Reference documentation for deal.II version 9.3.0
The step-5 tutorial program

This tutorial depends on step-4.

1. Introduction
2. The commented program

Introduction

Note
The material presented here is also discussed in video lecture 14. (All video lectures are also available here.)

This example does not show revolutionary new things, but it shows many small improvements over the previous examples, and also many small things that can usually be found in finite element programs. Among them are:

• Computations on successively refined grids. At least in the mathematical sciences, it is common to compute solutions on a hierarchy of grids, in order to get a feeling for the accuracy of the solution; if you only have one solution on a single grid, you usually can't guess the accuracy of the solution. Furthermore, deal.II is designed to support adaptive algorithms where iterative solution on successively refined grids is at the heart of algorithms. Although adaptive grids are not used in this example, the foundations for them is laid here.
• In practical applications, the domains are often subdivided into triangulations by automatic mesh generators. In order to use them, it is important to read coarse grids from a file. In this example, we will read a coarse grid in UCD (unstructured cell data) format. When this program was first written around 2000, UCD format was what the AVS Explorer used – a program reasonably widely used at the time but now no longer of importance. (Nonetheless, the file format has survived and is still understood by a number of programs.)
• Finite element programs usually use extensive amounts of computing time, so some optimizations are sometimes necessary. We will show some of them.
• On the other hand, finite element programs tend to be rather complex, so debugging is an important aspect. We support safe programming by using assertions that check the validity of parameters and internal states in a debug mode, but are removed in optimized mode. (See also video lecture 18.)
• Regarding the mathematical side, we show how to support a variable coefficient in the elliptic operator and how to use preconditioned iterative solvers for the linear systems of equations.

The equation to solve here is as follows:

\begin{align*} -\nabla \cdot a(\mathbf x) \nabla u(\mathbf x) &= 1 \qquad\qquad & \text{in}\ \Omega, \\ u &= 0 \qquad\qquad & \text{on}\ \partial\Omega. \end{align*}

If $$a(\mathbf x)$$ was a constant coefficient, this would simply be the Poisson equation. However, if it is indeed spatially variable, it is a more complex equation (often referred to as the "extended Poisson equation"). Depending on what the variable $$u$$ refers to it models a variety of situations with wide applicability:

• If $$u$$ is the electric potential, then $$-a\nabla u$$ is the electric current in a medium and the coefficient $$a$$ is the conductivity of the medium at any given point. (In this situation, the right hand side of the equation would be the electric source density and would usually be zero or consist of localized, Delta-like, functions.)
• If $$u$$ is the vertical deflection of a thin membrane, then $$a$$ would be a measure of the local stiffness. This is the interpretation that will allow us to interpret the images shown in the results section below.

Since the Laplace/Poisson equation appears in so many contexts, there are many more interpretations than just the two listed above.

When assembling the linear system for this equation, we need the weak form which here reads as follows:

\begin{align*} (a \nabla \varphi, \nabla u) &= (\varphi, 1) \qquad \qquad \forall \varphi. \end{align*}

The implementation in the assemble_system function follows immediately from this.

The commented program

Include files

Again, the first few include files are already known, so we won't comment on them:

This one is new. We want to read a triangulation from disk, and the class which does this is declared in the following file :

We will use a circular domain, and the object describing the boundary of it comes from this file :

This is C++ ...

#include <fstream>
#include <iostream>

Finally, this has been discussed in previous tutorial programs before:

using namespace dealii;

The Step5 class template

The main class is mostly as in the previous example. The most visible change is that the function make_grid has been removed, since creating the grid is now done in the run function and the rest of its functionality is now in setup_system. Apart from this, everything is as before.

template <int dim>
class Step5
{
public:
Step5();
void run();
private:
void setup_system();
void assemble_system();
void solve();
void output_results(const unsigned int cycle) const;
DoFHandler<dim> dof_handler;
SparsityPattern sparsity_pattern;
SparseMatrix<double> system_matrix;
Vector<double> solution;
Vector<double> system_rhs;
};

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)
{
if (p.square() < 0.5 * 0.5)
return 20;
else
return 1;
}

The Step5 class implementation

Step5::Step5

This function is as before.

template <int dim>
Step5<dim>::Step5()
: fe(1)
, dof_handler(triangulation)
{}

Step5::setup_system

This is the function make_grid 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);
std::cout << " Number of degrees of freedom: " << dof_handler.n_dofs()
<< std::endl;
DynamicSparsityPattern dsp(dof_handler.n_dofs());
sparsity_pattern.copy_from(dsp);
system_matrix.reinit(sparsity_pattern);
solution.reinit(dof_handler.n_dofs());
system_rhs.reinit(dof_handler.n_dofs());
}

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()
{
FEValues<dim> fe_values(fe,
const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
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);

Next is the typical loop over all cells to compute local contributions and then to transfer them into the global matrix and vector. The only change in this part, compared to step-4, is that we will use the coefficient() function defined above to compute the coefficient value at each quadrature point.

for (const auto &cell : dof_handler.active_cell_iterators())
{
cell_matrix = 0.;
cell_rhs = 0.;
fe_values.reinit(cell);
for (const unsigned int q_index : fe_values.quadrature_point_indices())
{
const double current_coefficient =
for (const unsigned int i : fe_values.dof_indices())
{
for (const unsigned int j : fe_values.dof_indices())
cell_matrix(i, j) +=
(current_coefficient * // a(x_q)
fe_values.JxW(q_index)); // dx
cell_rhs(i) += (fe_values.shape_value(i, q_index) * // phi_i(x_q)
1.0 * // f(x_q)
fe_values.JxW(q_index)); // dx
}
}
cell->get_dof_indices(local_dof_indices);
for (const unsigned int i : fe_values.dof_indices())
{
for (const unsigned int j : fe_values.dof_indices())
local_dof_indices[j],
cell_matrix(i, j));
system_rhs(local_dof_indices[i]) += cell_rhs(i);
}
}

With the matrix so built, we use zero boundary values again:

std::map<types::global_dof_index, double> boundary_values;
0,
boundary_values);
system_matrix,
solution,
system_rhs);
}

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 SparseMatrix 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 PreconditionSSOR class does this for us. (PreconditionSSOR class takes a template argument denoting the matrix type it is supposed to work on. The default value is SparseMatrix<double>, 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 PreconditionIdentity 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<Vector<double>> solver(solver_control);
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;
}

Step5::output_results and setting output flags

Writing output to a file is mostly the same as for the previous tutorial. The only difference is that we now need to construct a different filename for each refinement cycle.

The function writes the output in VTU format, a variation of the VTK format that requires less disk space because it compresses the data. Of course, there are many other formats supported by the DataOut class if you desire to use a program for visualization that doesn't understand VTK or VTU.

template <int dim>
void Step5<dim>::output_results(const unsigned int cycle) const
{
DataOut<dim> data_out;
data_out.attach_dof_handler(dof_handler);
data_out.build_patches();
std::ofstream output("solution-" + std::to_string(cycle) + ".vtu");
data_out.write_vtu(output);
}

Step5::run

The second to last thing in this program is the definition of the run() 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 GridGenerator 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>
{
GridIn<dim> grid_in;
grid_in.attach_triangulation(triangulation);
std::ifstream input_file("circle-grid.inp");

We would now like to read the file. However, the input file is only for a two-dimensional triangulation, while this function is a template for arbitrary dimension. Since this is only a demonstration program, we will not use different input files for the different dimensions, but rather quickly kill the whole program if we are not in 2D. Of course, since the main function below assumes that we are working in two dimensions we could skip this check, in this version of the program, without any ill effects.

It turns out that more than 90 per cent of programming errors are invalid function parameters such as invalid array sizes, etc, so we use assertions heavily throughout deal.II to catch such mistakes. For this, the Assert macro is a good choice, since it makes sure that the condition which is given as first argument is valid, and if not throws an exception (its second argument) which will usually terminate the program giving information where the error occurred and what the reason was. (A longer discussion of what exactly the Assert macro does can be found in the exception documentation module.) This generally reduces the time to find programming errors dramatically and we have found assertions an invaluable means to program fast.

On the other hand, all these checks (there are over 10,000 of them in the library at present) should not slow down the program too much if you want to do large computations. To this end, the Assert macro is only used in debug mode and expands to nothing if in optimized mode. Therefore, while you test your program on small problems and debug it, the assertions will tell you where the problems are. Once your program is stable, you can switch off debugging and the program will run your real computations without the assertions and at maximum speed. More precisely: turning off all the checks in the library (which prevent you from calling functions with wrong arguments, walking off of arrays, etc.) by compiling your program in optimized mode usually makes things run about four times faster. Even though optimized programs are more performant, we still recommend developing in debug mode since it allows the library to find lots of common programming errors automatically. For those who want to try: The way to switch from debug mode to optimized mode is to recompile your program with the command make release. The output of the make program should now indicate to you that the program is now compiled in optimized mode, and it will later also be linked to libraries that have been compiled for optimized mode. In order to switch back to debug mode, simply recompile with the command make debug.

Assert(dim == 2, ExcInternalError());

ExcInternalError is a globally defined exception, which may be thrown whenever something is terribly wrong. Usually, one would like to use more specific exceptions, and particular in this case one would of course try to do something else if dim is not equal to two, e.g. create a grid using library functions. Aborting a program is usually not a good idea and assertions should really only be used for exceptional cases which should not occur, but might due to stupidity of the programmer, user, or someone else. The situation above is not a very clever use of Assert, but again: this is a tutorial and it might be worth to show what not to do, after all.

So if we got past the assertion, we know that dim==2, and we can now actually read the grid. It is in UCD (unstructured cell data) format (though the convention is to use the suffix inp for UCD files):

If you like to use another input format, you have to use one of the other grid_in.read_xxx function. (See the documentation of the GridIn class to find out what input formats are presently supported.)

The grid in the file describes a circle. Therefore we have to use a manifold object which tells the triangulation where to put new points on the boundary when the grid is refined. Unlike step-1, since GridIn does not know that the domain has a circular boundary (unlike GridGenerator::hyper_shell) we have to explicitly attach a manifold to the boundary after creating the triangulation to get the correct result when we refine the mesh.

const SphericalManifold<dim> boundary;
triangulation.set_manifold(0, boundary);
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();
solve();
output_results(cycle);
}
}

The main 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;
}

Results

Here is the console output:

Cycle 0:
Number of active cells: 20
Total number of cells: 20
Number of degrees of freedom: 25
13 CG iterations needed to obtain convergence.
Cycle 1:
Number of active cells: 80
Total number of cells: 100
Number of degrees of freedom: 89
18 CG iterations needed to obtain convergence.
Cycle 2:
Number of active cells: 320
Total number of cells: 420
Number of degrees of freedom: 337
29 CG iterations needed to obtain convergence.
Cycle 3:
Number of active cells: 1280
Total number of cells: 1700
Number of degrees of freedom: 1313
52 CG iterations needed to obtain convergence.
Cycle 4:
Number of active cells: 5120
Total number of cells: 6820
Number of degrees of freedom: 5185
95 CG iterations needed to obtain convergence.
Cycle 5:
Number of active cells: 20480
Total number of cells: 27300
Number of degrees of freedom: 20609
182 CG iterations needed to obtain convergence.

In each cycle, the number of cells quadruples and the number of CG iterations roughly doubles. Also, in each cycle, the program writes one output graphic file in VTU format. They are depicted in the following:

Due to the variable coefficient (the curvature there is reduced by the same factor by which the coefficient is increased), the top region of the solution is flattened. The gradient of the solution is discontinuous along the interface, although this is not very clearly visible in the pictures above. We will look at this in more detail in the next example.

The pictures also show that the solution computed by this program is actually pretty wrong on a very coarse mesh (its magnitude is wrong). That's because no numerical method guarantees that the solution on a coarse mesh is particularly accurate – but we know that the solution converges to the exact solution, and indeed you can see how the solutions from one mesh to the next seem to not change very much any more at the end.

The plain program

/* ---------------------------------------------------------------------
*
* Copyright (C) 1999 - 2021 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
* 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
*/
#include <fstream>
#include <iostream>
using namespace dealii;
template <int dim>
class Step5
{
public:
Step5();
void run();
private:
void setup_system();
void assemble_system();
void solve();
void output_results(const unsigned int cycle) const;
DoFHandler<dim> dof_handler;
SparsityPattern sparsity_pattern;
SparseMatrix<double> system_matrix;
Vector<double> solution;
Vector<double> system_rhs;
};
template <int dim>
double coefficient(const Point<dim> &p)
{
if (p.square() < 0.5 * 0.5)
return 20;
else
return 1;
}
template <int dim>
Step5<dim>::Step5()
: fe(1)
, dof_handler(triangulation)
{}
template <int dim>
void Step5<dim>::setup_system()
{
dof_handler.distribute_dofs(fe);
std::cout << " Number of degrees of freedom: " << dof_handler.n_dofs()
<< std::endl;
DynamicSparsityPattern dsp(dof_handler.n_dofs());
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_system()
{
FEValues<dim> fe_values(fe,
const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
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);
for (const auto &cell : dof_handler.active_cell_iterators())
{
cell_matrix = 0.;
cell_rhs = 0.;
fe_values.reinit(cell);
for (const unsigned int q_index : fe_values.quadrature_point_indices())
{
const double current_coefficient =
for (const unsigned int i : fe_values.dof_indices())
{
for (const unsigned int j : fe_values.dof_indices())
cell_matrix(i, j) +=
(current_coefficient * // a(x_q)
fe_values.JxW(q_index)); // dx
cell_rhs(i) += (fe_values.shape_value(i, q_index) * // phi_i(x_q)
1.0 * // f(x_q)
fe_values.JxW(q_index)); // dx
}
}
cell->get_dof_indices(local_dof_indices);
for (const unsigned int i : fe_values.dof_indices())
{
for (const unsigned int j : fe_values.dof_indices())
local_dof_indices[j],
cell_matrix(i, j));
system_rhs(local_dof_indices[i]) += cell_rhs(i);
}
}
std::map<types::global_dof_index, double> boundary_values;
0,
boundary_values);
system_matrix,
solution,
system_rhs);
}
template <int dim>
void Step5<dim>::solve()
{
SolverControl solver_control(1000, 1e-12);
SolverCG<Vector<double>> solver(solver_control);
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>::output_results(const unsigned int cycle) const
{
DataOut<dim> data_out;
data_out.attach_dof_handler(dof_handler);
data_out.build_patches();
std::ofstream output("solution-" + std::to_string(cycle) + ".vtu");
data_out.write_vtu(output);
}
template <int dim>
{
GridIn<dim> grid_in;
grid_in.attach_triangulation(triangulation);
std::ifstream input_file("circle-grid.inp");
Assert(dim == 2, ExcInternalError());
const SphericalManifold<dim> boundary;
triangulation.set_manifold(0, boundary);
for (unsigned int cycle = 0; cycle < 6; ++cycle)
{
std::cout << "Cycle " << cycle << ':' << std::endl;
if (cycle != 0)
triangulation.refine_global(1);
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();
solve();
output_results(cycle);
}
}
int main()
{
Step5<2> laplace_problem_2d;
laplace_problem_2d.run();
return 0;
}