1327 *
const unsigned int )
const
1333 *
template <
int dim>
1334 *
void RightHandSide<dim>::vector_value(
const Point<dim> &p,
1337 *
for (
unsigned int c = 0; c < this->n_components; ++c)
1338 *
values(c) = RightHandSide<dim>::value(p, c);
1345 * <a name=
"Linearsolversandpreconditioners"></a>
1346 * <h3>Linear solvers and preconditioners</h3>
1350 * The linear solvers and preconditioners are discussed extensively in the
1351 * introduction. Here, we create the respective objects that will be used.
1356 * <a name=
"ThecodeInverseMatrixcodeclasstemplate"></a>
1357 * <h4>The <code>InverseMatrix</code>
class template</h4>
1358 * The <code>InverseMatrix</code>
class represents the data structure for an
1359 * inverse
matrix. Unlike @ref step_20
"step-20", we implement
this with a
class instead of
1360 * the helper function inverse_linear_operator() we will apply this class to
1361 * different kinds of matrices that will require different preconditioners
1362 * (in @ref step_20 "step-20" we only used a non-
identity preconditioner for the mass
1364 * via template parameters, and
matrix and preconditioner objects of these
1365 *
types will then be passed to the constructor when an
1366 * <code>InverseMatrix</code>
object is created. The member function
1367 * <code>vmult</code> is obtained by solving a linear system:
1370 * template <class MatrixType, class PreconditionerType>
1374 * InverseMatrix(
const MatrixType & m,
1375 *
const PreconditionerType &preconditioner);
1385 *
template <
class MatrixType,
class PreconditionerType>
1386 * InverseMatrix<MatrixType, PreconditionerType>::InverseMatrix(
1387 *
const MatrixType & m,
1388 *
const PreconditionerType &preconditioner)
1390 * , preconditioner(&preconditioner)
1396 * This is the implementation of the <code>vmult</code> function.
1400 * In
this class we use a rather large tolerance for the solver control. The
1401 * reason
for this is that the function is used very frequently, and hence,
1402 * any additional effort to make the residual in the CG solve smaller makes
1403 * the solution more expensive. Note that we
do not only use
this class as a
1404 * preconditioner
for the Schur complement, but also when forming the
1405 * inverse of the Laplace
matrix – which is hence directly responsible
1406 *
for the accuracy of the solution itself, so we can
't choose a too large
1407 * tolerance, either.
1410 * template <class MatrixType, class PreconditionerType>
1411 * void InverseMatrix<MatrixType, PreconditionerType>::vmult(
1412 * Vector<double> & dst,
1413 * const Vector<double> &src) const
1415 * SolverControl solver_control(src.size(), 1e-6 * src.l2_norm());
1416 * SolverCG<Vector<double>> cg(solver_control);
1420 * cg.solve(*matrix, dst, src, *preconditioner);
1427 * <a name="ThecodeSchurComplementcodeclasstemplate"></a>
1428 * <h4>The <code>SchurComplement</code> class template</h4>
1432 * This class implements the Schur complement discussed in the introduction.
1433 * It is in analogy to @ref step_20 "step-20". Though, we now call it with a template
1434 * parameter <code>PreconditionerType</code> in order to access that when
1435 * specifying the respective type of the inverse matrix class. As a
1436 * consequence of the definition above, the declaration
1437 * <code>InverseMatrix</code> now contains the second template parameter for
1438 * a preconditioner class as above, which affects the
1439 * <code>SmartPointer</code> object <code>m_inverse</code> as well.
1442 * template <class PreconditionerType>
1443 * class SchurComplement : public Subscriptor
1447 * const BlockSparseMatrix<double> &system_matrix,
1448 * const InverseMatrix<SparseMatrix<double>, PreconditionerType> &A_inverse);
1450 * void vmult(Vector<double> &dst, const Vector<double> &src) const;
1453 * const SmartPointer<const BlockSparseMatrix<double>> system_matrix;
1454 * const SmartPointer<
1455 * const InverseMatrix<SparseMatrix<double>, PreconditionerType>>
1458 * mutable Vector<double> tmp1, tmp2;
1463 * template <class PreconditionerType>
1464 * SchurComplement<PreconditionerType>::SchurComplement(
1465 * const BlockSparseMatrix<double> &system_matrix,
1466 * const InverseMatrix<SparseMatrix<double>, PreconditionerType> &A_inverse)
1467 * : system_matrix(&system_matrix)
1468 * , A_inverse(&A_inverse)
1469 * , tmp1(system_matrix.block(0, 0).m())
1470 * , tmp2(system_matrix.block(0, 0).m())
1474 * template <class PreconditionerType>
1476 * SchurComplement<PreconditionerType>::vmult(Vector<double> & dst,
1477 * const Vector<double> &src) const
1479 * system_matrix->block(0, 1).vmult(tmp1, src);
1480 * A_inverse->vmult(tmp2, tmp1);
1481 * system_matrix->block(1, 0).vmult(dst, tmp2);
1488 * <a name="StokesProblemclassimplementation"></a>
1489 * <h3>StokesProblem class implementation</h3>
1494 * <a name="StokesProblemStokesProblem"></a>
1495 * <h4>StokesProblem::StokesProblem</h4>
1499 * The constructor of this class looks very similar to the one of
1500 * @ref step_20 "step-20". The constructor initializes the variables for the polynomial
1501 * degree, triangulation, finite element system and the dof handler. The
1502 * underlying polynomial functions are of order <code>degree+1</code> for
1503 * the vector-valued velocity components and of order <code>degree</code>
1504 * for the pressure. This gives the LBB-stable element pair
1505 * @f$Q_{degree+1}^d\times Q_{degree}@f$, often referred to as the Taylor-Hood
1510 * Note that we initialize the triangulation with a MeshSmoothing argument,
1511 * which ensures that the refinement of cells is done in a way that the
1512 * approximation of the PDE solution remains well-behaved (problems arise if
1513 * grids are too unstructured), see the documentation of
1514 * <code>Triangulation::MeshSmoothing</code> for details.
1517 * template <int dim>
1518 * StokesProblem<dim>::StokesProblem(const unsigned int degree)
1520 * , triangulation(Triangulation<dim>::maximum_smoothing)
1521 * , fe(FE_Q<dim>(degree + 1), dim, FE_Q<dim>(degree), 1)
1522 * , dof_handler(triangulation)
1529 * <a name="StokesProblemsetup_dofs"></a>
1530 * <h4>StokesProblem::setup_dofs</h4>
1534 * Given a mesh, this function associates the degrees of freedom with it and
1535 * creates the corresponding matrices and vectors. At the beginning it also
1536 * releases the pointer to the preconditioner object (if the shared pointer
1537 * pointed at anything at all at this point) since it will definitely not be
1538 * needed any more after this point and will have to be re-computed after
1539 * assembling the matrix, and unties the sparse matrices from their sparsity
1544 * We then proceed with distributing degrees of freedom and renumbering
1545 * them: In order to make the ILU preconditioner (in 3D) work efficiently,
1546 * it is important to enumerate the degrees of freedom in such a way that it
1547 * reduces the bandwidth of the matrix, or maybe more importantly: in such a
1548 * way that the ILU is as close as possible to a real LU decomposition. On
1549 * the other hand, we need to preserve the block structure of velocity and
1550 * pressure already seen in @ref step_20 "step-20" and @ref step_21 "step-21". This is done in two
1551 * steps: First, all dofs are renumbered to improve the ILU and then we
1552 * renumber once again by components. Since
1553 * <code>DoFRenumbering::component_wise</code> does not touch the
1554 * renumbering within the individual blocks, the basic renumbering from the
1555 * first step remains. As for how the renumber degrees of freedom to improve
1556 * the ILU: deal.II has a number of algorithms that attempt to find
1557 * orderings to improve ILUs, or reduce the bandwidth of matrices, or
1558 * optimize some other aspect. The DoFRenumbering namespace shows a
1559 * comparison of the results we obtain with several of these algorithms
1560 * based on the testcase discussed here in this tutorial program. Here, we
1561 * will use the traditional Cuthill-McKee algorithm already used in some of
1562 * the previous tutorial programs. In the <a href="#improved-ilu">section
1563 * on improved ILU</a> we're going to discuss
this issue in more detail.
1567 * There is
one more change compared to previous tutorial programs: There is
1568 * no reason in sorting the <code>dim</code> velocity components
1569 * individually. In fact, rather than
first enumerating all @f$x@f$-velocities,
1570 * then all @f$y@f$-velocities, etc, we would like to keep all velocities at the
1571 * same location together and only separate between velocities (all
1572 * components) and pressures. By
default,
this is not what the
1574 * component separately; what we have to
do is group several components into
1575 *
"blocks" and pass
this block structure to that function. Consequently, we
1576 * allocate a vector <code>block_component</code> with as many elements as
1577 * there are components and describe all velocity components to correspond
1578 * to block 0,
while the pressure component will form block 1:
1581 *
template <
int dim>
1582 *
void StokesProblem<dim>::setup_dofs()
1584 * A_preconditioner.reset();
1585 * system_matrix.clear();
1586 * preconditioner_matrix.clear();
1588 * dof_handler.distribute_dofs(fe);
1591 * std::vector<unsigned int> block_component(dim + 1, 0);
1592 * block_component[dim] = 1;
1597 * Now comes the implementation of Dirichlet boundary conditions, which
1598 * should be evident after the discussion in the introduction. All that
1599 * changed is that the function already appears in the setup
functions,
1600 * whereas we were used to see it in some assembly routine. Further down
1601 * below where we
set up the mesh, we will associate the top boundary
1602 * where we impose Dirichlet boundary conditions with boundary indicator
1603 * 1. We will have to pass
this boundary indicator as
second argument to
1604 * the function below interpolating boundary
values. There is
one more
1605 * thing, though. The function describing the Dirichlet conditions was
1606 * defined
for all components, both velocity and pressure. However, the
1607 * Dirichlet conditions are to be
set for the velocity only. To
this end,
1608 * we use a
ComponentMask that only selects the velocity components. The
1609 * component mask is obtained from the finite element by specifying the
1610 * particular components we want. Since we use adaptively refined grids,
1611 * the
affine constraints
object needs to be
first filled with hanging node
1612 * constraints generated from the DoF handler. Note the order of the two
1613 *
functions — we
first compute the hanging node constraints, and
1614 * then
insert the boundary
values into the constraints
object. This makes
1615 * sure that we respect H<sup>1</sup> conformity on boundaries with
1616 * hanging nodes (in three space dimensions), where the hanging node needs
1617 * to dominate the Dirichlet boundary
values.
1621 * constraints.clear();
1627 * BoundaryValues<dim>(),
1629 * fe.component_mask(velocities));
1632 * constraints.close();
1636 * In analogy to @ref step_20
"step-20", we count the dofs in the individual components.
1637 * We could
do this in the same way as there, but we want to operate on
1638 * the block structure we used already
for the renumbering: The function
1641 * velocity and pressure block via <code>block_component</code>.
1644 *
const std::vector<types::global_dof_index> dofs_per_block =
1646 *
const unsigned int n_u = dofs_per_block[0];
1647 *
const unsigned int n_p = dofs_per_block[1];
1649 * std::cout <<
" Number of active cells: " <<
triangulation.n_active_cells()
1651 * <<
" Number of degrees of freedom: " << dof_handler.n_dofs()
1652 * <<
" (" << n_u <<
'+' << n_p <<
')' << std::endl;
1656 * The next task is to allocate a sparsity pattern
for the system
matrix we
1657 * will create and
one for the preconditioner
matrix. We could
do this in
1658 * the same way as in @ref step_20
"step-20", i.e. directly build an
object of type
1660 * is a major reason not to
do so:
1661 * In 3D, the function DoFTools::max_couplings_between_dofs yields a
1662 * conservative but rather large number
for the coupling between the
1663 * individual dofs, so that the memory initially provided
for the creation
1664 * of the sparsity pattern of the
matrix is far too much -- so much actually
1665 * that the
initial sparsity pattern won
't even fit into the physical memory
1666 * of most systems already for moderately-sized 3D problems, see also the
1667 * discussion in @ref step_18 "step-18". Instead, we first build temporary objects that use
1668 * a different data structure that doesn't require allocating more memory
1669 * than necessary but isn
't suitable for use as a basis of SparseMatrix or
1670 * BlockSparseMatrix objects; in a second step we then copy these objects
1671 * into objects of type BlockSparsityPattern. This is entirely analogous to
1672 * what we already did in @ref step_11 "step-11" and @ref step_18 "step-18". In particular, we make use of
1673 * the fact that we will never write into the @f$(1,1)@f$ block of the system
1674 * matrix and that this is the only block to be filled for the
1675 * preconditioner matrix.
1679 * All this is done inside new scopes, which means that the memory of
1680 * <code>dsp</code> will be released once the information has been copied to
1681 * <code>sparsity_pattern</code>.
1685 * BlockDynamicSparsityPattern dsp(2, 2);
1687 * dsp.block(0, 0).reinit(n_u, n_u);
1688 * dsp.block(1, 0).reinit(n_p, n_u);
1689 * dsp.block(0, 1).reinit(n_u, n_p);
1690 * dsp.block(1, 1).reinit(n_p, n_p);
1692 * dsp.collect_sizes();
1694 * Table<2, DoFTools::Coupling> coupling(dim + 1, dim + 1);
1696 * for (unsigned int c = 0; c < dim + 1; ++c)
1697 * for (unsigned int d = 0; d < dim + 1; ++d)
1698 * if (!((c == dim) && (d == dim)))
1699 * coupling[c][d] = DoFTools::always;
1701 * coupling[c][d] = DoFTools::none;
1703 * DoFTools::make_sparsity_pattern(
1704 * dof_handler, coupling, dsp, constraints, false);
1706 * sparsity_pattern.copy_from(dsp);
1710 * BlockDynamicSparsityPattern preconditioner_dsp(2, 2);
1712 * preconditioner_dsp.block(0, 0).reinit(n_u, n_u);
1713 * preconditioner_dsp.block(1, 0).reinit(n_p, n_u);
1714 * preconditioner_dsp.block(0, 1).reinit(n_u, n_p);
1715 * preconditioner_dsp.block(1, 1).reinit(n_p, n_p);
1717 * preconditioner_dsp.collect_sizes();
1719 * Table<2, DoFTools::Coupling> preconditioner_coupling(dim + 1, dim + 1);
1721 * for (unsigned int c = 0; c < dim + 1; ++c)
1722 * for (unsigned int d = 0; d < dim + 1; ++d)
1723 * if (((c == dim) && (d == dim)))
1724 * preconditioner_coupling[c][d] = DoFTools::always;
1726 * preconditioner_coupling[c][d] = DoFTools::none;
1728 * DoFTools::make_sparsity_pattern(dof_handler,
1729 * preconditioner_coupling,
1730 * preconditioner_dsp,
1734 * preconditioner_sparsity_pattern.copy_from(preconditioner_dsp);
1739 * Finally, the system matrix, the preconsitioner matrix, the solution and
1740 * the right hand side vector are created from the block structure similar
1741 * to the approach in @ref step_20 "step-20":
1744 * system_matrix.reinit(sparsity_pattern);
1745 * preconditioner_matrix.reinit(preconditioner_sparsity_pattern);
1747 * solution.reinit(2);
1748 * solution.block(0).reinit(n_u);
1749 * solution.block(1).reinit(n_p);
1750 * solution.collect_sizes();
1752 * system_rhs.reinit(2);
1753 * system_rhs.block(0).reinit(n_u);
1754 * system_rhs.block(1).reinit(n_p);
1755 * system_rhs.collect_sizes();
1762 * <a name="StokesProblemassemble_system"></a>
1763 * <h4>StokesProblem::assemble_system</h4>
1767 * The assembly process follows the discussion in @ref step_20 "step-20" and in the
1768 * introduction. We use the well-known abbreviations for the data structures
1769 * that hold the local matrices, right hand side, and global numbering of the
1770 * degrees of freedom for the present cell.
1773 * template <int dim>
1774 * void StokesProblem<dim>::assemble_system()
1776 * system_matrix = 0;
1778 * preconditioner_matrix = 0;
1780 * QGauss<dim> quadrature_formula(degree + 2);
1782 * FEValues<dim> fe_values(fe,
1783 * quadrature_formula,
1784 * update_values | update_quadrature_points |
1785 * update_JxW_values | update_gradients);
1787 * const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
1789 * const unsigned int n_q_points = quadrature_formula.size();
1791 * FullMatrix<double> local_matrix(dofs_per_cell, dofs_per_cell);
1792 * FullMatrix<double> local_preconditioner_matrix(dofs_per_cell,
1794 * Vector<double> local_rhs(dofs_per_cell);
1796 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
1798 * const RightHandSide<dim> right_hand_side;
1799 * std::vector<Vector<double>> rhs_values(n_q_points, Vector<double>(dim + 1));
1803 * Next, we need two objects that work as extractors for the FEValues
1804 * object. Their use is explained in detail in the report on @ref
1808 * const FEValuesExtractors::Vector velocities(0);
1809 * const FEValuesExtractors::Scalar pressure(dim);
1813 * As an extension over @ref step_20 "step-20" and @ref step_21 "step-21", we include a few optimizations
1814 * that make assembly much faster for this particular problem. The
1815 * improvements are based on the observation that we do a few calculations
1816 * too many times when we do as in @ref step_20 "step-20": The symmetric gradient actually
1817 * has <code>dofs_per_cell</code> different values per quadrature point, but
1818 * we extract it <code>dofs_per_cell*dofs_per_cell</code> times from the
1819 * FEValues object - for both the loop over <code>i</code> and the inner
1820 * loop over <code>j</code>. In 3d, that means evaluating it @f$89^2=7921@f$
1821 * instead of @f$89@f$ times, a not insignificant difference.
1825 * So what we're going to
do here is to avoid such repeated calculations
1826 * by getting a vector of rank-2 tensors (and similarly
for the divergence
1827 * and the basis function value on pressure) at the quadrature
point prior
1828 * to starting the
loop over the dofs on the cell. First, we create the
1829 * respective objects that will hold these
values. Then, we start the
loop
1830 * over all cells and the
loop over the quadrature points, where we
first
1833 * since all the operations involved are
symmetric with respect to @f$i@f$ and
1834 * @f$j@f$. This is implemented by simply running the inner
loop not to
1835 * <code>dofs_per_cell</code>, but only up to <code>i</code>, the index of
1839 * std::vector<SymmetricTensor<2, dim>> symgrad_phi_u(dofs_per_cell);
1840 * std::vector<double> div_phi_u(dofs_per_cell);
1841 * std::vector<double> phi_p(dofs_per_cell);
1843 *
for (
const auto &cell : dof_handler.active_cell_iterators())
1845 * fe_values.reinit(cell);
1847 * local_preconditioner_matrix = 0;
1850 * right_hand_side.vector_value_list(fe_values.get_quadrature_points(),
1853 *
for (
unsigned int q = 0; q < n_q_points; ++q)
1855 *
for (
unsigned int k = 0; k < dofs_per_cell; ++k)
1857 * symgrad_phi_u[k] =
1858 * fe_values[velocities].symmetric_gradient(k, q);
1859 * div_phi_u[k] = fe_values[velocities].divergence(k, q);
1860 * phi_p[k] = fe_values[pressure].value(k, q);
1865 * Now
finally for the bilinear forms of both the system
matrix and
1866 * the
matrix we use
for the preconditioner. Recall that the
1867 * formulas
for these two are
1869 * A_{ij} &= a(\varphi_i,\varphi_j)
1870 * \\ &= \underbrace{2(\varepsilon(\varphi_{i,\textbf{u}}),
1871 * \varepsilon(\varphi_{j,\textbf{u}}))_{\Omega}}
1874 * \underbrace{- (\textrm{div}\; \varphi_{i,\textbf{u}},
1875 * \varphi_{j,p})_{\Omega}}
1878 * \underbrace{- (\varphi_{i,p},
1880 * \varphi_{j,\textbf{u}})_{\Omega}}
1885 * M_{ij} &= \underbrace{(\varphi_{i,p},
1886 * \varphi_{j,p})_{\Omega}}
1889 * respectively, where @f$\varphi_{i,\textbf{u}}@f$ and @f$\varphi_{i,p}@f$
1890 * are the velocity and pressure components of the @f$i@f$th shape
1891 * function. The various terms above are then easily recognized in
1892 * the following implementation:
1895 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1897 *
for (
unsigned int j = 0; j <= i; ++j)
1899 * local_matrix(i, j) +=
1900 * (2 * (symgrad_phi_u[i] * symgrad_phi_u[j])
1901 * - div_phi_u[i] * phi_p[j]
1902 * - phi_p[i] * div_phi_u[j])
1903 * * fe_values.JxW(q);
1905 * local_preconditioner_matrix(i, j) +=
1906 * (phi_p[i] * phi_p[j])
1907 * * fe_values.JxW(q);
1911 * Note that in the implementation of (1) above, `operator*`
1912 * is overloaded for
symmetric tensors, yielding the scalar
1913 * product between the two tensors.
1917 * For the right-hand side we use the fact that the shape
1919 * elements are primitive). Instead of multiplying the tensor
1920 * representing the dim+1
values of shape function i with the
1921 * whole right-hand side vector, we only look at the only
1922 * non-
zero component. The function
1924 * which component this shape function lives in (0=x velocity,
1925 * 1=y velocity, 2=pressure in 2
d), which we use to pick out
1926 * the correct component of the right-hand side vector to
1930 * const
unsigned int component_i =
1931 * fe.system_to_component_index(i).
first;
1932 * local_rhs(i) += (fe_values.shape_value(i, q)
1933 * * rhs_values[q](component_i))
1934 * * fe_values.JxW(q);
1940 * Before we can write the local data into the global
matrix (and
1942 * Dirichlet boundary conditions and eliminate hanging node constraints,
1943 * as we discussed in the introduction), we have to be careful about
one
1944 * thing, though. We have only built half of the local matrices
1945 * because of symmetry, but we're going to save the full matrices
1946 * in order to use the standard
functions for solving. This is done
1947 * by flipping the indices in case we are pointing into the empty part
1948 * of the local matrices.
1951 * for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1952 * for (
unsigned int j = i + 1; j < dofs_per_cell; ++j)
1954 * local_matrix(i, j) = local_matrix(j, i);
1955 * local_preconditioner_matrix(i, j) =
1956 * local_preconditioner_matrix(j, i);
1959 * cell->get_dof_indices(local_dof_indices);
1960 * constraints.distribute_local_to_global(local_matrix,
1962 * local_dof_indices,
1965 * constraints.distribute_local_to_global(local_preconditioner_matrix,
1966 * local_dof_indices,
1967 * preconditioner_matrix);
1972 * Before we
're going to solve this linear system, we generate a
1973 * preconditioner for the velocity-velocity matrix, i.e.,
1974 * <code>block(0,0)</code> in the system matrix. As mentioned above, this
1975 * depends on the spatial dimension. Since the two classes described by
1976 * the <code>InnerPreconditioner::type</code> alias have the same
1977 * interface, we do not have to do anything different whether we want to
1978 * use a sparse direct solver or an ILU:
1981 * std::cout << " Computing preconditioner..." << std::endl << std::flush;
1983 * A_preconditioner =
1984 * std::make_shared<typename InnerPreconditioner<dim>::type>();
1985 * A_preconditioner->initialize(
1986 * system_matrix.block(0, 0),
1987 * typename InnerPreconditioner<dim>::type::AdditionalData());
1995 * <a name="StokesProblemsolve"></a>
1996 * <h4>StokesProblem::solve</h4>
2000 * After the discussion in the introduction and the definition of the
2001 * respective classes above, the implementation of the <code>solve</code>
2002 * function is rather straight-forward and done in a similar way as in
2003 * @ref step_20 "step-20". To start with, we need an object of the
2004 * <code>InverseMatrix</code> class that represents the inverse of the
2005 * matrix A. As described in the introduction, the inverse is generated with
2006 * the help of an inner preconditioner of type
2007 * <code>InnerPreconditioner::type</code>.
2010 * template <int dim>
2011 * void StokesProblem<dim>::solve()
2013 * const InverseMatrix<SparseMatrix<double>,
2014 * typename InnerPreconditioner<dim>::type>
2015 * A_inverse(system_matrix.block(0, 0), *A_preconditioner);
2016 * Vector<double> tmp(solution.block(0).size());
2020 * This is as in @ref step_20 "step-20". We generate the right hand side @f$B A^{-1} F - G@f$
2021 * for the Schur complement and an object that represents the respective
2022 * linear operation @f$B A^{-1} B^T@f$, now with a template parameter
2023 * indicating the preconditioner - in accordance with the definition of
2028 * Vector<double> schur_rhs(solution.block(1).size());
2029 * A_inverse.vmult(tmp, system_rhs.block(0));
2030 * system_matrix.block(1, 0).vmult(schur_rhs, tmp);
2031 * schur_rhs -= system_rhs.block(1);
2033 * SchurComplement<typename InnerPreconditioner<dim>::type> schur_complement(
2034 * system_matrix, A_inverse);
2038 * The usual control structures for the solver call are created...
2041 * SolverControl solver_control(solution.block(1).size(),
2042 * 1e-6 * schur_rhs.l2_norm());
2043 * SolverCG<Vector<double>> cg(solver_control);
2047 * Now to the preconditioner to the Schur complement. As explained in
2048 * the introduction, the preconditioning is done by a mass matrix in the
2049 * pressure variable.
2053 * Actually, the solver needs to have the preconditioner in the form
2054 * @f$P^{-1}@f$, so we need to create an inverse operation. Once again, we
2055 * use an object of the class <code>InverseMatrix</code>, which
2056 * implements the <code>vmult</code> operation that is needed by the
2057 * solver. In this case, we have to invert the pressure mass matrix. As
2058 * it already turned out in earlier tutorial programs, the inversion of
2059 * a mass matrix is a rather cheap and straight-forward operation
2060 * (compared to, e.g., a Laplace matrix). The CG method with ILU
2061 * preconditioning converges in 5-10 steps, independently on the mesh
2062 * size. This is precisely what we do here: We choose another ILU
2063 * preconditioner and take it along to the InverseMatrix object via the
2064 * corresponding template parameter. A CG solver is then called within
2065 * the vmult operation of the inverse matrix.
2069 * An alternative that is cheaper to build, but needs more iterations
2070 * afterwards, would be to choose a SSOR preconditioner with factor
2071 * 1.2. It needs about twice the number of iterations, but the costs for
2072 * its generation are almost negligible.
2075 * SparseILU<double> preconditioner;
2076 * preconditioner.initialize(preconditioner_matrix.block(1, 1),
2077 * SparseILU<double>::AdditionalData());
2079 * InverseMatrix<SparseMatrix<double>, SparseILU<double>> m_inverse(
2080 * preconditioner_matrix.block(1, 1), preconditioner);
2084 * With the Schur complement and an efficient preconditioner at hand, we
2085 * can solve the respective equation for the pressure (i.e. block 0 in
2086 * the solution vector) in the usual way:
2089 * cg.solve(schur_complement, solution.block(1), schur_rhs, m_inverse);
2093 * After this first solution step, the hanging node constraints have to
2094 * be distributed to the solution in order to achieve a consistent
2098 * constraints.distribute(solution);
2100 * std::cout << " " << solver_control.last_step()
2101 * << " outer CG Schur complement iterations for pressure"
2107 * As in @ref step_20 "step-20", we finally need to solve for the velocity equation where
2108 * we plug in the solution to the pressure equation. This involves only
2109 * objects we already know - so we simply multiply @f$p@f$ by @f$B^T@f$, subtract
2110 * the right hand side and multiply by the inverse of @f$A@f$. At the end, we
2111 * need to distribute the constraints from hanging nodes in order to
2112 * obtain a consistent flow field:
2116 * system_matrix.block(0, 1).vmult(tmp, solution.block(1));
2118 * tmp += system_rhs.block(0);
2120 * A_inverse.vmult(solution.block(0), tmp);
2122 * constraints.distribute(solution);
2130 * <a name="StokesProblemoutput_results"></a>
2131 * <h4>StokesProblem::output_results</h4>
2135 * The next function generates graphical output. In this example, we are
2136 * going to use the VTK file format. We attach names to the individual
2137 * variables in the problem: <code>velocity</code> to the <code>dim</code>
2138 * components of velocity and <code>pressure</code> to the pressure.
2142 * Not all visualization programs have the ability to group individual
2143 * vector components into a vector to provide vector plots; in particular,
2144 * this holds for some VTK-based visualization programs. In this case, the
2145 * logical grouping of components into vectors should already be described
2146 * in the file containing the data. In other words, what we need to do is
2147 * provide our output writers with a way to know which of the components of
2148 * the finite element logically form a vector (with @f$d@f$ components in @f$d@f$
2149 * space dimensions) rather than letting them assume that we simply have a
2150 * bunch of scalar fields. This is achieved using the members of the
2151 * <code>DataComponentInterpretation</code> namespace: as with the filename,
2152 * we create a vector in which the first <code>dim</code> components refer
2153 * to the velocities and are given the tag
2154 * DataComponentInterpretation::component_is_part_of_vector; we
2155 * finally push one tag
2156 * DataComponentInterpretation::component_is_scalar to describe
2157 * the grouping of the pressure variable.
2161 * The rest of the function is then the same as in @ref step_20 "step-20".
2164 * template <int dim>
2166 * StokesProblem<dim>::output_results(const unsigned int refinement_cycle) const
2168 * std::vector<std::string> solution_names(dim, "velocity");
2169 * solution_names.emplace_back("pressure");
2171 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
2172 * data_component_interpretation(
2173 * dim, DataComponentInterpretation::component_is_part_of_vector);
2174 * data_component_interpretation.push_back(
2175 * DataComponentInterpretation::component_is_scalar);
2177 * DataOut<dim> data_out;
2178 * data_out.attach_dof_handler(dof_handler);
2179 * data_out.add_data_vector(solution,
2181 * DataOut<dim>::type_dof_data,
2182 * data_component_interpretation);
2183 * data_out.build_patches();
2185 * std::ofstream output(
2186 * "solution-" + Utilities::int_to_string(refinement_cycle, 2) + ".vtk");
2187 * data_out.write_vtk(output);
2194 * <a name="StokesProblemrefine_mesh"></a>
2195 * <h4>StokesProblem::refine_mesh</h4>
2199 * This is the last interesting function of the <code>StokesProblem</code>
2200 * class. As indicated by its name, it takes the solution to the problem
2201 * and refines the mesh where this is needed. The procedure is the same as
2202 * in the respective step in @ref step_6 "step-6", with the exception that we base the
2203 * refinement only on the change in pressure, i.e., we call the Kelly error
2204 * estimator with a mask object of type ComponentMask that selects the
2205 * single scalar component for the pressure that we are interested in (we
2206 * get such a mask from the finite element class by specifying the component
2207 * we want). Additionally, we do not coarsen the grid again:
2210 * template <int dim>
2211 * void StokesProblem<dim>::refine_mesh()
2213 * Vector<float> estimated_error_per_cell(triangulation.n_active_cells());
2215 * FEValuesExtractors::Scalar pressure(dim);
2216 * KellyErrorEstimator<dim>::estimate(
2218 * QGauss<dim - 1>(degree + 1),
2219 * std::map<types::boundary_id, const Function<dim> *>(),
2221 * estimated_error_per_cell,
2222 * fe.component_mask(pressure));
2224 * GridRefinement::refine_and_coarsen_fixed_number(triangulation,
2225 * estimated_error_per_cell,
2228 * triangulation.execute_coarsening_and_refinement();
2235 * <a name="StokesProblemrun"></a>
2236 * <h4>StokesProblem::run</h4>
2240 * The last step in the Stokes class is, as usual, the function that
2241 * generates the initial grid and calls the other functions in the
2246 * We start off with a rectangle of size @f$4 \times 1@f$ (in 2d) or @f$4 \times 1
2247 * \times 1@f$ (in 3d), placed in @f$R^2/R^3@f$ as @f$(-2,2)\times(-1,0)@f$ or
2248 * @f$(-2,2)\times(0,1)\times(-1,0)@f$, respectively. It is natural to start
2249 * with equal mesh size in each direction, so we subdivide the initial
2250 * rectangle four times in the first coordinate direction. To limit the
2251 * scope of the variables involved in the creation of the mesh to the range
2252 * where we actually need them, we put the entire block between a pair of
2256 * template <int dim>
2257 * void StokesProblem<dim>::run()
2260 * std::vector<unsigned int> subdivisions(dim, 1);
2261 * subdivisions[0] = 4;
2263 * const Point<dim> bottom_left = (dim == 2 ?
2264 * Point<dim>(-2, -1) : // 2d case
2265 * Point<dim>(-2, 0, -1)); // 3d case
2267 * const Point<dim> top_right = (dim == 2 ?
2268 * Point<dim>(2, 0) : // 2d case
2269 * Point<dim>(2, 1, 0)); // 3d case
2271 * GridGenerator::subdivided_hyper_rectangle(triangulation,
2279 * A boundary indicator of 1 is set to all boundaries that are subject to
2280 * Dirichlet boundary conditions, i.e. to faces that are located at 0 in
2281 * the last coordinate direction. See the example description above for
2285 * for (const auto &cell : triangulation.active_cell_iterators())
2286 * for (const auto &face : cell->face_iterators())
2287 * if (face->center()[dim - 1] == 0)
2288 * face->set_all_boundary_ids(1);
2293 * We then apply an initial refinement before solving for the first
2294 * time. In 3D, there are going to be more degrees of freedom, so we
2295 * refine less there:
2298 * triangulation.refine_global(4 - dim);
2302 * As first seen in @ref step_6 "step-6", we cycle over the different refinement levels
2303 * and refine (except for the first cycle), setup the degrees of freedom
2304 * and matrices, assemble, solve and create output:
2307 * for (unsigned int refinement_cycle = 0; refinement_cycle < 6;
2308 * ++refinement_cycle)
2310 * std::cout << "Refinement cycle " << refinement_cycle << std::endl;
2312 * if (refinement_cycle > 0)
2317 * std::cout << " Assembling..." << std::endl << std::flush;
2318 * assemble_system();
2320 * std::cout << " Solving..." << std::flush;
2323 * output_results(refinement_cycle);
2325 * std::cout << std::endl;
2328 * } // namespace Step22
2334 * <a name="Thecodemaincodefunction"></a>
2335 * <h3>The <code>main</code> function</h3>
2339 * The main function is the same as in @ref step_20 "step-20". We pass the element degree as
2340 * a parameter and choose the space dimension at the well-known template slot.
2347 * using namespace Step22;
2349 * StokesProblem<2> flow_problem(1);
2350 * flow_problem.run();
2352 * catch (std::exception &exc)
2354 * std::cerr << std::endl
2356 * << "----------------------------------------------------"
2358 * std::cerr << "Exception on processing: " << std::endl
2359 * << exc.what() << std::endl
2360 * << "Aborting!" << std::endl
2361 * << "----------------------------------------------------"
2368 * std::cerr << std::endl
2370 * << "----------------------------------------------------"
2372 * std::cerr << "Unknown exception!" << std::endl
2373 * << "Aborting!" << std::endl
2374 * << "----------------------------------------------------"
2382<a name="Results"></a>
2383<a name="Results"></a><h1>Results</h1>
2386<a name="Outputoftheprogramandgraphicalvisualization"></a><h3>Output of the program and graphical visualization</h3>
2389<a name="2Dcalculations"></a><h4>2D calculations</h4>
2392Running the program with the space dimension set to 2 in the <code>main</code>
2393function yields the following output (in "release mode",
2394See also <a href="http://www.math.colostate.edu/~bangerth/videos.676.18.html">video lecture 18</a>.):
2396examples/step-22> make run
2398 Number of active cells: 64
2399 Number of degrees of freedom: 679 (594+85)
2401 Computing preconditioner...
2402 Solving... 11 outer CG Schur complement iterations for pressure
2405 Number of active cells: 160
2406 Number of degrees of freedom: 1683 (1482+201)
2408 Computing preconditioner...
2409 Solving... 11 outer CG Schur complement iterations for pressure
2412 Number of active cells: 376
2413 Number of degrees of freedom: 3813 (3370+443)
2415 Computing preconditioner...
2416 Solving... 11 outer CG Schur complement iterations for pressure
2419 Number of active cells: 880
2420 Number of degrees of freedom: 8723 (7722+1001)
2422 Computing preconditioner...
2423 Solving... 11 outer CG Schur complement iterations for pressure
2426 Number of active cells: 2008
2427 Number of degrees of freedom: 19383 (17186+2197)
2429 Computing preconditioner...
2430 Solving... 11 outer CG Schur complement iterations for pressure
2433 Number of active cells: 4288
2434 Number of degrees of freedom: 40855 (36250+4605)
2436 Computing preconditioner...
2437 Solving... 11 outer CG Schur complement iterations for pressure
2440The entire computation above takes about 2 seconds on a reasonably
2441quick (for 2015 standards) machine.
2443What we see immediately from this is that the number of (outer)
2444iterations does not increase as we refine the mesh. This confirms the
2445statement in the introduction that preconditioning the Schur
2446complement with the mass matrix indeed yields a matrix spectrally
2447equivalent to the identity matrix (i.e. with eigenvalues bounded above
2448and below independently of the mesh size or the relative sizes of
2449cells). In other words, the mass matrix and the Schur complement are
2450spectrally equivalent.
2452In the images below, we show the grids for the first six refinement
2453steps in the program. Observe how the grid is refined in regions
2454where the solution rapidly changes: On the upper boundary, we have
2455Dirichlet boundary conditions that are -1 in the left half of the line
2456and 1 in the right one, so there is an abrupt change at @f$x=0@f$. Likewise,
2457there are changes from Dirichlet to Neumann data in the two upper
2458corners, so there is need for refinement there as well:
2460<table width="60%" align="center">
2463 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-0.png" alt="">
2466 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-1.png" alt="">
2471 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-2.png" alt="">
2474 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-3.png" alt="">
2479 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-4.png" alt="">
2482 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-5.png" alt="">
2487Finally, following is a plot of the flow field. It shows fluid
2488transported along with the moving upper boundary and being replaced by
2489material coming from below:
2491<img src="https://www.dealii.org/images/steps/developer/step-22.2d.solution.png" alt="">
2493This plot uses the capability of VTK-based visualization programs (in
2494this case of VisIt) to show vector data; this is the result of us
2495declaring the velocity components of the finite element in use to be a
2496set of vector components, rather than independent scalar components in
2497the <code>StokesProblem@<dim@>::%output_results</code> function of this
2502<a name="3Dcalculations"></a><h4>3D calculations</h4>
2505In 3d, the screen output of the program looks like this:
2509 Number of active cells: 32
2510 Number of degrees of freedom: 1356 (1275+81)
2512 Computing preconditioner...
2513 Solving... 13 outer CG Schur complement iterations for pressure.
2516 Number of active cells: 144
2517 Number of degrees of freedom: 5088 (4827+261)
2519 Computing preconditioner...
2520 Solving... 14 outer CG Schur complement iterations for pressure.
2523 Number of active cells: 704
2524 Number of degrees of freedom: 22406 (21351+1055)
2526 Computing preconditioner...
2527 Solving... 14 outer CG Schur complement iterations for pressure.
2530 Number of active cells: 3168
2531 Number of degrees of freedom: 93176 (89043+4133)
2533 Computing preconditioner...
2534 Solving... 15 outer CG Schur complement iterations for pressure.
2537 Number of active cells: 11456
2538 Number of degrees of freedom: 327808 (313659+14149)
2540 Computing preconditioner...
2541 Solving... 15 outer CG Schur complement iterations for pressure.
2544 Number of active cells: 45056
2545 Number of degrees of freedom: 1254464 (1201371+53093)
2547 Computing preconditioner...
2548 Solving... 14 outer CG Schur complement iterations for pressure.
2551Again, we see that the number of outer iterations does not increase as
2552we refine the mesh. Nevertheless, the compute time increases
2553significantly: for each of the iterations above separately, it takes about
25540.14 seconds, 0.63 seconds, 4.8 seconds, 35 seconds, 2 minutes and 33 seconds,
2555and 13 minutes and 12 seconds. This overall superlinear (in the number of
2556unknowns) increase in runtime is due to the fact that our inner solver is not
2557@f${\cal O}(N)@f$: a simple experiment shows that as we keep refining the mesh, the
2558average number of ILU-preconditioned CG iterations to invert the
2559velocity-velocity block @f$A@f$ increases.
2561We will address the question of how possibly to improve our solver <a
2562href="#improved-solver">below</a>.
2564As for the graphical output, the grids generated during the solution
2567<table width="60%" align="center">
2570 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-0.png" alt="">
2573 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-1.png" alt="">
2578 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-2.png" alt="">
2581 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-3.png" alt="">
2586 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-4.png" alt="">
2589 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-5.png" alt="">
2594Again, they show essentially the location of singularities introduced
2595by boundary conditions. The vector field computed makes for an
2598<img src="https://www.dealii.org/images/steps/developer/step-22.3d.solution.png" alt="">
2600The isocontours shown here as well are those of the pressure
2601variable, showing the singularity at the point of discontinuous
2602velocity boundary conditions.
2606<a name="Sparsitypattern"></a><h3>Sparsity pattern</h3>
2609As explained during the generation of the sparsity pattern, it is
2610important to have the numbering of degrees of freedom in mind when
2611using preconditioners like incomplete LU decompositions. This is most
2612conveniently visualized using the distribution of nonzero elements in
2613the stiffness matrix.
2615If we don't
do anything special to renumber degrees of freedom (i.e.,
2618appropriately sorted into their corresponding blocks of the
matrix and
2619vector), then we get the following image after the
first adaptive
2620refinement in two dimensions:
2622<img src=
"https://www.dealii.org/images/steps/developer/step-22.2d.sparsity-nor.png" alt=
"">
2624In order to generate such a graph, you have to
insert a piece of
2625code like the following to the
end of the setup step.
2628 std::ofstream out (
"sparsity_pattern.gpl");
2629 sparsity_pattern.print_gnuplot(out);
2633It is clearly visible that the
nonzero entries are spread over almost the
2634whole
matrix. This makes preconditioning by ILU inefficient: ILU generates a
2635Gaussian elimination (LU decomposition) without fill-in elements, which means
2636that more tentative fill-ins left out will result in a worse approximation of
2637the complete decomposition.
2639In
this program, we have thus chosen a more advanced renumbering of
2641the components into velocity and pressure yields the following output:
2643<img src=
"https://www.dealii.org/images/steps/developer/step-22.2d.sparsity-ren.png" alt=
"">
2645It is apparent that the situation has improved a lot. Most of the elements are
2646now concentrated around the
diagonal in the (0,0) block in the
matrix. Similar
2647effects are also visible for the other blocks. In this case, the ILU
2648decomposition will be much closer to the full LU decomposition, which improves
2649the quality of the preconditioner. (It may be interesting to note that the
2650sparse direct solver UMFPACK does some %
internal renumbering of the equations
2651before actually generating a sparse LU decomposition; that procedure leads to
2652a very similar pattern to the
one we got from the Cuthill-McKee algorithm.)
2654Finally, we want to have a closer
2655look at a sparsity pattern in 3D. We show only the (0,0) block of the
2656matrix, again after
one adaptive refinement. Apart from the fact that the
matrix
2657size has increased, it is also visible that there are many more entries
2658in the
matrix. Moreover, even for the optimized renumbering, there will be a
2659considerable amount of tentative fill-in elements. This illustrates why UMFPACK
2660is not a good choice in 3D - a full decomposition needs many new entries that
2661 eventually won't fit into the physical memory (RAM):
2667<a name="Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
2670<a name="improved-solver">
2671<a name="Improvedlinearsolverin3D"></a><h4>Improved linear solver in 3D</h4>
2675We have seen in the section of computational results that the number of outer
2676iterations does not depend on the mesh size, which is optimal in a sense of
2677scalability. This does, however, not apply to the solver as a whole, as
2679We did not look at the number of inner iterations when generating the inverse of
2680the
matrix @f$A@f$ and the mass
matrix @f$M_p@f$. Of course, this is unproblematic in
2681the 2D case where we precondition @f$A@f$ with a direct solver and the
2682<code>vmult</code> operation of the inverse
matrix structure will converge in
2683one single CG step, but this changes in 3D where we only use an ILU
2684preconditioner. There, the number of required preconditioned CG steps to
2685invert @f$A@f$ increases as the mesh is refined, and each <code>vmult</code>
2686operation involves on average approximately 14, 23, 36, 59, 75 and 101 inner
2687CG iterations in the refinement steps shown above. (On the other hand,
2688the number of iterations for applying the inverse pressure mass
matrix is
2689always around five, both in two and three dimensions.) To summarize, most work
2690is spent on solving linear systems with the same
matrix @f$A@f$ over and over again.
2691What makes this look even worse is the fact that we
2692actually
invert a
matrix that is about 95 percent the size of the total system
2693matrix and stands for 85 percent of the non-
zero entries in the sparsity
2694pattern. Hence, the natural question is whether it is reasonable to solve a
2695linear system with
matrix @f$A@f$ for about 15 times when calculating the solution
2698The answer is, of course, that we can do that in a few other (most of the time
2700Nevertheless, it has to be remarked that an indefinite system as the
one
2701at hand puts indeed much higher
2702demands on the linear algebra than standard elliptic problems as we have seen
2703in the early tutorial programs. The improvements are still rather
2704unsatisfactory, if
one compares with an elliptic problem of similar
2705size. Either way, we will introduce below a number of improvements to the
2706linear solver, a discussion that we will re-consider again with additional
2707options in the @ref step_31 "step-31" program.
2709<a name="improved-ilu">
2710<a name="BetterILUdecompositionbysmartreordering"></a><h5>Better ILU decomposition by smart reordering</h5>
2713A first attempt to improve the speed of the linear solution process is to choose
2714a dof reordering that makes the ILU being closer to a full LU decomposition, as
2715already mentioned in the in-code comments. The
DoFRenumbering namespace compares
2716several choices for the renumbering of dofs for the Stokes equations. The best
2717result regarding the computing time was found for the King ordering, which is
2719program, the inner solver needs considerably less operations,
e.g. about 62
2720inner CG iterations for the inversion of @f$A@f$ at cycle 4 compared to about 75
2721iterations with the standard Cuthill-McKee-algorithm. Also, the computing time
2722at cycle 4 decreased from about 17 to 11 minutes for the <code>solve()</code>
2723call. However, the King ordering (and the orderings provided by the
2725much more memory than the in-build deal versions, since it acts on abstract
2726graphs rather than the geometry provided by the
triangulation. In the present
2727case, the renumbering takes about 5 times as much memory, which yields an
2728infeasible algorithm for the last cycle in 3D with 1.2 million
2731<a name="BetterpreconditionerfortheinnerCGsolver"></a><h5>Better preconditioner for the inner CG solver</h5>
2733Another idea to improve the situation even more would be to choose a
2734preconditioner that makes CG for the (0,0)
matrix @f$A@f$ converge in a
2735mesh-independent number of iterations, say 10 to 30. We have seen such a
2736candidate in @ref step_16 "step-16": multigrid.
2738<a name="BlockSchurcomplementpreconditioner"></a><h5>Block Schur complement preconditioner</h5>
2740<a name="block-schur"></a>
2741Even with a good preconditioner for @f$A@f$, we still
2742need to solve of the same linear system repeatedly (with different
2743right hand sides, though) in order to make the Schur complement solve
2744converge. The approach we are going to discuss here is how inner iteration
2745and outer iteration can be combined. If we persist in calculating the Schur
2746complement, there is no other possibility.
2748The alternative is to attack the block system at once and use an
approximate
2749Schur complement as efficient preconditioner. The idea is as
2750follows: If we find a block preconditioner @f$P@f$ such that the
matrix
2752 P^{-1}\left(\
begin{array}{cc}
2756is simple, then an iterative solver with that preconditioner will converge in a
2757few iterations. Using the Schur complement @f$S = B
A^{-1} B^
T@f$,
one finds that
2761 \left(\
begin{array}{cc}
2762 A^{-1} & 0 \\ S^{-1} B
A^{-1} & -S^{-1}
2765would appear to be a good choice since
2767 P^{-1}\left(\
begin{array}{cc}
2771 \left(\
begin{array}{cc}
2772 A^{-1} & 0 \\ S^{-1} B
A^{-1} & -S^{-1}
2773 \end{array}\right)\cdot \left(\
begin{array}{cc}
2777 \left(\
begin{array}{cc}
2778 I &
A^{-1} B^
T \\ 0 &
I
2781This is the approach taken by the paper by Silvester and Wathen referenced
2782to in the introduction (with the exception that Silvester and Wathen use
2783right preconditioning). In
this case, a Krylov-based iterative method would
2784converge in
one step only
if exact inverses of @f$A@f$ and @f$S@f$ were applied,
2785since all the
eigenvalues are
one (and the number of iterations in such a
2786method is bounded by the number of distinct
eigenvalues). Below, we will
2787discuss the choice of an adequate solver
for this problem. First, we are
2788going to have a closer look at the implementation of the preconditioner.
2790Since @f$P@f$ is aimed to be a preconditioner only, we shall use approximations to
2791the inverse of the Schur complement @f$S@f$ and the
matrix @f$A@f$. Hence, the Schur
2792complement will be approximated by the pressure mass
matrix @f$M_p@f$, and we use
2793a preconditioner to @f$A@f$ (without an InverseMatrix
class around it)
for
2794approximating @f$A^{-1}@f$.
2796Here comes the
class that
implements the block Schur
2797complement preconditioner. The <code>vmult</code> operation for block vectors
2798according to the derivation above can be specified by three successive
2801template <
class PreconditionerA,
class PreconditionerMp>
2802class BlockSchurPreconditioner :
public Subscriptor
2807 const PreconditionerA &Apreconditioner);
2815 PreconditionerMp > > m_inverse;
2816 const PreconditionerA &a_preconditioner;
2822template <
class PreconditionerA,
class PreconditionerMp>
2823BlockSchurPreconditioner<PreconditionerA, PreconditionerMp>::BlockSchurPreconditioner(
2826 const PreconditionerA &Apreconditioner
2831 a_preconditioner (Apreconditioner),
2832 tmp (S.block(1,1).m())
2837template <
class PreconditionerA,
class PreconditionerMp>
2838void BlockSchurPreconditioner<PreconditionerA, PreconditionerMp>::vmult (
2843 a_preconditioner.vmult (dst.
block(0), src.
block(0));
2847 system_matrix->block(1,0).residual(tmp, dst.
block(0), src.
block(1));
2852 m_inverse->vmult (dst.
block(1), tmp);
2856Since we act on the whole block system now, we have to live with
one
2857disadvantage: we need to perform the solver iterations on
2858the full block system instead of the smaller pressure space.
2860Now we turn to the question which solver we should use
for the block
2861system. The
first observation is that the resulting preconditioned
matrix cannot
2862be solved with CG since it is neither positive definite nor
symmetric.
2864The deal.II libraries implement several solvers that are appropriate
for the
2865problem at hand. One choice is the solver @ref
SolverBicgstab "BiCGStab", which
2866was used
for the solution of the unsymmetric advection problem in @ref step_9
"step-9". The
2868(generalized minimum residual). Both methods have their pros and cons - there
2869are problems where
one of the two candidates clearly outperforms the other, and
2871<a href=
"http://en.wikipedia.org/wiki/GMRES#Comparison_with_other_solvers">Wikipedia</a>
's
2872article on the GMRES method gives a comparative presentation.
2873A more comprehensive and well-founded comparison can be read e.g. in the book by
2874J.W. Demmel (Applied Numerical Linear Algebra, SIAM, 1997, section 6.6.6).
2876For our specific problem with the ILU preconditioner for @f$A@f$, we certainly need
2877to perform hundreds of iterations on the block system for large problem sizes
2878(we won't beat CG!). Actually,
this disfavors GMRES: During the GMRES
2879iterations, a basis of Krylov vectors is successively built up and some
2880operations are performed on these vectors. The more vectors are in
this basis,
2881the more operations and memory will be needed. The number of operations scales
2882as @f${\cal
O}(n + k^2)@f$ and memory as @f${\cal
O}(kn)@f$, where @f$k@f$ is the number of
2883vectors in the Krylov basis and @f$n@f$ the size of the (block)
matrix.
2884To not let these demands grow excessively, deal.II limits the size @f$k@f$ of the
2885basis to 30 vectors by
default.
2886Then, the basis is rebuilt. This implementation of the GMRES method is called
2887GMRES(k), with
default @f$k=30@f$. What we have gained by
this restriction,
2888namely a bound on operations and memory requirements, will be compensated by
2889the fact that we use an incomplete basis -
this will increase the number of
2892BiCGStab, on the other hand, won
't get slower when many iterations are needed
2893(one iteration uses only results from one preceding step and
2894not all the steps as GMRES). Besides the fact the BiCGStab is more expensive per
2895step since two matrix-vector products are needed (compared to one for
2896CG or GMRES), there is one main reason which makes BiCGStab not appropriate for
2897this problem: The preconditioner applies the inverse of the pressure
2898mass matrix by using the InverseMatrix class. Since the application of the
2899inverse matrix to a vector is done only in approximative way (an exact inverse
2900is too expensive), this will also affect the solver. In the case of BiCGStab,
2901the Krylov vectors will not be orthogonal due to that perturbation. While
2902this is uncritical for a small number of steps (up to about 50), it ruins the
2903performance of the solver when these perturbations have grown to a significant
2904magnitude in the coarse of iterations.
2906We did some experiments with BiCGStab and found it to
2907be faster than GMRES up to refinement cycle 3 (in 3D), but it became very slow
2908for cycles 4 and 5 (even slower than the original Schur complement), so the
2909solver is useless in this situation. Choosing a sharper tolerance for the
2910inverse matrix class (<code>1e-10*src.l2_norm()</code> instead of
2911<code>1e-6*src.l2_norm()</code>) made BiCGStab perform well also for cycle 4,
2912but did not change the failure on the very large problems.
2914GMRES is of course also effected by the approximate inverses, but it is not as
2915sensitive to orthogonality and retains a relatively good performance also for
2916large sizes, see the results below.
2918With this said, we turn to the realization of the solver call with GMRES with
2919@f$k=100@f$ temporary vectors:
2922 const SparseMatrix<double> &pressure_mass_matrix
2923 = preconditioner_matrix.block(1,1);
2924 SparseILU<double> pmass_preconditioner;
2925 pmass_preconditioner.initialize (pressure_mass_matrix,
2926 SparseILU<double>::AdditionalData());
2928 InverseMatrix<SparseMatrix<double>,SparseILU<double> >
2929 m_inverse (pressure_mass_matrix, pmass_preconditioner);
2931 BlockSchurPreconditioner<typename InnerPreconditioner<dim>::type,
2933 preconditioner (system_matrix, m_inverse, *A_preconditioner);
2935 SolverControl solver_control (system_matrix.m(),
2936 1e-6*system_rhs.l2_norm());
2937 GrowingVectorMemory<BlockVector<double> > vector_memory;
2938 SolverGMRES<BlockVector<double> >::AdditionalData gmres_data;
2939 gmres_data.max_n_tmp_vectors = 100;
2941 SolverGMRES<BlockVector<double> > gmres(solver_control, vector_memory,
2944 gmres.solve(system_matrix, solution, system_rhs,
2947 constraints.distribute (solution);
2950 << solver_control.last_step()
2951 << " block GMRES iterations";
2954Obviously, one needs to add the include file @ref SolverGMRES
2955"<lac/solver_gmres.h>" in order to make this run.
2956We call the solver with a BlockVector template in order to enable
2957GMRES to operate on block vectors and matrices.
2958Note also that we need to set the (1,1) block in the system
2959matrix to zero (we saved the pressure mass matrix there which is not part of the
2960problem) after we copied the information to another matrix.
2962Using the Timer class, we collect some statistics that compare the runtime
2963of the block solver with the one from the problem implementation above.
2964Besides the solution with the two options we also check if the solutions
2965of the two variants are close to each other (i.e. this solver gives indeed the
2966same solution as we had before) and calculate the infinity
2967norm of the vector difference.
2969Let's
first see the results in 2D:
2972 Number of active cells: 64
2973 Number of degrees of freedom: 679 (594+85) [0.00162792 s]
2974 Assembling... [0.00108981 s]
2975 Computing preconditioner... [0.0025959 s]
2977 Schur complement: 11 outer CG iterations
for p [0.00479603s ]
2978 Block Schur preconditioner: 12 GMRES iterations [0.00441718 s]
2979 l_infinity difference between solution vectors: 5.38258e-07
2982 Number of active cells: 160
2983 Number of degrees of freedom: 1683 (1482+201) [0.00345707 s]
2984 Assembling... [0.00237417 s]
2985 Computing preconditioner... [0.00605702 s]
2987 Schur complement: 11 outer CG iterations
for p [0.0123992s ]
2988 Block Schur preconditioner: 12 GMRES iterations [0.011909 s]
2989 l_infinity difference between solution vectors: 1.74658e-05
2992 Number of active cells: 376
2993 Number of degrees of freedom: 3813 (3370+443) [0.00729299 s]
2994 Assembling... [0.00529909 s]
2995 Computing preconditioner... [0.0167508 s]
2997 Schur complement: 11 outer CG iterations
for p [0.031672s ]
2998 Block Schur preconditioner: 12 GMRES iterations [0.029232 s]
2999 l_infinity difference between solution vectors: 7.81569e-06
3002 Number of active cells: 880
3003 Number of degrees of freedom: 8723 (7722+1001) [0.017709 s]
3004 Assembling... [0.0126002 s]
3005 Computing preconditioner... [0.0435679 s]
3007 Schur complement: 11 outer CG iterations
for p [0.0971651s ]
3008 Block Schur preconditioner: 12 GMRES iterations [0.0992041 s]
3009 l_infinity difference between solution vectors: 1.87249e-05
3012 Number of active cells: 2008
3013 Number of degrees of freedom: 19383 (17186+2197) [0.039988 s]
3014 Assembling... [0.028281 s]
3015 Computing preconditioner... [0.118314 s]
3017 Schur complement: 11 outer CG iterations
for p [0.252133s ]
3018 Block Schur preconditioner: 13 GMRES iterations [0.269125 s]
3019 l_infinity difference between solution vectors: 6.38657e-05
3022 Number of active cells: 4288
3023 Number of degrees of freedom: 40855 (36250+4605) [0.0880702 s]
3024 Assembling... [0.0603511 s]
3025 Computing preconditioner... [0.278339 s]
3027 Schur complement: 11 outer CG iterations
for p [0.53846s ]
3028 Block Schur preconditioner: 13 GMRES iterations [0.578667 s]
3029 l_infinity difference between solution vectors: 0.000173363
3032We see that there is no huge difference in the solution time between the
3033block Schur complement preconditioner solver and the Schur complement
3034itself. The reason is simple: we used a direct solve as preconditioner
for
3035@f$A@f$ - so we cannot expect any gain by avoiding the inner iterations. We see
3036that the number of iterations has slightly increased
for GMRES, but all in
3037all the two choices are fairly similar.
3039The picture of course changes in 3D:
3043 Number of active cells: 32
3044 Number of degrees of freedom: 1356 (1275+81) [0.00845218 s]
3045 Assembling... [0.019372 s]
3046 Computing preconditioner... [0.00712395 s]
3048 Schur complement: 13 outer CG iterations
for p [0.0320101s ]
3049 Block Schur preconditioner: 22 GMRES iterations [0.0048759 s]
3050 l_infinity difference between solution vectors: 2.15942e-05
3053 Number of active cells: 144
3054 Number of degrees of freedom: 5088 (4827+261) [0.0346942 s]
3055 Assembling... [0.0857739 s]
3056 Computing preconditioner... [0.0465031 s]
3058 Schur complement: 14 outer CG iterations
for p [0.349258s ]
3059 Block Schur preconditioner: 35 GMRES iterations [0.048759 s]
3060 l_infinity difference between solution vectors: 1.77657e-05
3063 Number of active cells: 704
3064 Number of degrees of freedom: 22406 (21351+1055) [0.175669 s]
3065 Assembling... [0.437447 s]
3066 Computing preconditioner... [0.286435 s]
3068 Schur complement: 14 outer CG iterations
for p [3.65519s ]
3069 Block Schur preconditioner: 63 GMRES iterations [0.497787 s]
3070 l_infinity difference between solution vectors: 5.08078e-05
3073 Number of active cells: 3168
3074 Number of degrees of freedom: 93176 (89043+4133) [0.790985 s]
3075 Assembling... [1.97598 s]
3076 Computing preconditioner... [1.4325 s]
3078 Schur complement: 15 outer CG iterations
for p [29.9666s ]
3079 Block Schur preconditioner: 128 GMRES iterations [5.02645 s]
3080 l_infinity difference between solution vectors: 0.000119671
3083 Number of active cells: 11456
3084 Number of degrees of freedom: 327808 (313659+14149) [3.44995 s]
3085 Assembling... [7.54772 s]
3086 Computing preconditioner... [5.46306 s]
3088 Schur complement: 15 outer CG iterations
for p [139.987s ]
3089 Block Schur preconditioner: 255 GMRES iterations [38.0946 s]
3090 l_infinity difference between solution vectors: 0.00020793
3093 Number of active cells: 45056
3094 Number of degrees of freedom: 1254464 (1201371+53093) [19.6795 s]
3095 Assembling... [28.6586 s]
3096 Computing preconditioner... [22.401 s]
3098 Schur complement: 14 outer CG iterations
for p [796.767s ]
3099 Block Schur preconditioner: 524 GMRES iterations [355.597 s]
3100 l_infinity difference between solution vectors: 0.000501219
3103Here, the block preconditioned solver is clearly superior to the Schur
3104complement, but the advantage gets less
for more mesh points. This is
3105because GMRES(k) scales worse with the problem size than CG, as we discussed
3106above. Nonetheless, the improvement by a factor of 3-6
for moderate problem
3107sizes is quite impressive.
3110<a name=
"Combiningtheblockpreconditionerandmultigrid"></a><h5>Combining the block preconditioner and multigrid</h5>
3112An ultimate linear solver
for this problem could be imagined as a
3113combination of an optimal
3114preconditioner
for @f$A@f$ (
e.g. multigrid) and the block preconditioner
3115described above, which is the approach taken in the @ref step_31
"step-31"
3116and @ref step_32
"step-32" tutorial programs (where we use an algebraic multigrid
3117method) and @ref step_56
"step-56" (where we use a geometric multigrid method).
3120<a name=
"Noblockmatricesandvectors"></a><h5>No block matrices and vectors</h5>
3122Another possibility that can be taken into account is to not
set up a block
3123system, but rather solve the system of velocity and pressure all at once. The
3124options are direct solve with UMFPACK (2D) or GMRES with ILU
3125preconditioning (3D). It should be straightforward to
try that.
3129<a name=
"Moreinterestingtestcases"></a><h4>More interesting testcases</h4>
3132The program can of course also serve as a basis to compute the flow in more
3133interesting cases. The original motivation to write
this program was
for it to
3134be a starting
point for some geophysical flow problems, such as the
3135movement of magma under places where continental plates drift apart (
for
3136example mid-ocean ridges). Of course, in such places, the geometry is more
3137complicated than the examples shown above, but it is not hard to accommodate
3140For example, by
using the following modification of the boundary
values
3145BoundaryValues<dim>::value (
const Point<dim> &p,
3146 const unsigned int component)
const
3148 Assert (component < this->n_components,
3151 const double x_offset =
std::atan(p[1]*4)/3;
3154 return (p[0] < x_offset ? -1 : (p[0] > x_offset ? 1 : 0));
3158and the following way to generate the mesh as the domain
3159@f$[-2,2]\times[-2,2]\times[-1,0]@f$
3161 std::vector<unsigned int> subdivisions (dim, 1);
3162 subdivisions[0] = 4;
3164 subdivisions[1] = 4;
3168 Point<dim>(-2,-2,-1));
3178then we get images where the fault line is curved:
3179<table width=
"60%" align=
"center">
3182 <img src=
"https://www.dealii.org/images/steps/developer/step-22.3d-extension.png" alt=
"">
3185 <img src=
"https://www.dealii.org/images/steps/developer/step-22.3d-grid-extension.png" alt=
"">
3191<a name=
"PlainProg"></a>
3192<h1> The plain program</h1>
3193@include
"step-22.cc"
constexpr SymmetricTensor< 2, dim, Number > invert(const SymmetricTensor< 2, dim, Number > &t)
std::array< Number, 1 > eigenvalues(const SymmetricTensor< 2, 1, Number > &T)
__global__ void set(Number *val, const Number s, const size_type N)
static ::ExceptionBase & ExcIndexRange(int arg1, int arg2, int arg3)
#define Assert(cond, exc)
BlockType & block(const unsigned int i)
void loop(ITERATOR begin, typename identity< ITERATOR >::type end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(DOFINFO &, DOFINFO &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, ASSEMBLER &assembler, const LoopControl &lctrl=LoopControl())
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
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)
void approximate(SynchronousIterators< std::tuple< typename DoFHandler< dim, spacedim >::active_cell_iterator, Vector< float >::iterator > > const &cell, const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof_handler, const InputVector &solution, const unsigned int component)
Expression atan(const Expression &x)
void king_ordering(DoFHandler< dim, spacedim > &dof_handler, const bool reversed_numbering=false, const bool use_constraints=false)
void component_wise(DoFHandler< dim, spacedim > &dof_handler, const std::vector< unsigned int > &target_component=std::vector< unsigned int >())
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 >())
void subdivided_hyper_rectangle(Triangulation< dim, spacedim > &tria, const std::vector< unsigned int > &repetitions, const Point< dim > &p1, const Point< dim > &p2, const bool colorize=false)
static const types::blas_int zero
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
static const types::blas_int one
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
constexpr ReturnType< rank, T >::value_type & extract(T &t, const ArrayType &indices)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
VectorType::value_type * end(VectorType &V)
VectorType::value_type * begin(VectorType &V)
int(&) functions(const void *v1, const void *v2)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation