vector_fe_ex4.C File Reference

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

Function Documentation

main()

int main	(	int	argc,
		char **	argv
	)

Definition at line 49 of file vector_fe_ex4.C.

{
  // Initialize libMesh.
  LibMeshInit init (argc, argv);
 
  // This example requires a linear solver package.
  libmesh_example_requires(libMesh::default_solver_package() != INVALID_SOLVER_PACKAGE,
                           "--enable-petsc, --enable-trilinos, or --enable-eigen");
 
  // Parse the input file
  GetPot infile("vector_fe_ex4.in");
 
  // But allow the command line to override it.
  infile.parse_command_line(argc, argv);
 
  // hypre AMS requires PETSc 3.12.2 or above with hypre support enabled
#if PETSC_VERSION_LESS_THAN(3, 12, 2) || !defined(LIBMESH_HAVE_PETSC_HYPRE)
  libmesh_example_requires(!infile.search("ams"),
                           "PETSc 3.12.2 or above with hypre support enabled");
#endif
 
  // Read in parameters from the input file
  const unsigned int grid_size = infile("grid_size", 2);
 
  // Skip higher-dimensional examples on a lower-dimensional libMesh build
  libmesh_example_requires(3 <= LIBMESH_DIM, "3D support");
 
  // Create a mesh, with dimension to be overridden later, on the
  // default MPI communicator.
  Mesh mesh(init.comm());
 
  // Use the MeshTools::Generation mesh generator to create a uniform
  // grid on the square [-1,1]^D. We must use at least TET10 or HEX20 elements
  // for the Nedelec tetrahedral or hexahedral elements, respectively.
  std::string elem_str =
    command_line_value(std::string("element_type"),
                       std::string("HEX27"));
 
  // In general we expect O(h) convergence in *both* the L2 and
  // H(curl) norms when using Nedelec elements. For more information
  // on this topic, see the relevant results in other software [1-3]
  // and the descriptions in [4,5]. In this particular example, we
  // have observed O(h^2) convergence in L2 for HEX20/HEX27s, since the
  // particular solution we seek is, just like the basis functions, constant
  // along each of the cartesian axes. The basis functions are linear in the
  // other two dimensions, improving the convergence speed.
  //
  // [1]: deal.ii, https://www.dealii.org/reports/nedelec/nedelec.pdf
  // [2]: FEMPAR, https://www.sciencedirect.com/science/article/pii/S096599781831113X
  // [3]: FEniCS, https://fenicsproject.org/pub/book/book/fenics-book-2011-06-24.pdf
  // [4]: Monk, https://icerm.brown.edu/materials/Slides/tw-18-7/Finite_Element_Methods_for_Maxwells_Equations_%5D_Peter_Monk,_University_of_Delaware.pdf
  // [5]: Hiptmair at al., https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-04_fp.pdf
  libmesh_error_msg_if(elem_str != "TET10" && elem_str != "TET14" && elem_str != "HEX20" && elem_str != "HEX27",
                       "You entered: " << elem_str <<
                       " but this example must be run with TET10, TET14, HEX20 or HEX27.");
 
  MeshTools::Generation::build_cube (mesh,
                                     grid_size,
                                     grid_size,
                                     grid_size,
                                     -1., 1.,
                                     -1., 1.,
                                     -1., 1.,
                                     Utility::string_to_enum<ElemType>(elem_str));
 
 
  // Print information about the mesh to the screen.
  mesh.print_info();
 
  // Create an equation systems object.
  EquationSystems equation_systems (mesh);
 
  // Declare the system "CurlCurl" and its variables.
  CurlCurlSystem & system =
    equation_systems.add_system<CurlCurlSystem> ("CurlCurl");
 
  // This example only implements the steady-state problem
  system.time_solver = std::make_unique<SteadySolver>(system);
 
  // Initialize the system
  equation_systems.init();
 
  // And the nonlinear solver options
  DiffSolver & solver = *(system.time_solver->diff_solver().get());
  solver.quiet = infile("solver_quiet", true);
  solver.verbose = !solver.quiet;
  solver.max_nonlinear_iterations = infile("max_nonlinear_iterations", 15);
  solver.relative_step_tolerance = infile("relative_step_tolerance", 1.e-3);
  solver.relative_residual_tolerance = infile("relative_residual_tolerance", 1.0e-13);
  solver.absolute_residual_tolerance = infile("absolute_residual_tolerance", 0.0);
 
  // And the linear solver options
  solver.max_linear_iterations = infile("max_linear_iterations", 50000);
  solver.initial_linear_tolerance = infile("initial_linear_tolerance", 1.e-10);
 
  // Print information about the system to the screen.
  equation_systems.print_info();
 
  // Solve the system.
  system.solve();
 
  ExactSolution exact_sol(equation_systems);
 
  SolutionFunction soln_func(system.variable_number("u"));
  SolutionGradient soln_grad(system.variable_number("u"));
 
  // Build FunctionBase* containers to attach to the ExactSolution object.
  std::vector<FunctionBase<Number> *> sols(1, &soln_func);
  std::vector<FunctionBase<Gradient> *> grads(1, &soln_grad);
 
  exact_sol.attach_exact_values(sols);
  exact_sol.attach_exact_derivs(grads);
 
  // Use higher quadrature order for more accurate error results
  int extra_error_quadrature = infile("extra_error_quadrature", 2);
  exact_sol.extra_quadrature_order(extra_error_quadrature);
 
  // Compute the error.
  exact_sol.compute_error("CurlCurl", "u");
 
  // Print out the error values
  libMesh::out << "L2-Error is: "
               << exact_sol.l2_error("CurlCurl", "u")
               << std::endl;
  libMesh::out << "HCurl semi-norm error is: "
               << exact_sol.error_norm("CurlCurl", "u", HCURL_SEMINORM)
               << std::endl;
  libMesh::out << "HCurl-Error is: "
               << exact_sol.hcurl_error("CurlCurl", "u")
               << std::endl;
 
#ifdef LIBMESH_HAVE_EXODUS_API
 
  // We write the file in the ExodusII format.
  ExodusII_IO(mesh).write_equation_systems("out.e", equation_systems);
 
#endif // #ifdef LIBMESH_HAVE_EXODUS_API
 
  // All done.
  return 0;
}

References libMesh::DiffSolver::absolute_residual_tolerance, libMesh::EquationSystems::add_system(), libMesh::ExactSolution::attach_exact_derivs(), libMesh::ExactSolution::attach_exact_values(), libMesh::MeshTools::Generation::build_cube(), libMesh::command_line_value(), libMesh::ExactSolution::compute_error(), libMesh::default_solver_package(), libMesh::ExactSolution::error_norm(), libMesh::ExactSolution::extra_quadrature_order(), libMesh::ExactSolution::hcurl_error(), libMesh::HCURL_SEMINORM, libMesh::EquationSystems::init(), libMesh::DiffSolver::initial_linear_tolerance, libMesh::INVALID_SOLVER_PACKAGE, libMesh::ExactSolution::l2_error(), main(), libMesh::DiffSolver::max_linear_iterations, libMesh::DiffSolver::max_nonlinear_iterations, mesh, libMesh::out, libMesh::EquationSystems::print_info(), libMesh::MeshBase::print_info(), libMesh::DiffSolver::quiet, libMesh::DiffSolver::relative_residual_tolerance, libMesh::DiffSolver::relative_step_tolerance, libMesh::FEMSystem::solve(), libMesh::DifferentiableSystem::time_solver, libMesh::System::variable_number(), libMesh::DiffSolver::verbose, and libMesh::ExodusII_IO::write_equation_systems().