I wrote a test program, based on step-5, for testing unrelated things, and
checked the results. But when I increased the degree of the finite elements
from 1 to 3 (or higher), the results were wrong, while the change from 1 to
2 just improved convergence, as expected. Did I make a mistake in my code
here?
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/* ---------------------------------------------------------------------
*
* Copyright (C) 1999 - 2018 by the deal.II authors
*
* This file is part of the deal.II library.
*
* The deal.II library is free software; you can use it, redistribute
* it, and/or modify it under the terms of the GNU Lesser General
* Public License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
* The full text of the license can be found in the file LICENSE.md at
* the top level directory of deal.II.
*
* ---------------------------------------------------------------------
*
* Author: Wolfgang Bangerth, University of Heidelberg, 1999
*/
// @sect3{Include files}
// Again, the first few include files are already known, so we won't comment
// on them:
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/base/function.h>
#include <deal.II/base/logstream.h>
#include <deal.II/base/convergence_table.h>
#include <deal.II/lac/vector.h>
#include <deal.II/lac/full_matrix.h>
#include <deal.II/lac/sparse_matrix.h>
#include <deal.II/lac/sparse_direct.h>
#include <deal.II/lac/dynamic_sparsity_pattern.h>
#include <deal.II/lac/solver_cg.h>
#include <deal.II/lac/precondition.h>
#include <deal.II/lac/solver_gmres.h>
#include <deal.II/grid/tria.h>
#include <deal.II/dofs/dof_handler.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/tria_accessor.h>
#include <deal.II/grid/tria_iterator.h>
#include <deal.II/dofs/dof_accessor.h>
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/numerics/vector_tools.h>
#include <deal.II/numerics/matrix_tools.h>
#include <deal.II/numerics/data_out.h>
// This one is new. We want to read a triangulation from disk, and the class
// which does this is declared in the following file:
#include <deal.II/grid/grid_in.h>
// We will use a circular domain, and the object describing the boundary of it
// comes from this file:
#include <deal.II/grid/manifold_lib.h>
// This is C++ ...
#include <fstream>
#include <iostream>
// Finally, this has been discussed in previous tutorial programs before:
using namespace dealii;
template <int dim>
class Solution : public Function<dim>
{
public:
Solution() : Function<dim>(1)
{
}
virtual double value(const Point<dim> &p, const unsigned int component) const override;
virtual Tensor<1, dim> gradient(const Point<dim> &p, const unsigned int component) const override;
private:
};
template <int dim>
double Solution<dim>::value(const Point<dim> &p, const unsigned int) const
{
const double x = p[0];
const double y = p[1];
return sin(M_PI * x) * sin(M_PI * y);
}
template<int dim>
Tensor<1, dim> Solution<dim>::gradient(const Point<dim> &p, const unsigned int) const
{
Tensor<1, dim> return_value;
AssertThrow(dim == 2, ExcNotImplemented());
const double x = p[0];
const double y = p[1];
return_value[0] = M_PI * cos(M_PI * x) * sin(M_PI * y);
return_value[1] = M_PI * cos(M_PI * y) * sin(M_PI * x);
return return_value;
}
// @sect3{The <code>Step5</code> class template}
// The main class is mostly as in the previous example. The most visible
// change is that the function <code>make_grid_and_dofs</code> has been
// removed, since creating the grid is now done in the <code>run</code>
// function and the rest of its functionality is now in
// <code>setup_system</code>. Apart from this, everything is as before.
template <int dim>
class Step5
{
public:
Step5();
~Step5();
void run();
private:
void setup_system();
void assemble_system();
void calculate_residual_multi_val(const Vector<double> &epsilon_v, const Vector<double> &u, Vector<double> &residual);
void calculate_residual(const Vector<double> &u, Vector<double> &residual);
void solve();
void solve_with_matrix_jacobian();
void process_solution();
void output_results(const unsigned int cycle);
void assemble_rhs(double &return_value,
const double factor,
const double &val_u,
const Tensor<1, dim> &grad_u,
const Point<dim> p,
const double &fe_values_value,
const Tensor<1, dim> &fe_gradient_value,
const double JxW_value);
Triangulation<dim> triangulation;
FE_Q<dim> fe;
DoFHandler<dim> dof_handler;
SparsityPattern sparsity_pattern;
SparseMatrix<double> system_matrix;
AffineConstraints<double> hanging_node_constraints;
ConvergenceTable convergence_table;
Vector<double> solution;
Vector<double> system_rhs;
};
// @sect3{Working with nonconstant coefficients}
// In step-4, we showed how to use non-constant boundary values and right hand
// side. In this example, we want to use a variable coefficient in the
// elliptic operator instead. Since we have a function which just depends on
// the point in space we can do things a bit more simply and use a plain
// function instead of inheriting from Function.
// This is the implementation of the coefficient function for a single
// point. We let it return 20 if the distance to the origin is less than 0.5,
// and 1 otherwise.
template <int dim>
double coefficient(const Point<dim> &p)
{
return -2 * M_PI * M_PI * sin(M_PI * p[0]) * sin(M_PI * p[1]);
}
// @sect3{The <code>Step5</code> class implementation}
// @sect4{Step5::Step5}
// This function is as before.
template <int dim>
Step5<dim>::Step5()
: fe(2)
, dof_handler(triangulation)
{}
template <int dim>
Step5<dim>::~Step5<dim>()
{
dof_handler.clear();
}
// @sect4{Step5::setup_system}
// This is the function <code>make_grid_and_dofs</code> from the previous
// example, minus the generation of the grid. Everything else is unchanged:
template <int dim>
void Step5<dim>::setup_system()
{
dof_handler.distribute_dofs(fe);
hanging_node_constraints.clear();
DoFTools::make_zero_boundary_constraints(dof_handler,
0,
hanging_node_constraints);
hanging_node_constraints.close();
std::cout << " Number of degrees of freedom: " << dof_handler.n_dofs()
<< std::endl;
DynamicSparsityPattern dsp(dof_handler.n_dofs());
DoFTools::make_sparsity_pattern(dof_handler, dsp);
sparsity_pattern.copy_from(dsp);
system_matrix.reinit(sparsity_pattern);
solution.reinit(dof_handler.n_dofs());
system_rhs.reinit(dof_handler.n_dofs());
}
template <int dim>
void Step5<dim>::assemble_rhs(double &return_value,
const double factor,
const double &val_u,
const Tensor<1, dim> &grad_u,
const Point<dim> p,
const double &fe_values_value,
const Tensor<1, dim> &fe_gradient_value,
const double JxW_value)
{
(void) val_u;
return_value += factor
* (coefficient(p)
* fe_values_value
- grad_u
* fe_gradient_value
)
* JxW_value;
}
// @sect4{Step5::assemble_system}
// As in the previous examples, this function is not changed much with regard
// to its functionality, but there are still some optimizations which we will
// show. For this, it is important to note that if efficient solvers are used
// (such as the preconditioned CG method), assembling the matrix and right hand
// side can take a comparable time, and you should think about using one or
// two optimizations at some places.
//
// The first parts of the function are completely unchanged from before:
template <int dim>
void Step5<dim>::assemble_system()
{
QGauss<dim> quadrature_formula(2);
FEValues<dim> fe_values(fe,
quadrature_formula,
update_values | update_gradients |
update_quadrature_points | update_JxW_values);
const unsigned int dofs_per_cell = fe.dofs_per_cell;
const unsigned int n_q_points = quadrature_formula.size();
FullMatrix<double> cell_matrix(dofs_per_cell, dofs_per_cell);
Vector<double> cell_rhs(dofs_per_cell);
std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
std::vector<Tensor<1, dim>> grad_u(quadrature_formula.size());
std::vector<double> val_u(quadrature_formula.size());
for (const auto &cell : dof_handler.active_cell_iterators())
{
cell_matrix = 0.;
cell_rhs = 0.;
fe_values.reinit(cell);
fe_values.get_function_values(solution, val_u);
fe_values.get_function_gradients(solution, grad_u);
for (unsigned int q = 0; q < n_q_points; ++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)
* fe_values.shape_grad(j, q)
* fe_values.JxW(q);
cell_rhs(i) += coefficient<dim>(fe_values.quadrature_point(q))
* fe_values.shape_value(i, q)
* fe_values.JxW(q);
}
}
cell->get_dof_indices(local_dof_indices);
hanging_node_constraints.distribute_local_to_global(cell_matrix,
cell_rhs,
local_dof_indices,
system_matrix,
system_rhs);
}
}
template <int dim>
void Step5<dim>::calculate_residual_multi_val(const Vector<double> &epsilon_v, const Vector<double> &u, Vector<double> &residual)
{
Vector<double> u_new(u);
u_new += epsilon_v;
calculate_residual(u_new, residual);
}
template <int dim>
void Step5<dim>::calculate_residual(const Vector<double> &u, Vector<double> &residual)
{
QGauss<dim> quadrature_formula(2);
FEValues<dim> fe_values(fe,
quadrature_formula,
update_values | update_gradients |
update_quadrature_points | update_JxW_values);
const unsigned int dofs_per_cell = fe.dofs_per_cell;
const unsigned int n_q_points = quadrature_formula.size();
residual.reinit(dof_handler.n_dofs());
std::vector<Tensor<1, dim>> grad_u(quadrature_formula.size());
std::vector<double> val_u(quadrature_formula.size());
Vector<double> evaluation_point(u.size());
evaluation_point = u;
Vector<double> cell_rhs(dofs_per_cell);
std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
for (const auto &cell : dof_handler.active_cell_iterators())
{
cell_rhs = 0.;
fe_values.reinit(cell);
fe_values.get_function_values(evaluation_point, val_u);
fe_values.get_function_gradients(evaluation_point, grad_u);
for (unsigned int q = 0; q < n_q_points; ++q)
{
for (unsigned int i = 0; i < dofs_per_cell; ++i)
{
assemble_rhs(cell_rhs(i),
1,
val_u[q],
grad_u[q],
fe_values.quadrature_point(q),
fe_values.shape_value(i, q),
fe_values.shape_grad(i, q),
fe_values.JxW(q));
}
}
cell->get_dof_indices(local_dof_indices);
for (unsigned int i = 0; i < dofs_per_cell; ++i)
{
residual(local_dof_indices[i]) += cell_rhs(i);
}
}
}
template <int dim>
void Step5<dim>::process_solution(void)
{
Vector<double> difference_per_cell(triangulation.n_active_cells());
VectorTools::integrate_difference(dof_handler,
solution,
Solution<dim>(),
difference_per_cell,
QGauss<dim>(fe.degree * 2 + 1),
VectorTools::L2_norm);
const double L2_error = VectorTools::compute_global_error(triangulation, difference_per_cell, VectorTools::L2_norm);
VectorTools::integrate_difference(dof_handler,
solution,
Solution<dim>(),
difference_per_cell,
QGauss<dim>(fe.degree * 2 + 1),
VectorTools::H1_seminorm);
const double H1_error = VectorTools::compute_global_error(triangulation, difference_per_cell, VectorTools::H1_seminorm);
const QTrapez<1> q_trapez;
const QIterated<dim> q_iterated(q_trapez, fe.degree * 2 + 1);
VectorTools::integrate_difference(dof_handler,
solution,
Solution<dim>(),
difference_per_cell,
q_iterated,
VectorTools::Linfty_norm);
const double Linfty_error = VectorTools::compute_global_error(triangulation, difference_per_cell, VectorTools::Linfty_norm);
const unsigned int n_global_active_cells = triangulation.n_global_active_cells();
const unsigned int n_dofs = dof_handler.n_dofs();
convergence_table.add_value("cells", n_global_active_cells);
convergence_table.add_value("dofs", n_dofs);
convergence_table.add_value("L2", L2_error);
convergence_table.add_value("H1", H1_error);
convergence_table.add_value("Linfty", Linfty_error);
convergence_table.add_value("RHS-L2", system_rhs.l2_norm());
}
// @sect4{Step5::solve}
// The solution process again looks mostly like in the previous
// examples. However, we will now use a preconditioned conjugate gradient
// algorithm. It is not very difficult to make this change. In fact, the only
// thing we have to alter is that we need an object which will act as a
// preconditioner. We will use SSOR (symmetric successive overrelaxation),
// with a relaxation factor of 1.2. For this purpose, the
// <code>SparseMatrix</code> class has a function which does one SSOR step,
// and we need to package the address of this function together with the
// matrix on which it should act (which is the matrix to be inverted) and the
// relaxation factor into one object. The <code>PreconditionSSOR</code> class
// does this for us. (<code>PreconditionSSOR</code> class takes a template
// argument denoting the matrix type it is supposed to work on. The default
// value is <code>SparseMatrix@<double@></code>, which is exactly what we need
// here, so we simply stick with the default and do not specify anything in
// the angle brackets.)
//
// Note that for the present case, SSOR doesn't really perform much better
// than most other preconditioners (though better than no preconditioning at
// all). A brief comparison of different preconditioners is presented in the
// Results section of the next tutorial program, step-6.
//
// With this, the rest of the function is trivial: instead of the
// <code>PreconditionIdentity</code> object we have created before, we now use
// the preconditioner we have declared, and the CG solver will do the rest for
// us:
template <int dim>
void Step5<dim>::solve()
{
SolverControl solver_control(1000, 1e-12);
SolverCG<> solver(solver_control);
PreconditionSSOR<> preconditioner;
preconditioner.initialize(system_matrix, 1.2);
solver.solve(system_matrix, solution, system_rhs, preconditioner);
std::cout << " " << solver_control.last_step()
<< " CG iterations needed to obtain convergence." << std::endl;
}
template <int dim>
void Step5<dim>::solve_with_matrix_jacobian()
{
SolverControl solver_control(dof_handler.n_dofs(), 1e-12);
SolverGMRES<> solver(solver_control);
PreconditionSSOR<> preconditioner;
preconditioner.initialize(system_matrix, 1.2);
Vector<double> local_residual(dof_handler.n_dofs()), local_solution(dof_handler.n_dofs()), newton_update(dof_handler.n_dofs());
calculate_residual_multi_val(local_solution, solution, local_residual);
std::cout << "L2-norm before resetting: " << local_residual.l2_norm() << '\n';
solution = 0;
calculate_residual_multi_val(local_solution, solution, local_residual);
std::cout << "L2-norm before solving: " << local_residual.l2_norm() << '\n';
solver.solve(system_matrix, solution, system_rhs, preconditioner);
calculate_residual_multi_val(local_solution, solution, local_residual);
std::cout << "L2-norm after solving: " << local_residual.l2_norm() << '\n';
}
// @sect4{Step5::output_results and setting output flags}
// Writing output to a file is mostly the same as for the previous example,
// but here we will show how to modify some output options and how to
// construct a different filename for each refinement cycle.
template <int dim>
void Step5<dim>::output_results(const unsigned int cycle)
{
DataOut<dim> data_out;
data_out.attach_dof_handler(dof_handler);
data_out.add_data_vector(solution, "Calculated_solution");
data_out.build_patches();
std::ofstream output("solution-" + std::to_string(cycle) + ".vtu");
data_out.write_vtu(output);
convergence_table.set_precision("L2", 3);
convergence_table.set_precision("H1", 3);
convergence_table.set_precision("Linfty", 3);
convergence_table.set_precision("RHS-L2", 3);
convergence_table.set_scientific("L2", true);
convergence_table.set_scientific("H1", true);
convergence_table.set_scientific("Linfty", true);
convergence_table.set_scientific("RHS-L2", true);
convergence_table.set_tex_caption("cells", "\\# cells");
convergence_table.set_tex_caption("dofs", "\\# dofs");
convergence_table.set_tex_caption("L2", "$L^2$-error");
convergence_table.set_tex_caption("H1", "$H^1$-error");
convergence_table.set_tex_caption("Linfty", "$L^\\infty$-error");
convergence_table.set_tex_caption("RHS-L2", "$L^2$-norm of RHS");
convergence_table.add_column_to_supercolumn("cells", "n cells");
std::vector<std::string> new_order;
new_order.emplace_back("n cells");
new_order.emplace_back("H1");
new_order.emplace_back("L2");
new_order.emplace_back("Linfty");
convergence_table.set_column_order(new_order);
convergence_table.evaluate_convergence_rates(
"L2", ConvergenceTable::reduction_rate);
convergence_table.evaluate_convergence_rates(
"L2", ConvergenceTable::reduction_rate_log2);
convergence_table.evaluate_convergence_rates(
"H1", ConvergenceTable::reduction_rate);
convergence_table.evaluate_convergence_rates(
"H1", ConvergenceTable::reduction_rate_log2);
std::cout << "\n";
convergence_table.write_text(std::cout);
std::cout << "\n\n";
convergence_table.clear();
}
// @sect4{Step5::run}
// The second to last thing in this program is the definition of the
// <code>run()</code> function. In contrast to the previous programs, we will
// compute on a sequence of meshes that after each iteration is globally
// refined. The function therefore consists of a loop over 6 cycles. In each
// cycle, we first print the cycle number, and then have to decide what to do
// with the mesh. If this is not the first cycle, we simply refine the
// existing mesh once globally. Before running through these cycles, however,
// we have to generate a mesh:
// In previous examples, we have already used some of the functions from the
// <code>GridGenerator</code> class. Here we would like to read a grid from a
// file where the cells are stored and which may originate from someone else,
// or may be the product of a mesh generator tool.
//
// In order to read a grid from a file, we generate an object of data type
// GridIn and associate the triangulation to it (i.e. we tell it to fill our
// triangulation object when we ask it to read the file). Then we open the
// respective file and initialize the triangulation with the data in the file:
template <int dim>
void Step5<dim>::run()
{
GridGenerator::hyper_cube(triangulation);
for (unsigned int cycle = 0; cycle < 6; ++cycle)
{
std::cout << "Cycle " << cycle << ':' << std::endl;
if (cycle != 0)
triangulation.refine_global(1);
// Now that we have a mesh for sure, we write some output and do all the
// things that we have already seen in the previous examples.
std::cout << " Number of active cells: " //
<< triangulation.n_active_cells() //
<< std::endl //
<< " Total number of cells: " //
<< triangulation.n_cells() //
<< std::endl;
setup_system();
assemble_system();
process_solution();
solve_with_matrix_jacobian();
process_solution();
output_results(cycle);
}
}
// @sect3{The <code>main</code> function}
// The main function looks mostly like the one in the previous example, so we
// won't comment on it further:
int main()
{
Step5<2> laplace_problem_2d;
laplace_problem_2d.run();
return 0;
}
