it looks as follows:
<table width="60%" align="center">
  <tr>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-77.solution.png" alt="">
    </td>
  </tr>
</table>
 
 
The key takeaway messages of this program are the following:
 
- The solution is the same as the one we computed in @ref step_15 "step-15", i.e., the
  interfaces to %SUNDIALS' KINSOL package really did what they were supposed
  to do. This should not come as a surprise, but the important point is that
  we don't have to spend the time implementing the complex algorithms that
  underlie advanced nonlinear solvers ourselves.
 
- KINSOL is able to avoid all sorts of operations such as rebuilding the
  Jacobian matrix when that is not actually necessary. Comparing the
  number of linear solves in the output above with the number of times
  we rebuild the Jacobian and compute its factorization should make it
  clear that this leads to very substantial savings in terms of compute
  times, without us having to implement the intricacies of algorithms
  that determine when we need to rebuild this information.
 
<a name="extensions"></a>
<a name="Possibilitiesforextensions"></a><h3> Possibilities for extensions </h3>
 
 
For all but the small problems we consider here, a sparse direct solver
requires too much time and memory -- we need an iterative solver like
we use in many other programs. The trade-off between constructing an
expensive preconditioner (say, a geometric or algebraic multigrid method)
is different in the current case, however: Since we can re-use the same
matrix for numerous linear solves, we can do the same for the preconditioner
and putting more work into building a good preconditioner can more easily
be justified than if we used it only for a single linear solve as one
does for many other situations.
 
But iterative solvers also afford other opportunities. For example (and as
discussed briefly in the introduction), we may not need to solve to
very high accuracy (small tolerances) in early nonlinear iterations as long
as we are still far away from the actual solution. This was the basis of the
Eisenstat-Walker trick mentioned there.
 
KINSOL provides the function that does the linear solution with a target
tolerance that needs to be reached. We ignore it in the program above
because the direct solver we use does not need a tolerance and instead
solves the linear system exactly (up to round-off, of course), but iterative
solvers could make use of this kind of information -- and, in fact, should.
 *
 *
<a name="PlainProg"></a>
<h1> The plain program</h1>
@include "step-77.cc"
*/