 Reference documentation for deal.II version GIT 2c37b3e359 2022-05-26 00:10:02+00:00
The step-56 tutorial program

This tutorial depends on step-16, step-22.

1. Introduction
2. The commented program
1. Results
2. The plain program
This program was contributed by Ryan Grove and Timo Heister.

This material is based upon work partially supported by National Science Foundation grant DMS1522191 and the Computational Infrastructure in Geodynamics initiative (CIG), through the National Science Foundation under Award No. EAR-0949446 and The University of California-Davis.

The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Melt in the Mantle where work on this tutorial was undertaken. This work was supported by EPSRC grant no EP/K032208/1.

Note
If you use this program as a basis for your own work, please consider citing it in your list of references. The initial version of this work was contributed to the deal.II project by the authors listed in the following citation: # Introduction

### Stokes Problem

The purpose of this tutorial is to create an efficient linear solver for the Stokes equation and compare it to alternative approaches. Here, we will use FGMRES with geometric multigrid as a preconditioner velocity block, and we will show in the results section that this is a fundamentally better approach than the linear solvers used in step-22 (including the scheme described in "Possible Extensions"). Fundamentally, this is because only with multigrid it is possible to get $$O(n)$$ solve time, where $$n$$ is the number of unknowns of the linear system. Using the Timer class, we collect some statistics to compare setup times, solve times, and number of iterations. We also compute errors to make sure that what we have implemented is correct.

Let $$u \in H_0^1 = \{ u \in H^1(\Omega), u|_{\partial \Omega} = 0 \}$$ and $$p \in L_*^2 = \{ p \in L^2(\Omega), \int_\Omega p = 0 \}$$. The Stokes equations read as follows in non-dimensionalized form:

\begin{eqnarray*} - 2 \text{div} \frac {1}{2} \left[ (\nabla \textbf{u}) + (\nabla \textbf{u})^T\right] + \nabla p & =& f \\ - \nabla \cdot u &=& 0 \end{eqnarray*}

Note that we are using the deformation tensor instead of $$\Delta u$$ (a detailed description of the difference between the two can be found in step-22, but in summary, the deformation tensor is more physical as well as more expensive).

### Linear Solver and Preconditioning Issues

The weak form of the discrete equations naturally leads to the following linear system for the nodal values of the velocity and pressure fields:

\begin{eqnarray*} \left(\begin{array}{cc} A & B^T \\ B & 0 \end{array}\right) \left(\begin{array}{c} U \\ P \end{array}\right) = \left(\begin{array}{c} F \\ 0 \end{array}\right). \end{eqnarray*}

Our goal is to compare several solution approaches. While step-22 solves the linear system using a "Schur complement approach" in two separate steps, we instead attack the block system at once using FMGRES with an efficient preconditioner, in the spirit of the approach outlined in the "Results" section of step-22. The idea is as follows: if we find a block preconditioner $$P$$ such that the matrix

\begin{eqnarray*} \left(\begin{array}{cc} A & B^T \\ B & 0 \end{array}\right) P^{-1} \end{eqnarray*}

is simple, then an iterative solver with that preconditioner will converge in a few iterations. Notice that we are doing right preconditioning here. Using the Schur complement $$S=BA^{-1}B^T$$, we find that

\begin{eqnarray*} P^{-1} = \left(\begin{array}{cc} A & B^T \\ 0 & S \end{array}\right)^{-1} \end{eqnarray*}

is a good choice. Let $$\widetilde{A^{-1}}$$ be an approximation of $$A^{-1}$$ and $$\widetilde{S^{-1}}$$ of $$S^{-1}$$, we see

\begin{eqnarray*} P^{-1} = \left(\begin{array}{cc} A^{-1} & 0 \\ 0 & I \end{array}\right) \left(\begin{array}{cc} I & B^T \\ 0 & -I \end{array}\right) \left(\begin{array}{cc} I & 0 \\ 0 & S^{-1} \end{array}\right) \approx \left(\begin{array}{cc} \widetilde{A^{-1}} & 0 \\ 0 & I \end{array}\right) \left(\begin{array}{cc} I & B^T \\ 0 & -I \end{array}\right) \left(\begin{array}{cc} I & 0 \\ 0 & \widetilde{S^{-1}} \end{array}\right). \end{eqnarray*}

Since $$P$$ is aimed to be a preconditioner only, we shall use the approximations on the right in the equation above.

As discussed in step-22, $$-M_p^{-1}=:\widetilde{S^{-1}} \approx S^{-1}$$, where $$M_p$$ is the pressure mass matrix and is solved approximately by using CG with ILU as a preconditioner, and $$\widetilde{A^{-1}}$$ is obtained by one of multiple methods: solving a linear system with CG and ILU as preconditioner, just using one application of an ILU, solving a linear system with CG and GMG (Geometric Multigrid as described in step-16) as a preconditioner, or just performing a single V-cycle of GMG.

As a comparison, instead of FGMRES, we also use the direct solver UMFPACK on the whole system to compare our results with. If you want to use a direct solver (like UMFPACK), the system needs to be invertible. To avoid the one dimensional null space given by the constant pressures, we fix the first pressure unknown to zero. This is not necessary for the iterative solvers.

### Reference Solution

The test problem is a "Manufactured Solution" (see step-7 for details), and we choose $$u=(u_1,u_2,u_3)=(2\sin (\pi x), - \pi y \cos (\pi x),- \pi z \cos (\pi x))$$ and $$p = \sin (\pi x)\cos (\pi y)\sin (\pi z)$$. We apply Dirichlet boundary conditions for the velocity on the whole boundary of the domain $$\Omega=[0,1]\times[0,1]\times[0,1]$$. To enforce the boundary conditions we can just use our reference solution.

If you look up in the deal.II manual what is needed to create a class derived from Function<dim>, you will find that this class has numerous virtual functions, including Function::value(), Function::vector_value(), Function::value_list(), etc., all of which can be overloaded. Different parts of deal.II will require different ones of these particular functions. This can be confusing at first, but luckily the only thing you actually have to implement is value(). The other virtual functions in the Function class have default implementations inside that will call your implementation of value by default.

Notice that our reference solution fulfills $$\nabla \cdot u = 0$$. In addition, the pressure is chosen to have a mean value of zero. For the "Method of Manufactured Solutions" of step-7, we need to find $$\bf f$$ such that:

\begin{align*} {\bf f} = - 2 \text{div} \frac {1}{2} \left[ (\nabla \textbf{u}) + (\nabla \textbf{u})^T\right] + \nabla p. \end{align*}

Using the reference solution above, we obtain:

\begin{eqnarray*} {\bf f} &=& (2 \pi^2 \sin (\pi x),- \pi^3 y \cos(\pi x),- \pi^3 z \cos(\pi x))\\ & & + (\pi \cos(\pi x) \cos(\pi y) \sin(\pi z) ,- \pi \sin(\pi y) \sin(\pi x) \sin(\pi z), \pi \cos(\pi z) \sin(\pi x) \cos(\pi y)) \end{eqnarray*}

### Computing Errors

Because we do not enforce the mean pressure to be zero for our numerical solution in the linear system, we need to post process the solution after solving. To do this we use the VectorTools::compute_mean_value() function to compute the mean value of the pressure to subtract it from the pressure.

### DoF Handlers

The way we implement geometric multigrid here only executes it on the velocity variables (i.e., the $$A$$ matrix described above) but not the pressure. One could implement this in different ways, including one in which one considers all coarse grid operations as acting on $$2\times 2$$ block systems where we only consider the top left block. Alternatively, we can implement things by really only considering a linear system on the velocity part of the overall finite element discretization. The latter is the way we want to use here.

To implement this, one would need to be able to ask questions such as "May I have just part of a DoFHandler?". This is not possible at the time when this program was written, so in order to answer this request for our needs, we simply create a separate, second DoFHandler for just the velocities. We then build linear systems for the multigrid preconditioner based on only this second DoFHandler, and simply transfer the first block of (overall) vectors into corresponding vectors for the entire second DoFHandler. To make this work, we have to assure that the order in which the (velocity) degrees of freedom are ordered in the two DoFHandler objects is the same. This is in fact the case by first distributing degrees of freedom on both, and then using the same sequence of DoFRenumbering operations on both.

### Differences from the Step 22 tutorial

The main difference between step-56 and step-22 is that we use block solvers instead of the Schur Complement approach used in step-22. Details of this approach can be found under the "Block Schur complement preconditioner" subsection of the "Possible Extensions" section of step-22. For the preconditioner of the velocity block, we borrow a class from ASPECT called BlockSchurPreconditioner that has the option to solve for the inverse of $$A$$ or just apply one preconditioner sweep for it instead, which provides us with an expensive and cheap approach, respectively.

# The commented program

### Include files

We need to include the following file to do timings:

This includes the files necessary for us to use geometric Multigrid

In order to make it easy to switch between the different solvers that are being used, we declare an enum that can be passed as an argument to the constructor of the main class.

enum class SolverType
{
FGMRES_ILU,
FGMRES_GMG,
UMFPACK
};

### Functions for Solution and Righthand side

The class Solution is used to define the boundary conditions and to compute errors of the numerical solution. Note that we need to define the values and gradients in order to compute L2 and H1 errors. Here we decided to separate the implementations for 2d and 3d using template specialization.

Note that the first dim components are the velocity components and the last is the pressure.

template <int dim>
class Solution : public Function<dim>
{
public:
Solution()
: Function<dim>(dim + 1)
{}
virtual double value(const Point<dim> & p,
const unsigned int component = 0) const override;
const unsigned int component = 0) const override;
};
template <>
double Solution<2>::value(const Point<2> & p,
const unsigned int component) const
{
Assert(component <= 2 + 1, ExcIndexRange(component, 0, 2 + 1));
using numbers::PI;
const double x = p(0);
const double y = p(1);
if (component == 0)
return sin(PI * x);
if (component == 1)
return -PI * y * cos(PI * x);
if (component == 2)
return sin(PI * x) * cos(PI * y);
return 0;
}
template <>
double Solution<3>::value(const Point<3> & p,
const unsigned int component) const
{
Assert(component <= 3 + 1, ExcIndexRange(component, 0, 3 + 1));
using numbers::PI;
const double x = p(0);
const double y = p(1);
const double z = p(2);
if (component == 0)
return 2.0 * sin(PI * x);
if (component == 1)
return -PI * y * cos(PI * x);
if (component == 2)
return -PI * z * cos(PI * x);
if (component == 3)
return sin(PI * x) * cos(PI * y) * sin(PI * z);
return 0;
}
virtual Tensor< 1, dim, RangeNumberType > gradient(const Point< dim > &p, const unsigned int component=0) const
virtual RangeNumberType value(const Point< dim > &p, const unsigned int component=0) const
Definition: point.h:111
VectorizedArray< Number, width > cos(const ::VectorizedArray< Number, width > &x)
VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &x)
#define Assert(cond, exc)
Definition: exceptions.h:1473
static ::ExceptionBase & ExcIndexRange(std::size_t arg1, std::size_t arg2, std::size_t arg3)
static constexpr double PI
Definition: numbers.h:233

Note that for the gradient we need to return a Tensor<1,dim>

template <>
Tensor<1, 2> Solution<2>::gradient(const Point<2> & p,
const unsigned int component) const
{
Assert(component <= 2, ExcIndexRange(component, 0, 2 + 1));
using numbers::PI;
const double x = p(0);
const double y = p(1);
Tensor<1, 2> return_value;
if (component == 0)
{
return_value = PI * cos(PI * x);
return_value = 0.0;
}
else if (component == 1)
{
return_value = y * PI * PI * sin(PI * x);
return_value = -PI * cos(PI * x);
}
else if (component == 2)
{
return_value = PI * cos(PI * x) * cos(PI * y);
return_value = -PI * sin(PI * x) * sin(PI * y);
}
return return_value;
}
template <>
Tensor<1, 3> Solution<3>::gradient(const Point<3> & p,
const unsigned int component) const
{
Assert(component <= 3, ExcIndexRange(component, 0, 3 + 1));
using numbers::PI;
const double x = p(0);
const double y = p(1);
const double z = p(2);
Tensor<1, 3> return_value;
if (component == 0)
{
return_value = 2 * PI * cos(PI * x);
return_value = 0.0;
return_value = 0.0;
}
else if (component == 1)
{
return_value = y * PI * PI * sin(PI * x);
return_value = -PI * cos(PI * x);
return_value = 0.0;
}
else if (component == 2)
{
return_value = z * PI * PI * sin(PI * x);
return_value = 0.0;
return_value = -PI * cos(PI * x);
}
else if (component == 3)
{
return_value = PI * cos(PI * x) * cos(PI * y) * sin(PI * z);
return_value = -PI * sin(PI * x) * sin(PI * y) * sin(PI * z);
return_value = PI * sin(PI * x) * cos(PI * y) * cos(PI * z);
}
return return_value;
}

Implementation of $$f$$. See the introduction for more information.

template <int dim>
class RightHandSide : public Function<dim>
{
public:
RightHandSide()
: Function<dim>(dim + 1)
{}
virtual double value(const Point<dim> & p,
const unsigned int component = 0) const override;
};
template <>
double RightHandSide<2>::value(const Point<2> & p,
const unsigned int component) const
{
Assert(component <= 2, ExcIndexRange(component, 0, 2 + 1));
using numbers::PI;
double x = p(0);
double y = p(1);
if (component == 0)
return PI * PI * sin(PI * x) + PI * cos(PI * x) * cos(PI * y);
if (component == 1)
return -PI * PI * PI * y * cos(PI * x) - PI * sin(PI * y) * sin(PI * x);
if (component == 2)
return 0;
return 0;
}
template <>
double RightHandSide<3>::value(const Point<3> & p,
const unsigned int component) const
{
Assert(component <= 3, ExcIndexRange(component, 0, 3 + 1));
using numbers::PI;
double x = p(0);
double y = p(1);
double z = p(2);
if (component == 0)
return 2 * PI * PI * sin(PI * x) +
PI * cos(PI * x) * cos(PI * y) * sin(PI * z);
if (component == 1)
return -PI * PI * PI * y * cos(PI * x) +
PI * (-1) * sin(PI * y) * sin(PI * x) * sin(PI * z);
if (component == 2)
return -PI * PI * PI * z * cos(PI * x) +
PI * cos(PI * z) * sin(PI * x) * cos(PI * y);
if (component == 3)
return 0;
return 0;
}

### ASPECT BlockSchurPreconditioner

In the following, we will implement a preconditioner that expands on the ideas discussed in the Results section of step-22. Specifically, we

1. use an upper block-triangular preconditioner because we want to use right preconditioning.
2. optionally allow using an inner solver for the velocity block instead of a single preconditioner application.
3. do not use InverseMatrix but explicitly call SolverCG. This approach is also used in the ASPECT code (see https://aspect.geodynamics.org) that solves the Stokes equations in the context of simulating convection in the earth mantle, and which has been used to solve problems on many thousands of processors.

The bool flag do_solve_A in the constructor allows us to either apply the preconditioner for the velocity block once or use an inner iterative solver for a more accurate approximation instead.

Notice how we keep track of the sum of the inner iterations (preconditioner applications).

template <class PreconditionerAType, class PreconditionerSType>
class BlockSchurPreconditioner : public Subscriptor
{
public:
BlockSchurPreconditioner(
const BlockSparseMatrix<double> &system_matrix,
const SparseMatrix<double> & schur_complement_matrix,
const PreconditionerAType & preconditioner_A,
const PreconditionerSType & preconditioner_S,
const bool do_solve_A);
void vmult(BlockVector<double> &dst, const BlockVector<double> &src) const;
mutable unsigned int n_iterations_A;
mutable unsigned int n_iterations_S;
private:
const BlockSparseMatrix<double> &system_matrix;
const SparseMatrix<double> & schur_complement_matrix;
const PreconditionerAType & preconditioner_A;
const PreconditionerSType & preconditioner_S;
const bool do_solve_A;
};
template <class PreconditionerAType, class PreconditionerSType>
BlockSchurPreconditioner<PreconditionerAType, PreconditionerSType>::
BlockSchurPreconditioner(
const BlockSparseMatrix<double> &system_matrix,
const SparseMatrix<double> & schur_complement_matrix,
const PreconditionerAType & preconditioner_A,
const PreconditionerSType & preconditioner_S,
const bool do_solve_A)
: n_iterations_A(0)
, n_iterations_S(0)
, system_matrix(system_matrix)
, schur_complement_matrix(schur_complement_matrix)
, preconditioner_A(preconditioner_A)
, preconditioner_S(preconditioner_S)
, do_solve_A(do_solve_A)
{}
template <class PreconditionerAType, class PreconditionerSType>
void
BlockSchurPreconditioner<PreconditionerAType, PreconditionerSType>::vmult(
const BlockVector<double> &src) const
{
Vector<double> utmp(src.block(0));
BlockType & block(const unsigned int i)

First solve with the approximation for S

{
SolverControl solver_control(1000, 1e-6 * src.block(1).l2_norm());
SolverCG<Vector<double>> cg(solver_control);
dst.block(1) = 0.0;
cg.solve(schur_complement_matrix,
dst.block(1),
src.block(1),
preconditioner_S);
n_iterations_S += solver_control.last_step();
dst.block(1) *= -1.0;
}
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)

Second, apply the top right block (B^T)

{
system_matrix.block(0, 1).vmult(utmp, dst.block(1));
utmp *= -1.0;
utmp += src.block(0);
}

Finally, either solve with the top left block or just apply one preconditioner sweep

if (do_solve_A == true)
{
SolverControl solver_control(10000, utmp.l2_norm() * 1e-4);
SolverCG<Vector<double>> cg(solver_control);
dst.block(0) = 0.0;
cg.solve(system_matrix.block(0, 0),
dst.block(0),
utmp,
preconditioner_A);
n_iterations_A += solver_control.last_step();
}
else
{
preconditioner_A.vmult(dst.block(0), utmp);
n_iterations_A += 1;
}
}

### The StokesProblem class

This is the main class of the problem.

template <int dim>
class StokesProblem
{
public:
StokesProblem(const unsigned int pressure_degree,
const SolverType solver_type);
void run();
private:
void setup_dofs();
void assemble_system();
void assemble_multigrid();
void solve();
void compute_errors();
void output_results(const unsigned int refinement_cycle) const;
const unsigned int pressure_degree;
const SolverType solver_type;
FESystem<dim> velocity_fe;
DoFHandler<dim> dof_handler;
DoFHandler<dim> velocity_dof_handler;
BlockSparsityPattern sparsity_pattern;
SparseMatrix<double> pressure_mass_matrix;
BlockVector<double> system_rhs;
MGLevelObject<SparsityPattern> mg_sparsity_patterns;
MGLevelObject<SparseMatrix<double>> mg_interface_matrices;
MGConstrainedDoFs mg_constrained_dofs;
TimerOutput computing_timer;
};
template <int dim>
StokesProblem<dim>::StokesProblem(const unsigned int pressure_degree,
const SolverType solver_type)
: pressure_degree(pressure_degree)
, solver_type(solver_type)
, triangulation(Triangulation<dim>::maximum_smoothing)
,
void run(const Iterator &begin, const typename identity< Iterator >::type &end, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length, const unsigned int chunk_size)
Definition: work_stream.h:474
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation

Finite element for the velocity only:

velocity_fe(FE_Q<dim>(pressure_degree + 1), dim)
,
Definition: fe_q.h:549

Finite element for the whole system:

fe(velocity_fe, 1, FE_Q<dim>(pressure_degree), 1)
, dof_handler(triangulation)
, velocity_dof_handler(triangulation)
, computing_timer(std::cout, TimerOutput::never, TimerOutput::wall_times)
{}
@ wall_times
Definition: timer.h:653

#### StokesProblem::setup_dofs

This function sets up the DoFHandler, matrices, vectors, and Multigrid structures (if needed).

template <int dim>
void StokesProblem<dim>::setup_dofs()
{
TimerOutput::Scope scope(computing_timer, "Setup");
system_matrix.clear();
pressure_mass_matrix.clear();

The main DoFHandler only needs active DoFs, so we are not calling distribute_mg_dofs() here

dof_handler.distribute_dofs(fe);

This block structure separates the dim velocity components from the pressure component (used for reordering). Note that we have 2 instead of dim+1 blocks like in step-22, because our FESystem is nested and the dim velocity components appear as one block.

std::vector<unsigned int> block_component(2);
block_component = 0;
block_component = 1;

Velocities start at component 0:

const FEValuesExtractors::Vector velocities(0);

ILU behaves better if we apply a reordering to reduce fillin. There is no advantage in doing this for the other solvers.

if (solver_type == SolverType::FGMRES_ILU)
{
TimerOutput::Scope ilu_specific(computing_timer, "(ILU specific)");
}
void Cuthill_McKee(DoFHandler< dim, spacedim > &dof_handler, const bool reversed_numbering=false, const bool use_constraints=false, const std::vector< types::global_dof_index > &starting_indices=std::vector< types::global_dof_index >())

This ensures that all velocities DoFs are enumerated before the pressure unknowns. This allows us to use blocks for vectors and matrices and allows us to get the same DoF numbering for dof_handler and velocity_dof_handler.

if (solver_type == SolverType::FGMRES_GMG)
{
TimerOutput::Scope multigrid_specific(computing_timer,
"(Multigrid specific)");
TimerOutput::Scope setup_multigrid(computing_timer,
"Setup - Multigrid");
void block_wise(DoFHandler< dim, spacedim > &dof_handler)

This distributes the active dofs and multigrid dofs for the velocity space in a separate DoFHandler as described in the introduction.

velocity_dof_handler.distribute_dofs(velocity_fe);
velocity_dof_handler.distribute_mg_dofs();

The following block of code initializes the MGConstrainedDofs (using the boundary conditions for the velocity), and the sparsity patterns and matrices for each level. The resize() function of MGLevelObject<T> will destroy all existing contained objects.

const std::set<types::boundary_id> zero_boundary_ids = {0};
mg_constrained_dofs.clear();
mg_constrained_dofs.initialize(velocity_dof_handler);
mg_constrained_dofs.make_zero_boundary_constraints(velocity_dof_handler,
zero_boundary_ids);
const unsigned int n_levels = triangulation.n_levels();
mg_interface_matrices.resize(0, n_levels - 1);
mg_matrices.resize(0, n_levels - 1);
mg_sparsity_patterns.resize(0, n_levels - 1);
for (unsigned int level = 0; level < n_levels; ++level)
{
DynamicSparsityPattern csp(velocity_dof_handler.n_dofs(level),
velocity_dof_handler.n_dofs(level));
MGTools::make_sparsity_pattern(velocity_dof_handler, csp, level);
mg_sparsity_patterns[level].copy_from(csp);
mg_matrices[level].reinit(mg_sparsity_patterns[level]);
mg_interface_matrices[level].reinit(mg_sparsity_patterns[level]);
}
}
const std::vector<types::global_dof_index> dofs_per_block =
DoFTools::count_dofs_per_fe_block(dof_handler, block_component);
const unsigned int n_u = dofs_per_block;
const unsigned int n_p = dofs_per_block;
{
constraints.clear();
unsigned int level
Definition: grid_out.cc:4606
std::vector< types::global_dof_index > count_dofs_per_fe_block(const DoFHandler< dim, spacedim > &dof, const std::vector< unsigned int > &target_block=std::vector< unsigned int >())
Definition: dof_tools.cc:1980
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternType &sparsity, const unsigned int level, const AffineConstraints< number > &constraints=AffineConstraints< number >(), const bool keep_constrained_dofs=true)
Definition: mg_tools.cc:595

The following makes use of a component mask for interpolation of the boundary values for the velocity only, which is further explained in the vector valued dealii step-20 tutorial.

DoFTools::make_hanging_node_constraints(dof_handler, constraints);
0,
Solution<dim>(),
constraints,
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
void interpolate_boundary_values(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const std::map< types::boundary_id, const Function< spacedim, number > * > &function_map, std::map< types::global_dof_index, number > &boundary_values, const ComponentMask &component_mask=ComponentMask())

As discussed in the introduction, we need to fix one degree of freedom of the pressure variable to ensure solvability of the problem. We do this here by marking the first pressure dof, which has index n_u as a constrained dof.

if (solver_type == SolverType::UMFPACK)
constraints.close();
}
std::cout << "\tNumber of active cells: " << triangulation.n_active_cells()
<< std::endl
<< "\tNumber of degrees of freedom: " << dof_handler.n_dofs()
<< " (" << n_u << '+' << n_p << ')' << std::endl;
{
BlockDynamicSparsityPattern csp(dofs_per_block, dofs_per_block);
DoFTools::make_sparsity_pattern(dof_handler, csp, constraints, false);
sparsity_pattern.copy_from(csp);
}
system_matrix.reinit(sparsity_pattern);
solution.reinit(dofs_per_block);
system_rhs.reinit(dofs_per_block);
}
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternType &sparsity_pattern, const AffineConstraints< number > &constraints=AffineConstraints< number >(), const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)

#### StokesProblem::assemble_system

In this function, the system matrix is assembled. We assemble the pressure mass matrix in the (1,1) block (if needed) and move it out of this location at the end of this function.

template <int dim>
void StokesProblem<dim>::assemble_system()
{
TimerOutput::Scope assemble(computing_timer, "Assemble");
system_matrix = 0;
system_rhs = 0;
void assemble(const MeshWorker::DoFInfoBox< dim, DOFINFO > &dinfo, A *assembler)
Definition: loop.h:71

If true, we will assemble the pressure mass matrix in the (1,1) block:

const bool assemble_pressure_mass_matrix =
(solver_type == SolverType::UMFPACK) ? false : true;
FEValues<dim> fe_values(fe,
const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
const unsigned int n_q_points = quadrature_formula.size();
FullMatrix<double> local_matrix(dofs_per_cell, dofs_per_cell);
Vector<double> local_rhs(dofs_per_cell);
std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
const RightHandSide<dim> right_hand_side;
std::vector<Vector<double>> rhs_values(n_q_points, Vector<double>(dim + 1));
const FEValuesExtractors::Vector velocities(0);
const FEValuesExtractors::Scalar pressure(dim);
std::vector<double> div_phi_u(dofs_per_cell);
std::vector<double> phi_p(dofs_per_cell);
for (const auto &cell : dof_handler.active_cell_iterators())
{
fe_values.reinit(cell);
local_matrix = 0;
local_rhs = 0;
rhs_values);
for (unsigned int q = 0; q < n_q_points; ++q)
{
for (unsigned int k = 0; k < dofs_per_cell; ++k)
{
div_phi_u[k] = fe_values[velocities].divergence(k, q);
phi_p[k] = fe_values[pressure].value(k, q);
}
for (unsigned int i = 0; i < dofs_per_cell; ++i)
{
for (unsigned int j = 0; j <= i; ++j)
{
local_matrix(i, j) +=
div_phi_u[i] * phi_p[j] - phi_p[i] * div_phi_u[j] +
(assemble_pressure_mass_matrix ? phi_p[i] * phi_p[j] :
0)) *
fe_values.JxW(q);
}
const unsigned int component_i =
fe.system_to_component_index(i).first;
local_rhs(i) += fe_values.shape_value(i, q) *
rhs_values[q](component_i) * fe_values.JxW(q);
}
}
for (unsigned int i = 0; i < dofs_per_cell; ++i)
for (unsigned int j = i + 1; j < dofs_per_cell; ++j)
local_matrix(i, j) = local_matrix(j, i);
cell->get_dof_indices(local_dof_indices);
constraints.distribute_local_to_global(local_matrix,
local_rhs,
local_dof_indices,
system_matrix,
system_rhs);
}
if (solver_type != SolverType::UMFPACK)
{
pressure_mass_matrix.reinit(sparsity_pattern.block(1, 1));
pressure_mass_matrix.copy_from(system_matrix.block(1, 1));
system_matrix.block(1, 1) = 0;
}
}
@ update_values
Shape function values.
@ update_JxW_values

#### StokesProblem::assemble_multigrid

Here, like in step-16, we have a function that assembles the level and interface matrices necessary for the multigrid preconditioner.

template <int dim>
void StokesProblem<dim>::assemble_multigrid()
{
TimerOutput::Scope multigrid_specific(computing_timer,
"(Multigrid specific)");
TimerOutput::Scope assemble_multigrid(computing_timer,
"Assemble Multigrid");
mg_matrices = 0.;
FEValues<dim> fe_values(velocity_fe,
const unsigned int dofs_per_cell = velocity_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<types::global_dof_index> local_dof_indices(dofs_per_cell);
const FEValuesExtractors::Vector velocities(0);
std::vector<AffineConstraints<double>> boundary_constraints(
triangulation.n_levels());
std::vector<AffineConstraints<double>> boundary_interface_constraints(
triangulation.n_levels());
for (unsigned int level = 0; level < triangulation.n_levels(); ++level)
{
mg_constrained_dofs.get_refinement_edge_indices(level));
mg_constrained_dofs.get_boundary_indices(level));
boundary_constraints[level].close();
IndexSet idx = mg_constrained_dofs.get_refinement_edge_indices(level) &
mg_constrained_dofs.get_boundary_indices(level);
boundary_interface_constraints[level].close();
}
void cell_matrix(FullMatrix< double > &M, const FEValuesBase< dim > &fe, const FEValuesBase< dim > &fetest, const ArrayView< const std::vector< double >> &velocity, const double factor=1.)

This iterator goes over all cells (not just active)

for (const auto &cell : velocity_dof_handler.cell_iterators())
{
fe_values.reinit(cell);
for (unsigned int q = 0; q < n_q_points; ++q)
{
for (unsigned int k = 0; k < dofs_per_cell; ++k)
for (unsigned int i = 0; i < dofs_per_cell; ++i)
for (unsigned int j = 0; j <= i; ++j)
{
cell_matrix(i, j) +=
}
}
for (unsigned int i = 0; i < dofs_per_cell; ++i)
for (unsigned int j = i + 1; j < dofs_per_cell; ++j)
cell_matrix(i, j) = cell_matrix(j, i);
cell->get_mg_dof_indices(local_dof_indices);
boundary_constraints[cell->level()].distribute_local_to_global(
cell_matrix, local_dof_indices, mg_matrices[cell->level()]);
for (unsigned int i = 0; i < dofs_per_cell; ++i)
for (unsigned int j = 0; j < dofs_per_cell; ++j)
if (!mg_constrained_dofs.at_refinement_edge(cell->level(),
local_dof_indices[i]) ||
mg_constrained_dofs.at_refinement_edge(cell->level(),
local_dof_indices[j]))
cell_matrix(i, j) = 0;
boundary_interface_constraints[cell->level()]
.distribute_local_to_global(cell_matrix,
local_dof_indices,
mg_interface_matrices[cell->level()]);
}
}

#### StokesProblem::solve

This function sets up things differently based on if you want to use ILU or GMG as a preconditioner. Both methods share the same solver (FGMRES) but require a different preconditioner to be initialized. Here we time not only the entire solve function, but we separately time the setup of the preconditioner as well as the solve itself.

template <int dim>
void StokesProblem<dim>::solve()
{
TimerOutput::Scope solve(computing_timer, "Solve");
constraints.set_zero(solution);
if (solver_type == SolverType::UMFPACK)
{
computing_timer.enter_subsection("(UMFPACK specific)");
computing_timer.enter_subsection("Solve - Initialize");
A_direct.initialize(system_matrix);
computing_timer.leave_subsection();
computing_timer.leave_subsection();
{
TimerOutput::Scope solve_backslash(computing_timer,
"Solve - Backslash");
A_direct.vmult(solution, system_rhs);
}
constraints.distribute(solution);
return;
}
void initialize(const SparsityPattern &sparsity_pattern)
void vmult(Vector< double > &dst, const Vector< double > &src) const

Here we must make sure to solve for the residual with "good enough" accuracy

SolverControl solver_control(system_matrix.m(),
1e-10 * system_rhs.l2_norm());
unsigned int n_iterations_A;
unsigned int n_iterations_S;

This is used to pass whether or not we want to solve for A inside the preconditioner. One could change this to false to see if there is still convergence and if so does the program then run faster or slower

const bool use_expensive = true;
SolverFGMRES<BlockVector<double>> solver(solver_control);
if (solver_type == SolverType::FGMRES_ILU)
{
computing_timer.enter_subsection("(ILU specific)");
computing_timer.enter_subsection("Solve - Set-up Preconditioner");
std::cout << " Computing preconditioner..." << std::endl
<< std::flush;
SparseILU<double> A_preconditioner;
A_preconditioner.initialize(system_matrix.block(0, 0));
SparseILU<double> S_preconditioner;
S_preconditioner.initialize(pressure_mass_matrix);
const BlockSchurPreconditioner<SparseILU<double>, SparseILU<double>>
preconditioner(system_matrix,
pressure_mass_matrix,
A_preconditioner,
S_preconditioner,
use_expensive);
computing_timer.leave_subsection();
computing_timer.leave_subsection();
{
TimerOutput::Scope solve_fmgres(computing_timer, "Solve - FGMRES");
solver.solve(system_matrix, solution, system_rhs, preconditioner);
n_iterations_A = preconditioner.n_iterations_A;
n_iterations_S = preconditioner.n_iterations_S;
}
}
else
{
computing_timer.enter_subsection("(Multigrid specific)");
computing_timer.enter_subsection("Solve - Set-up Preconditioner");

Transfer operators between levels

MGTransferPrebuilt<Vector<double>> mg_transfer(mg_constrained_dofs);
mg_transfer.build(velocity_dof_handler);

Setup coarse grid solver

FullMatrix<double> coarse_matrix;
coarse_matrix.copy_from(mg_matrices);
coarse_grid_solver.initialize(coarse_matrix);
mg_smoother.initialize(mg_matrices);
mg_smoother.set_steps(2);
void copy_from(const MatrixType &)
void initialize(const FullMatrix< number > &A)
void set_steps(const unsigned int)

Multigrid, when used as a preconditioner for CG, needs to be a symmetric operator, so the smoother must be symmetric

mg_smoother.set_symmetric(true);
mg::Matrix<Vector<double>> mg_matrix(mg_matrices);
mg::Matrix<Vector<double>> mg_interface_up(mg_interface_matrices);
mg::Matrix<Vector<double>> mg_interface_down(mg_interface_matrices);
void set_symmetric(const bool)

Now, we are ready to set up the V-cycle operator and the multilevel preconditioner.

mg_matrix, coarse_grid_solver, mg_transfer, mg_smoother, mg_smoother);
mg.set_edge_matrices(mg_interface_down, mg_interface_up);
A_Multigrid(velocity_dof_handler, mg, mg_transfer);
SparseILU<double> S_preconditioner;
S_preconditioner.initialize(pressure_mass_matrix,
const BlockSchurPreconditioner<
preconditioner(system_matrix,
pressure_mass_matrix,
A_Multigrid,
S_preconditioner,
use_expensive);
computing_timer.leave_subsection();
computing_timer.leave_subsection();
{
TimerOutput::Scope solve_fmgres(computing_timer, "Solve - FGMRES");
solver.solve(system_matrix, solution, system_rhs, preconditioner);
n_iterations_A = preconditioner.n_iterations_A;
n_iterations_S = preconditioner.n_iterations_S;
}
}
constraints.distribute(solution);
std::cout
<< std::endl
<< "\tNumber of FGMRES iterations: " << solver_control.last_step()
<< std::endl
<< "\tTotal number of iterations used for approximation of A inverse: "
<< n_iterations_A << std::endl
<< "\tTotal number of iterations used for approximation of S inverse: "
<< n_iterations_S << std::endl
<< std::endl;
}
Definition: sparse_ilu.h:79
Definition: mg.h:82

#### StokesProblem::process_solution

This function computes the L2 and H1 errors of the solution. For this, we need to make sure the pressure has mean zero.

template <int dim>
void StokesProblem<dim>::compute_errors()
{

Compute the mean pressure $$\frac{1}{\Omega} \int_{\Omega} p(x) dx$$ and then subtract it from each pressure coefficient. This will result in a pressure with mean value zero. Here we make use of the fact that the pressure is component $$dim$$ and that the finite element space is nodal.

const double mean_pressure = VectorTools::compute_mean_value(
dof_handler, QGauss<dim>(pressure_degree + 2), solution, dim);
std::cout << " Note: The mean value was adjusted by " << -mean_pressure
<< std::endl;
const ComponentSelectFunction<dim> pressure_mask(dim, dim + 1);
dim + 1);
Vector<float> difference_per_cell(triangulation.n_active_cells());
solution,
Solution<dim>(),
difference_per_cell,
QGauss<dim>(pressure_degree + 2),
const double Velocity_L2_error =
difference_per_cell,
solution,
Solution<dim>(),
difference_per_cell,
QGauss<dim>(pressure_degree + 2),
const double Pressure_L2_error =
difference_per_cell,
solution,
Solution<dim>(),
difference_per_cell,
QGauss<dim>(pressure_degree + 2),
const double Velocity_H1_error =
difference_per_cell,
std::cout << std::endl
<< " Velocity L2 Error: " << Velocity_L2_error << std::endl
<< " Pressure L2 Error: " << Pressure_L2_error << std::endl
<< " Velocity H1 Error: " << Velocity_H1_error << std::endl;
}
VectorType::value_type compute_mean_value(const hp::MappingCollection< dim, spacedim > &mapping_collection, const DoFHandler< dim, spacedim > &dof, const hp::QCollection< dim > &q_collection, const VectorType &v, const unsigned int component)
double compute_global_error(const Triangulation< dim, spacedim > &tria, const InVector &cellwise_error, const NormType &norm, const double exponent=2.)
void integrate_difference(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const InVector &fe_function, const Function< spacedim, typename InVector::value_type > &exact_solution, OutVector &difference, const Quadrature< dim > &q, const NormType &norm, const Function< spacedim, double > *weight=nullptr, const double exponent=2.)

#### StokesProblem::output_results

This function generates graphical output like it is done in step-22.

template <int dim>
void
StokesProblem<dim>::output_results(const unsigned int refinement_cycle) const
{
std::vector<std::string> solution_names(dim, "velocity");
solution_names.emplace_back("pressure");
std::vector<DataComponentInterpretation::DataComponentInterpretation>
data_component_interpretation(
data_component_interpretation.push_back(
DataOut<dim> data_out;
data_out.attach_dof_handler(dof_handler);
solution_names,
data_component_interpretation);
data_out.build_patches();
std::ofstream output(
"solution-" + Utilities::int_to_string(refinement_cycle, 2) + ".vtk");
data_out.write_vtk(output);
}
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
void add_data_vector(const VectorType &data, const std::vector< std::string > &names, const DataVectorType type=type_automatic, const std::vector< DataComponentInterpretation::DataComponentInterpretation > &data_component_interpretation={})
virtual void build_patches(const unsigned int n_subdivisions=0)
Definition: data_out.cc:1064
void write_vtk(std::ostream &out) const
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
Definition: utilities.cc:473

#### StokesProblem::run

The last step in the Stokes class is, as usual, the function that generates the initial grid and calls the other functions in the respective order.

template <int dim>
{
triangulation.refine_global(6 - dim);
if (solver_type == SolverType::FGMRES_ILU)
std::cout << "Now running with ILU" << std::endl;
else if (solver_type == SolverType::FGMRES_GMG)
std::cout << "Now running with Multigrid" << std::endl;
else
std::cout << "Now running with UMFPACK" << std::endl;
for (unsigned int refinement_cycle = 0; refinement_cycle < 3;
++refinement_cycle)
{
std::cout << "Refinement cycle " << refinement_cycle << std::endl;
if (refinement_cycle > 0)
triangulation.refine_global(1);
std::cout << " Set-up..." << std::endl;
setup_dofs();
std::cout << " Assembling..." << std::endl;
assemble_system();
if (solver_type == SolverType::FGMRES_GMG)
{
std::cout << " Assembling Multigrid..." << std::endl;
assemble_multigrid();
}
std::cout << " Solving..." << std::flush;
solve();
compute_errors();
output_results(refinement_cycle);
std::cout << " VM Peak: " << mem.VmPeak << std::endl;
computing_timer.print_summary();
computing_timer.reset();
}
}
} // namespace Step56
void hyper_cube(Triangulation< dim, spacedim > &tria, const double left=0., const double right=1., const bool colorize=false)
void get_memory_stats(MemoryStats &stats)
Definition: utilities.cc:967
unsigned long int VmPeak
Definition: utilities.h:867

### The main function

int main()
{
try
{
using namespace Step56;
const int degree = 1;
const int dim = 3;

options for SolverType: UMFPACK FGMRES_ILU FGMRES_GMG

StokesProblem<dim> flow_problem(degree, SolverType::FGMRES_GMG);
flow_problem.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;
}

# Results

### Errors

We first run the code and confirm that the finite element solution converges with the correct rates as predicted by the error analysis of mixed finite element problems. Given sufficiently smooth exact solutions $$u$$ and $$p$$, the errors of the Taylor-Hood element $$Q_k \times Q_{k-1}$$ should be

$\| u -u_h \|_0 + h ( \| u- u_h\|_1 + \|p - p_h \|_0) \leq C h^{k+1} ( \|u \|_{k+1} + \| p \|_k )$

see for example Ern/Guermond "Theory and Practice of Finite Elements", Section 4.2.5 p195. This is indeed what we observe, using the $$Q_2 \times Q_1$$ element as an example (this is what is done in the code, but is easily changed in main()):

L2 Velocity Reduction L2 Pressure Reduction H1 Velocity Reduction
3D, 3 global refinements 0.000670888 - 0.0036533 - 0.0414704 -
3D, 4 global refinements 8.38E-005 8.0 0.00088494 4.1 0.0103781 4.0
3D, 5 global refinements 1.05E-005 8.0 0.000220253 4.0 0.00259519 4.0

### Timing Results

Let us compare the direct solver approach using UMFPACK to the two methods in which we choose $$\widetilde {A^{-1}}=A^{-1}$$ and $$\widetilde{S^{-1}}=S^{-1}$$ by solving linear systems with $$A,S$$ using CG. The preconditioner for CG is then either ILU or GMG. The following table summarizes solver iterations, timings, and virtual memory (VM) peak usage:

General GMG ILU UMFPACK
Timings Timings Iterations Timings Iterations Timings
Cycle DoFs Setup Assembly Setup Solve Outer Inner (A) Inner (S) VM Peak Setup Solve Outer Inner (A) Inner (S) VM Peak Setup Solve VM Peak
0 15468 0.1s 0.3s 0.3s 1.3s 21 67 22 4805 0.3s 0.6s 21 180 22 4783 2.65s 2.8s 5054
1 112724 1.0s 2.4s 2.6s 14s 21 67 22 5441 2.8s 15.8s 21 320 22 5125 236s 237s 11288
2 859812 9.0s 20s 20s 101s 20 65 21 10641 27s 268s 21 592 22 8307 - - -

As can be seen from the table:

1. UMFPACK uses large amounts of memory, especially in 3d. Also, UMFPACK timings do not scale favorably with problem size.
2. Because we are using inner solvers for $$A$$ and $$S$$, ILU and GMG require the same number of outer iterations.
3. The number of (inner) iterations for $$A$$ increases for ILU with refinement, leading to worse than linear scaling in solve time. In contrast, the number of inner iterations for $$A$$ stays constant with GMG leading to nearly perfect scaling in solve time.
4. GMG needs slightly more memory than ILU to store the level and interface matrices.

### Possibilities for extensions

#### Check higher order discretizations

Experiment with higher order stable FE pairs and check that you observe the correct convergence rates.

#### Compare with cheap preconditioner

The introduction also outlined another option to precondition the overall system, namely one in which we do not choose $$\widetilde {A^{-1}}=A^{-1}$$ as in the table above, but in which $$\widetilde{A^{-1}}$$ is only a single preconditioner application with GMG or ILU, respectively.

This is in fact implemented in the code: Currently, the boolean use_expensive in solve() is set to true. The option mentioned above is obtained by setting it to false.

What you will find is that the number of FGMRES iterations stays constant under refinement if you use GMG this way. This means that the Multigrid is optimal and independent of $$h$$.

# The plain program

/* ---------------------------------------------------------------------
*
* Copyright (C) 2016 - 2022 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: Ryan Grove, Clemson University
* Timo Heister, Clemson University
*/
#include <iostream>
#include <fstream>
namespace Step56
{
using namespace dealii;
enum class SolverType
{
FGMRES_ILU,
FGMRES_GMG,
UMFPACK
};
template <int dim>
class Solution : public Function<dim>
{
public:
Solution()
: Function<dim>(dim + 1)
{}
virtual double value(const Point<dim> & p,
const unsigned int component = 0) const override;
const unsigned int component = 0) const override;
};
template <>
double Solution<2>::value(const Point<2> & p,
const unsigned int component) const
{
Assert(component <= 2 + 1, ExcIndexRange(component, 0, 2 + 1));
using numbers::PI;
const double x = p(0);
const double y = p(1);
if (component == 0)
return sin(PI * x);
if (component == 1)
return -PI * y * cos(PI * x);
if (component == 2)
return sin(PI * x) * cos(PI * y);
return 0;
}
template <>
double Solution<3>::value(const Point<3> & p,
const unsigned int component) const
{
Assert(component <= 3 + 1, ExcIndexRange(component, 0, 3 + 1));
using numbers::PI;
const double x = p(0);
const double y = p(1);
const double z = p(2);
if (component == 0)
return 2.0 * sin(PI * x);
if (component == 1)
return -PI * y * cos(PI * x);
if (component == 2)
return -PI * z * cos(PI * x);
if (component == 3)
return sin(PI * x) * cos(PI * y) * sin(PI * z);
return 0;
}
template <>
Tensor<1, 2> Solution<2>::gradient(const Point<2> & p,
const unsigned int component) const
{
Assert(component <= 2, ExcIndexRange(component, 0, 2 + 1));
using numbers::PI;
const double x = p(0);
const double y = p(1);
Tensor<1, 2> return_value;
if (component == 0)
{
return_value = PI * cos(PI * x);
return_value = 0.0;
}
else if (component == 1)
{
return_value = y * PI * PI * sin(PI * x);
return_value = -PI * cos(PI * x);
}
else if (component == 2)
{
return_value = PI * cos(PI * x) * cos(PI * y);
return_value = -PI * sin(PI * x) * sin(PI * y);
}
return return_value;
}
template <>
Tensor<1, 3> Solution<3>::gradient(const Point<3> & p,
const unsigned int component) const
{
Assert(component <= 3, ExcIndexRange(component, 0, 3 + 1));
using numbers::PI;
const double x = p(0);
const double y = p(1);
const double z = p(2);
Tensor<1, 3> return_value;
if (component == 0)
{
return_value = 2 * PI * cos(PI * x);
return_value = 0.0;
return_value = 0.0;
}
else if (component == 1)
{
return_value = y * PI * PI * sin(PI * x);
return_value = -PI * cos(PI * x);
return_value = 0.0;
}
else if (component == 2)
{
return_value = z * PI * PI * sin(PI * x);
return_value = 0.0;
return_value = -PI * cos(PI * x);
}
else if (component == 3)
{
return_value = PI * cos(PI * x) * cos(PI * y) * sin(PI * z);
return_value = -PI * sin(PI * x) * sin(PI * y) * sin(PI * z);
return_value = PI * sin(PI * x) * cos(PI * y) * cos(PI * z);
}
return return_value;
}
template <int dim>
class RightHandSide : public Function<dim>
{
public:
RightHandSide()
: Function<dim>(dim + 1)
{}
virtual double value(const Point<dim> & p,
const unsigned int component = 0) const override;
};
template <>
double RightHandSide<2>::value(const Point<2> & p,
const unsigned int component) const
{
Assert(component <= 2, ExcIndexRange(component, 0, 2 + 1));
using numbers::PI;
double x = p(0);
double y = p(1);
if (component == 0)
return PI * PI * sin(PI * x) + PI * cos(PI * x) * cos(PI * y);
if (component == 1)
return -PI * PI * PI * y * cos(PI * x) - PI * sin(PI * y) * sin(PI * x);
if (component == 2)
return 0;
return 0;
}
template <>
double RightHandSide<3>::value(const Point<3> & p,
const unsigned int component) const
{
Assert(component <= 3, ExcIndexRange(component, 0, 3 + 1));
using numbers::PI;
double x = p(0);
double y = p(1);
double z = p(2);
if (component == 0)
return 2 * PI * PI * sin(PI * x) +
PI * cos(PI * x) * cos(PI * y) * sin(PI * z);
if (component == 1)
return -PI * PI * PI * y * cos(PI * x) +
PI * (-1) * sin(PI * y) * sin(PI * x) * sin(PI * z);
if (component == 2)
return -PI * PI * PI * z * cos(PI * x) +
PI * cos(PI * z) * sin(PI * x) * cos(PI * y);
if (component == 3)
return 0;
return 0;
}
template <class PreconditionerAType, class PreconditionerSType>
class BlockSchurPreconditioner : public Subscriptor
{
public:
BlockSchurPreconditioner(
const BlockSparseMatrix<double> &system_matrix,
const SparseMatrix<double> & schur_complement_matrix,
const PreconditionerAType & preconditioner_A,
const PreconditionerSType & preconditioner_S,
const bool do_solve_A);
void vmult(BlockVector<double> &dst, const BlockVector<double> &src) const;
mutable unsigned int n_iterations_A;
mutable unsigned int n_iterations_S;
private:
const BlockSparseMatrix<double> &system_matrix;
const SparseMatrix<double> & schur_complement_matrix;
const PreconditionerAType & preconditioner_A;
const PreconditionerSType & preconditioner_S;
const bool do_solve_A;
};
template <class PreconditionerAType, class PreconditionerSType>
BlockSchurPreconditioner<PreconditionerAType, PreconditionerSType>::
BlockSchurPreconditioner(
const BlockSparseMatrix<double> &system_matrix,
const SparseMatrix<double> & schur_complement_matrix,
const PreconditionerAType & preconditioner_A,
const PreconditionerSType & preconditioner_S,
const bool do_solve_A)
: n_iterations_A(0)
, n_iterations_S(0)
, system_matrix(system_matrix)
, schur_complement_matrix(schur_complement_matrix)
, preconditioner_A(preconditioner_A)
, preconditioner_S(preconditioner_S)
, do_solve_A(do_solve_A)
{}
template <class PreconditionerAType, class PreconditionerSType>
void
BlockSchurPreconditioner<PreconditionerAType, PreconditionerSType>::vmult(
const BlockVector<double> &src) const
{
Vector<double> utmp(src.block(0));
{
SolverControl solver_control(1000, 1e-6 * src.block(1).l2_norm());
SolverCG<Vector<double>> cg(solver_control);
dst.block(1) = 0.0;
cg.solve(schur_complement_matrix,
dst.block(1),
src.block(1),
preconditioner_S);
n_iterations_S += solver_control.last_step();
dst.block(1) *= -1.0;
}
{
system_matrix.block(0, 1).vmult(utmp, dst.block(1));
utmp *= -1.0;
utmp += src.block(0);
}
if (do_solve_A == true)
{
SolverControl solver_control(10000, utmp.l2_norm() * 1e-4);
SolverCG<Vector<double>> cg(solver_control);
dst.block(0) = 0.0;
cg.solve(system_matrix.block(0, 0),
dst.block(0),
utmp,
preconditioner_A);
n_iterations_A += solver_control.last_step();
}
else
{
preconditioner_A.vmult(dst.block(0), utmp);
n_iterations_A += 1;
}
}
template <int dim>
class StokesProblem
{
public:
StokesProblem(const unsigned int pressure_degree,
const SolverType solver_type);
void run();
private:
void setup_dofs();
void assemble_system();
void assemble_multigrid();
void solve();
void compute_errors();
void output_results(const unsigned int refinement_cycle) const;
const unsigned int pressure_degree;
const SolverType solver_type;
FESystem<dim> velocity_fe;
DoFHandler<dim> dof_handler;
DoFHandler<dim> velocity_dof_handler;
BlockSparsityPattern sparsity_pattern;
SparseMatrix<double> pressure_mass_matrix;
BlockVector<double> system_rhs;
MGLevelObject<SparsityPattern> mg_sparsity_patterns;
MGLevelObject<SparseMatrix<double>> mg_interface_matrices;
MGConstrainedDoFs mg_constrained_dofs;
TimerOutput computing_timer;
};
template <int dim>
StokesProblem<dim>::StokesProblem(const unsigned int pressure_degree,
const SolverType solver_type)
: pressure_degree(pressure_degree)
, solver_type(solver_type)
, triangulation(Triangulation<dim>::maximum_smoothing)
,
velocity_fe(FE_Q<dim>(pressure_degree + 1), dim)
,
fe(velocity_fe, 1, FE_Q<dim>(pressure_degree), 1)
, dof_handler(triangulation)
, velocity_dof_handler(triangulation)
, computing_timer(std::cout, TimerOutput::never, TimerOutput::wall_times)
{}
template <int dim>
void StokesProblem<dim>::setup_dofs()
{
TimerOutput::Scope scope(computing_timer, "Setup");
system_matrix.clear();
pressure_mass_matrix.clear();
dof_handler.distribute_dofs(fe);
std::vector<unsigned int> block_component(2);
block_component = 0;
block_component = 1;
const FEValuesExtractors::Vector velocities(0);
if (solver_type == SolverType::FGMRES_ILU)
{
TimerOutput::Scope ilu_specific(computing_timer, "(ILU specific)");
}
if (solver_type == SolverType::FGMRES_GMG)
{
TimerOutput::Scope multigrid_specific(computing_timer,
"(Multigrid specific)");
TimerOutput::Scope setup_multigrid(computing_timer,
"Setup - Multigrid");
velocity_dof_handler.distribute_dofs(velocity_fe);
velocity_dof_handler.distribute_mg_dofs();
const std::set<types::boundary_id> zero_boundary_ids = {0};
mg_constrained_dofs.clear();
mg_constrained_dofs.initialize(velocity_dof_handler);
mg_constrained_dofs.make_zero_boundary_constraints(velocity_dof_handler,
zero_boundary_ids);
const unsigned int n_levels = triangulation.n_levels();
mg_interface_matrices.resize(0, n_levels - 1);
mg_matrices.resize(0, n_levels - 1);
mg_sparsity_patterns.resize(0, n_levels - 1);
for (unsigned int level = 0; level < n_levels; ++level)
{
DynamicSparsityPattern csp(velocity_dof_handler.n_dofs(level),
velocity_dof_handler.n_dofs(level));
MGTools::make_sparsity_pattern(velocity_dof_handler, csp, level);
mg_sparsity_patterns[level].copy_from(csp);
mg_matrices[level].reinit(mg_sparsity_patterns[level]);
mg_interface_matrices[level].reinit(mg_sparsity_patterns[level]);
}
}
const std::vector<types::global_dof_index> dofs_per_block =
DoFTools::count_dofs_per_fe_block(dof_handler, block_component);
const unsigned int n_u = dofs_per_block;
const unsigned int n_p = dofs_per_block;
{
constraints.clear();
DoFTools::make_hanging_node_constraints(dof_handler, constraints);
0,
Solution<dim>(),
constraints,
if (solver_type == SolverType::UMFPACK)
constraints.close();
}
std::cout << "\tNumber of active cells: " << triangulation.n_active_cells()
<< std::endl
<< "\tNumber of degrees of freedom: " << dof_handler.n_dofs()
<< " (" << n_u << '+' << n_p << ')' << std::endl;
{
BlockDynamicSparsityPattern csp(dofs_per_block, dofs_per_block);
DoFTools::make_sparsity_pattern(dof_handler, csp, constraints, false);
sparsity_pattern.copy_from(csp);
}
system_matrix.reinit(sparsity_pattern);
solution.reinit(dofs_per_block);
system_rhs.reinit(dofs_per_block);
}
template <int dim>
void StokesProblem<dim>::assemble_system()
{
TimerOutput::Scope assemble(computing_timer, "Assemble");
system_matrix = 0;
system_rhs = 0;
const bool assemble_pressure_mass_matrix =
(solver_type == SolverType::UMFPACK) ? false : true;
FEValues<dim> fe_values(fe,
const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
const unsigned int n_q_points = quadrature_formula.size();
FullMatrix<double> local_matrix(dofs_per_cell, dofs_per_cell);
Vector<double> local_rhs(dofs_per_cell);
std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
const RightHandSide<dim> right_hand_side;
std::vector<Vector<double>> rhs_values(n_q_points, Vector<double>(dim + 1));
const FEValuesExtractors::Vector velocities(0);
const FEValuesExtractors::Scalar pressure(dim);
std::vector<double> div_phi_u(dofs_per_cell);
std::vector<double> phi_p(dofs_per_cell);
for (const auto &cell : dof_handler.active_cell_iterators())
{
fe_values.reinit(cell);
local_matrix = 0;
local_rhs = 0;
rhs_values);
for (unsigned int q = 0; q < n_q_points; ++q)
{
for (unsigned int k = 0; k < dofs_per_cell; ++k)
{
div_phi_u[k] = fe_values[velocities].divergence(k, q);
phi_p[k] = fe_values[pressure].value(k, q);
}
for (unsigned int i = 0; i < dofs_per_cell; ++i)
{
for (unsigned int j = 0; j <= i; ++j)
{
local_matrix(i, j) +=
div_phi_u[i] * phi_p[j] - phi_p[i] * div_phi_u[j] +
(assemble_pressure_mass_matrix ? phi_p[i] * phi_p[j] :
0)) *
fe_values.JxW(q);
}
const unsigned int component_i =
fe.system_to_component_index(i).first;
local_rhs(i) += fe_values.shape_value(i, q) *
rhs_values[q](component_i) * fe_values.JxW(q);
}
}
for (unsigned int i = 0; i < dofs_per_cell; ++i)
for (unsigned int j = i + 1; j < dofs_per_cell; ++j)
local_matrix(i, j) = local_matrix(j, i);
cell->get_dof_indices(local_dof_indices);
constraints.distribute_local_to_global(local_matrix,
local_rhs,
local_dof_indices,
system_matrix,
system_rhs);
}
if (solver_type != SolverType::UMFPACK)
{
pressure_mass_matrix.reinit(sparsity_pattern.block(1, 1));
pressure_mass_matrix.copy_from(system_matrix.block(1, 1));
system_matrix.block(1, 1) = 0;
}
}
template <int dim>
void StokesProblem<dim>::assemble_multigrid()
{
TimerOutput::Scope multigrid_specific(computing_timer,
"(Multigrid specific)");
TimerOutput::Scope assemble_multigrid(computing_timer,
"Assemble Multigrid");
mg_matrices = 0.;
FEValues<dim> fe_values(velocity_fe,
const unsigned int dofs_per_cell = velocity_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<types::global_dof_index> local_dof_indices(dofs_per_cell);
const FEValuesExtractors::Vector velocities(0);
std::vector<AffineConstraints<double>> boundary_constraints(
triangulation.n_levels());
std::vector<AffineConstraints<double>> boundary_interface_constraints(
triangulation.n_levels());
for (unsigned int level = 0; level < triangulation.n_levels(); ++level)
{
mg_constrained_dofs.get_refinement_edge_indices(level));
mg_constrained_dofs.get_boundary_indices(level));
boundary_constraints[level].close();
IndexSet idx = mg_constrained_dofs.get_refinement_edge_indices(level) &
mg_constrained_dofs.get_boundary_indices(level);
boundary_interface_constraints[level].close();
}
for (const auto &cell : velocity_dof_handler.cell_iterators())
{
fe_values.reinit(cell);
for (unsigned int q = 0; q < n_q_points; ++q)
{
for (unsigned int k = 0; k < dofs_per_cell; ++k)
for (unsigned int i = 0; i < dofs_per_cell; ++i)
for (unsigned int j = 0; j <= i; ++j)
{
cell_matrix(i, j) +=
}
}
for (unsigned int i = 0; i < dofs_per_cell; ++i)
for (unsigned int j = i + 1; j < dofs_per_cell; ++j)
cell_matrix(i, j) = cell_matrix(j, i);
cell->get_mg_dof_indices(local_dof_indices);
boundary_constraints[cell->level()].distribute_local_to_global(
cell_matrix, local_dof_indices, mg_matrices[cell->level()]);
for (unsigned int i = 0; i < dofs_per_cell; ++i)
for (unsigned int j = 0; j < dofs_per_cell; ++j)
if (!mg_constrained_dofs.at_refinement_edge(cell->level(),
local_dof_indices[i]) ||
mg_constrained_dofs.at_refinement_edge(cell->level(),
local_dof_indices[j]))
cell_matrix(i, j) = 0;
boundary_interface_constraints[cell->level()]
.distribute_local_to_global(cell_matrix,
local_dof_indices,
mg_interface_matrices[cell->level()]);
}
}
template <int dim>
void StokesProblem<dim>::solve()
{
TimerOutput::Scope solve(computing_timer, "Solve");
constraints.set_zero(solution);
if (solver_type == SolverType::UMFPACK)
{
computing_timer.enter_subsection("(UMFPACK specific)");
computing_timer.enter_subsection("Solve - Initialize");
A_direct.initialize(system_matrix);
computing_timer.leave_subsection();
computing_timer.leave_subsection();
{
TimerOutput::Scope solve_backslash(computing_timer,
"Solve - Backslash");
A_direct.vmult(solution, system_rhs);
}
constraints.distribute(solution);
return;
}
SolverControl solver_control(system_matrix.m(),
1e-10 * system_rhs.l2_norm());
unsigned int n_iterations_A;
unsigned int n_iterations_S;
const bool use_expensive = true;
SolverFGMRES<BlockVector<double>> solver(solver_control);
if (solver_type == SolverType::FGMRES_ILU)
{
computing_timer.enter_subsection("(ILU specific)");
computing_timer.enter_subsection("Solve - Set-up Preconditioner");
std::cout << " Computing preconditioner..." << std::endl
<< std::flush;
SparseILU<double> A_preconditioner;
A_preconditioner.initialize(system_matrix.block(0, 0));
SparseILU<double> S_preconditioner;
S_preconditioner.initialize(pressure_mass_matrix);
const BlockSchurPreconditioner<SparseILU<double>, SparseILU<double>>
preconditioner(system_matrix,
pressure_mass_matrix,
A_preconditioner,
S_preconditioner,
use_expensive);
computing_timer.leave_subsection();
computing_timer.leave_subsection();
{
TimerOutput::Scope solve_fmgres(computing_timer, "Solve - FGMRES");
solver.solve(system_matrix, solution, system_rhs, preconditioner);
n_iterations_A = preconditioner.n_iterations_A;
n_iterations_S = preconditioner.n_iterations_S;
}
}
else
{
computing_timer.enter_subsection("(Multigrid specific)");
computing_timer.enter_subsection("Solve - Set-up Preconditioner");
MGTransferPrebuilt<Vector<double>> mg_transfer(mg_constrained_dofs);
mg_transfer.build(velocity_dof_handler);
FullMatrix<double> coarse_matrix;
coarse_matrix.copy_from(mg_matrices);
coarse_grid_solver.initialize(coarse_matrix);
mg_smoother.initialize(mg_matrices);
mg_smoother.set_steps(2);
mg_smoother.set_symmetric(true);
mg::Matrix<Vector<double>> mg_matrix(mg_matrices);
mg::Matrix<Vector<double>> mg_interface_up(mg_interface_matrices);
mg::Matrix<Vector<double>> mg_interface_down(mg_interface_matrices);
mg_matrix, coarse_grid_solver, mg_transfer, mg_smoother, mg_smoother);
mg.set_edge_matrices(mg_interface_down, mg_interface_up);
A_Multigrid(velocity_dof_handler, mg, mg_transfer);
SparseILU<double> S_preconditioner;
S_preconditioner.initialize(pressure_mass_matrix,
const BlockSchurPreconditioner<
preconditioner(system_matrix,
pressure_mass_matrix,
A_Multigrid,
S_preconditioner,
use_expensive);
computing_timer.leave_subsection();
computing_timer.leave_subsection();
{
TimerOutput::Scope solve_fmgres(computing_timer, "Solve - FGMRES");
solver.solve(system_matrix, solution, system_rhs, preconditioner);
n_iterations_A = preconditioner.n_iterations_A;
n_iterations_S = preconditioner.n_iterations_S;
}
}
constraints.distribute(solution);
std::cout
<< std::endl
<< "\tNumber of FGMRES iterations: " << solver_control.last_step()
<< std::endl
<< "\tTotal number of iterations used for approximation of A inverse: "
<< n_iterations_A << std::endl
<< "\tTotal number of iterations used for approximation of S inverse: "
<< n_iterations_S << std::endl
<< std::endl;
}
template <int dim>
void StokesProblem<dim>::compute_errors()
{
const double mean_pressure = VectorTools::compute_mean_value(
dof_handler, QGauss<dim>(pressure_degree + 2), solution, dim);
std::cout << " Note: The mean value was adjusted by " << -mean_pressure
<< std::endl;
const ComponentSelectFunction<dim> pressure_mask(dim, dim + 1);
dim + 1);
Vector<float> difference_per_cell(triangulation.n_active_cells());
solution,
Solution<dim>(),
difference_per_cell,
QGauss<dim>(pressure_degree + 2),
const double Velocity_L2_error =
difference_per_cell,
solution,
Solution<dim>(),
difference_per_cell,
QGauss<dim>(pressure_degree + 2),
const double Pressure_L2_error =
difference_per_cell,
solution,
Solution<dim>(),
difference_per_cell,
QGauss<dim>(pressure_degree + 2),
const double Velocity_H1_error =
difference_per_cell,
std::cout << std::endl
<< " Velocity L2 Error: " << Velocity_L2_error << std::endl
<< " Pressure L2 Error: " << Pressure_L2_error << std::endl
<< " Velocity H1 Error: " << Velocity_H1_error << std::endl;
}
template <int dim>
void
StokesProblem<dim>::output_results(const unsigned int refinement_cycle) const
{
std::vector<std::string> solution_names(dim, "velocity");
solution_names.emplace_back("pressure");
std::vector<DataComponentInterpretation::DataComponentInterpretation>
data_component_interpretation(
data_component_interpretation.push_back(
DataOut<dim> data_out;
data_out.attach_dof_handler(dof_handler);
solution_names,
data_component_interpretation);
data_out.build_patches();
std::ofstream output(
"solution-" + Utilities::int_to_string(refinement_cycle, 2) + ".vtk");
data_out.write_vtk(output);
}
template <int dim>
{
triangulation.refine_global(6 - dim);
if (solver_type == SolverType::FGMRES_ILU)
std::cout << "Now running with ILU" << std::endl;
else if (solver_type == SolverType::FGMRES_GMG)
std::cout << "Now running with Multigrid" << std::endl;
else
std::cout << "Now running with UMFPACK" << std::endl;
for (unsigned int refinement_cycle = 0; refinement_cycle < 3;
++refinement_cycle)
{
std::cout << "Refinement cycle " << refinement_cycle << std::endl;
if (refinement_cycle > 0)
triangulation.refine_global(1);
std::cout << " Set-up..." << std::endl;
setup_dofs();
std::cout << " Assembling..." << std::endl;
assemble_system();
if (solver_type == SolverType::FGMRES_GMG)
{
std::cout << " Assembling Multigrid..." << std::endl;
assemble_multigrid();
}
std::cout << " Solving..." << std::flush;
solve();
compute_errors();
output_results(refinement_cycle);
std::cout << " VM Peak: " << mem.VmPeak << std::endl;
computing_timer.print_summary();
computing_timer.reset();
}
}
} // namespace Step56
int main()
{
try
{
using namespace Step56;
const int degree = 1;
const int dim = 3;
StokesProblem<dim> flow_problem(degree, SolverType::FGMRES_GMG);
flow_problem.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;
}
::VectorizedArray< Number, width > cos(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)