Reference documentation for deal.II version 9.5.0
\(\newcommand{\dealvcentcolon}{\mathrel{\mathop{:}}}\) \(\newcommand{\dealcoloneq}{\dealvcentcolon\mathrel{\mkern-1.2mu}=}\) \(\newcommand{\jump}[1]{\left[\!\left[ #1 \right]\!\right]}\) \(\newcommand{\average}[1]{\left\{\!\left\{ #1 \right\}\!\right\}}\)
Loading...
Searching...
No Matches
step-22.h
Go to the documentation of this file.
1) const
1343 *   {
1344 *   return Tensor<1, dim>();
1345 *   }
1346 *  
1347 *  
1348 *   template <int dim>
1349 *   void RightHandSide<dim>::value_list(const std::vector<Point<dim>> &vp,
1350 *   std::vector<Tensor<1, dim>> &values) const
1351 *   {
1352 *   for (unsigned int c = 0; c < vp.size(); ++c)
1353 *   {
1354 *   values[c] = RightHandSide<dim>::value(vp[c]);
1355 *   }
1356 *   }
1357 *  
1358 *  
1359 * @endcode
1360 *
1361 *
1362 * <a name="Linearsolversandpreconditioners"></a>
1363 * <h3>Linear solvers and preconditioners</h3>
1364 *
1365
1366 *
1367 * The linear solvers and preconditioners are discussed extensively in the
1368 * introduction. Here, we create the respective objects that will be used.
1369 *
1370
1371 *
1372 *
1373 * <a name="ThecodeInverseMatrixcodeclasstemplate"></a>
1374 * <h4>The <code>InverseMatrix</code> class template</h4>
1375 * The <code>InverseMatrix</code> class represents the data structure for an
1376 * inverse matrix. Unlike @ref step_20 "step-20", we implement this with a class instead of
1377 * the helper function inverse_linear_operator() we will apply this class to
1378 * different kinds of matrices that will require different preconditioners
1379 * (in @ref step_20 "step-20" we only used a non-identity preconditioner for the mass
1380 * matrix). The types of matrix and preconditioner are passed to this class
1381 * via template parameters, and matrix and preconditioner objects of these
1382 * types will then be passed to the constructor when an
1383 * <code>InverseMatrix</code> object is created. The member function
1384 * <code>vmult</code> is obtained by solving a linear system:
1385 *
1386 * @code
1387 *   template <class MatrixType, class PreconditionerType>
1388 *   class InverseMatrix : public Subscriptor
1389 *   {
1390 *   public:
1391 *   InverseMatrix(const MatrixType & m,
1392 *   const PreconditionerType &preconditioner);
1393 *  
1394 *   void vmult(Vector<double> &dst, const Vector<double> &src) const;
1395 *  
1396 *   private:
1398 *   const SmartPointer<const PreconditionerType> preconditioner;
1399 *   };
1400 *  
1401 *  
1402 *   template <class MatrixType, class PreconditionerType>
1403 *   InverseMatrix<MatrixType, PreconditionerType>::InverseMatrix(
1404 *   const MatrixType & m,
1405 *   const PreconditionerType &preconditioner)
1406 *   : matrix(&m)
1407 *   , preconditioner(&preconditioner)
1408 *   {}
1409 *  
1410 *  
1411 * @endcode
1412 *
1413 * This is the implementation of the <code>vmult</code> function.
1414 *
1415
1416 *
1417 * In this class we use a rather large tolerance for the solver control. The
1418 * reason for this is that the function is used very frequently, and hence,
1419 * any additional effort to make the residual in the CG solve smaller makes
1420 * the solution more expensive. Note that we do not only use this class as a
1421 * preconditioner for the Schur complement, but also when forming the
1422 * inverse of the Laplace matrix &ndash; which is hence directly responsible
1423 * for the accuracy of the solution itself, so we can't choose a too large
1424 * tolerance, either.
1425 *
1426 * @code
1427 *   template <class MatrixType, class PreconditionerType>
1428 *   void InverseMatrix<MatrixType, PreconditionerType>::vmult(
1429 *   Vector<double> & dst,
1430 *   const Vector<double> &src) const
1431 *   {
1432 *   SolverControl solver_control(src.size(), 1e-6 * src.l2_norm());
1433 *   SolverCG<Vector<double>> cg(solver_control);
1434 *  
1435 *   dst = 0;
1436 *  
1437 *   cg.solve(*matrix, dst, src, *preconditioner);
1438 *   }
1439 *  
1440 *  
1441 * @endcode
1442 *
1443 *
1444 * <a name="ThecodeSchurComplementcodeclasstemplate"></a>
1445 * <h4>The <code>SchurComplement</code> class template</h4>
1446 *
1447
1448 *
1449 * This class implements the Schur complement discussed in the introduction.
1450 * It is in analogy to @ref step_20 "step-20". Though, we now call it with a template
1451 * parameter <code>PreconditionerType</code> in order to access that when
1452 * specifying the respective type of the inverse matrix class. As a
1453 * consequence of the definition above, the declaration
1454 * <code>InverseMatrix</code> now contains the second template parameter for
1455 * a preconditioner class as above, which affects the
1456 * <code>SmartPointer</code> object <code>m_inverse</code> as well.
1457 *
1458 * @code
1459 *   template <class PreconditionerType>
1460 *   class SchurComplement : public Subscriptor
1461 *   {
1462 *   public:
1463 *   SchurComplement(
1464 *   const BlockSparseMatrix<double> &system_matrix,
1465 *   const InverseMatrix<SparseMatrix<double>, PreconditionerType> &A_inverse);
1466 *  
1467 *   void vmult(Vector<double> &dst, const Vector<double> &src) const;
1468 *  
1469 *   private:
1470 *   const SmartPointer<const BlockSparseMatrix<double>> system_matrix;
1471 *   const SmartPointer<
1472 *   const InverseMatrix<SparseMatrix<double>, PreconditionerType>>
1473 *   A_inverse;
1474 *  
1475 *   mutable Vector<double> tmp1, tmp2;
1476 *   };
1477 *  
1478 *  
1479 *  
1480 *   template <class PreconditionerType>
1481 *   SchurComplement<PreconditionerType>::SchurComplement(
1482 *   const BlockSparseMatrix<double> &system_matrix,
1483 *   const InverseMatrix<SparseMatrix<double>, PreconditionerType> &A_inverse)
1484 *   : system_matrix(&system_matrix)
1485 *   , A_inverse(&A_inverse)
1486 *   , tmp1(system_matrix.block(0, 0).m())
1487 *   , tmp2(system_matrix.block(0, 0).m())
1488 *   {}
1489 *  
1490 *  
1491 *   template <class PreconditionerType>
1492 *   void
1493 *   SchurComplement<PreconditionerType>::vmult(Vector<double> & dst,
1494 *   const Vector<double> &src) const
1495 *   {
1496 *   system_matrix->block(0, 1).vmult(tmp1, src);
1497 *   A_inverse->vmult(tmp2, tmp1);
1498 *   system_matrix->block(1, 0).vmult(dst, tmp2);
1499 *   }
1500 *  
1501 *  
1502 * @endcode
1503 *
1504 *
1505 * <a name="StokesProblemclassimplementation"></a>
1506 * <h3>StokesProblem class implementation</h3>
1507 *
1508
1509 *
1510 *
1511 * <a name="StokesProblemStokesProblem"></a>
1512 * <h4>StokesProblem::StokesProblem</h4>
1513 *
1514
1515 *
1516 * The constructor of this class looks very similar to the one of
1517 * @ref step_20 "step-20". The constructor initializes the variables for the polynomial
1518 * degree, triangulation, finite element system and the dof handler. The
1519 * underlying polynomial functions are of order <code>degree+1</code> for
1520 * the vector-valued velocity components and of order <code>degree</code>
1521 * for the pressure. This gives the LBB-stable element pair
1522 * @f$Q_{degree+1}^d\times Q_{degree}@f$, often referred to as the Taylor-Hood
1523 * element.
1524 *
1525
1526 *
1527 * Note that we initialize the triangulation with a MeshSmoothing argument,
1528 * which ensures that the refinement of cells is done in a way that the
1529 * approximation of the PDE solution remains well-behaved (problems arise if
1530 * grids are too unstructured), see the documentation of
1531 * <code>Triangulation::MeshSmoothing</code> for details.
1532 *
1533 * @code
1534 *   template <int dim>
1535 *   StokesProblem<dim>::StokesProblem(const unsigned int degree)
1536 *   : degree(degree)
1537 *   , triangulation(Triangulation<dim>::maximum_smoothing)
1538 *   , fe(FE_Q<dim>(degree + 1), dim, FE_Q<dim>(degree), 1)
1539 *   , dof_handler(triangulation)
1540 *   {}
1541 *  
1542 *  
1543 * @endcode
1544 *
1545 *
1546 * <a name="StokesProblemsetup_dofs"></a>
1547 * <h4>StokesProblem::setup_dofs</h4>
1548 *
1549
1550 *
1551 * Given a mesh, this function associates the degrees of freedom with it and
1552 * creates the corresponding matrices and vectors. At the beginning it also
1553 * releases the pointer to the preconditioner object (if the shared pointer
1554 * pointed at anything at all at this point) since it will definitely not be
1555 * needed any more after this point and will have to be re-computed after
1556 * assembling the matrix, and unties the sparse matrices from their sparsity
1557 * pattern objects.
1558 *
1559
1560 *
1561 * We then proceed with distributing degrees of freedom and renumbering
1562 * them: In order to make the ILU preconditioner (in 3d) work efficiently,
1563 * it is important to enumerate the degrees of freedom in such a way that it
1564 * reduces the bandwidth of the matrix, or maybe more importantly: in such a
1565 * way that the ILU is as close as possible to a real LU decomposition. On
1566 * the other hand, we need to preserve the block structure of velocity and
1567 * pressure already seen in @ref step_20 "step-20" and @ref step_21 "step-21". This is done in two
1568 * steps: First, all dofs are renumbered to improve the ILU and then we
1569 * renumber once again by components. Since
1570 * <code>DoFRenumbering::component_wise</code> does not touch the
1571 * renumbering within the individual blocks, the basic renumbering from the
1572 * first step remains. As for how the renumber degrees of freedom to improve
1573 * the ILU: deal.II has a number of algorithms that attempt to find
1574 * orderings to improve ILUs, or reduce the bandwidth of matrices, or
1575 * optimize some other aspect. The DoFRenumbering namespace shows a
1576 * comparison of the results we obtain with several of these algorithms
1577 * based on the testcase discussed here in this tutorial program. Here, we
1578 * will use the traditional Cuthill-McKee algorithm already used in some of
1579 * the previous tutorial programs. In the <a href="#improved-ilu">section
1580 * on improved ILU</a> we're going to discuss this issue in more detail.
1581 *
1582
1583 *
1584 * There is one more change compared to previous tutorial programs: There is
1585 * no reason in sorting the <code>dim</code> velocity components
1586 * individually. In fact, rather than first enumerating all @f$x@f$-velocities,
1587 * then all @f$y@f$-velocities, etc, we would like to keep all velocities at the
1588 * same location together and only separate between velocities (all
1589 * components) and pressures. By default, this is not what the
1590 * DoFRenumbering::component_wise function does: it treats each vector
1591 * component separately; what we have to do is group several components into
1592 * "blocks" and pass this block structure to that function. Consequently, we
1593 * allocate a vector <code>block_component</code> with as many elements as
1594 * there are components and describe all velocity components to correspond
1595 * to block 0, while the pressure component will form block 1:
1596 *
1597 * @code
1598 *   template <int dim>
1599 *   void StokesProblem<dim>::setup_dofs()
1600 *   {
1601 *   A_preconditioner.reset();
1602 *   system_matrix.clear();
1603 *   preconditioner_matrix.clear();
1604 *  
1605 *   dof_handler.distribute_dofs(fe);
1606 *   DoFRenumbering::Cuthill_McKee(dof_handler);
1607 *  
1608 *   std::vector<unsigned int> block_component(dim + 1, 0);
1609 *   block_component[dim] = 1;
1610 *   DoFRenumbering::component_wise(dof_handler, block_component);
1611 *  
1612 * @endcode
1613 *
1614 * Now comes the implementation of Dirichlet boundary conditions, which
1615 * should be evident after the discussion in the introduction. All that
1616 * changed is that the function already appears in the setup functions,
1617 * whereas we were used to see it in some assembly routine. Further down
1618 * below where we set up the mesh, we will associate the top boundary
1619 * where we impose Dirichlet boundary conditions with boundary indicator
1620 * 1. We will have to pass this boundary indicator as second argument to
1621 * the function below interpolating boundary values. There is one more
1622 * thing, though. The function describing the Dirichlet conditions was
1623 * defined for all components, both velocity and pressure. However, the
1624 * Dirichlet conditions are to be set for the velocity only. To this end,
1625 * we use a ComponentMask that only selects the velocity components. The
1626 * component mask is obtained from the finite element by specifying the
1627 * particular components we want. Since we use adaptively refined grids,
1628 * the affine constraints object needs to be first filled with hanging node
1629 * constraints generated from the DoF handler. Note the order of the two
1630 * functions &mdash; we first compute the hanging node constraints, and
1631 * then insert the boundary values into the constraints object. This makes
1632 * sure that we respect H<sup>1</sup> conformity on boundaries with
1633 * hanging nodes (in three space dimensions), where the hanging node needs
1634 * to dominate the Dirichlet boundary values.
1635 *
1636 * @code
1637 *   {
1638 *   constraints.clear();
1639 *  
1640 *   const FEValuesExtractors::Vector velocities(0);
1641 *   DoFTools::make_hanging_node_constraints(dof_handler, constraints);
1643 *   1,
1644 *   BoundaryValues<dim>(),
1645 *   constraints,
1646 *   fe.component_mask(velocities));
1647 *   }
1648 *  
1649 *   constraints.close();
1650 *  
1651 * @endcode
1652 *
1653 * In analogy to @ref step_20 "step-20", we count the dofs in the individual components.
1654 * We could do this in the same way as there, but we want to operate on
1655 * the block structure we used already for the renumbering: The function
1656 * <code>DoFTools::count_dofs_per_fe_block</code> does the same as
1657 * <code>DoFTools::count_dofs_per_fe_component</code>, but now grouped as
1658 * velocity and pressure block via <code>block_component</code>.
1659 *
1660 * @code
1661 *   const std::vector<types::global_dof_index> dofs_per_block =
1662 *   DoFTools::count_dofs_per_fe_block(dof_handler, block_component);
1663 *   const types::global_dof_index n_u = dofs_per_block[0];
1664 *   const types::global_dof_index n_p = dofs_per_block[1];
1665 *  
1666 *   std::cout << " Number of active cells: " << triangulation.n_active_cells()
1667 *   << std::endl
1668 *   << " Number of degrees of freedom: " << dof_handler.n_dofs()
1669 *   << " (" << n_u << '+' << n_p << ')' << std::endl;
1670 *  
1671 * @endcode
1672 *
1673 * The next task is to allocate a sparsity pattern for the system matrix we
1674 * will create and one for the preconditioner matrix. We could do this in
1675 * the same way as in @ref step_20 "step-20", i.e. directly build an object of type
1676 * SparsityPattern through DoFTools::make_sparsity_pattern. However, there
1677 * is a major reason not to do so:
1678 * In 3d, the function DoFTools::max_couplings_between_dofs yields a
1679 * conservative but rather large number for the coupling between the
1680 * individual dofs, so that the memory initially provided for the creation
1681 * of the sparsity pattern of the matrix is far too much -- so much actually
1682 * that the initial sparsity pattern won't even fit into the physical memory
1683 * of most systems already for moderately-sized 3d problems, see also the
1684 * discussion in @ref step_18 "step-18". Instead, we first build temporary objects that use
1685 * a different data structure that doesn't require allocating more memory
1686 * than necessary but isn't suitable for use as a basis of SparseMatrix or
1687 * BlockSparseMatrix objects; in a second step we then copy these objects
1688 * into objects of type BlockSparsityPattern. This is entirely analogous to
1689 * what we already did in @ref step_11 "step-11" and @ref step_18 "step-18". In particular, we make use of
1690 * the fact that we will never write into the @f$(1,1)@f$ block of the system
1691 * matrix and that this is the only block to be filled for the
1692 * preconditioner matrix.
1693 *
1694
1695 *
1696 * All this is done inside new scopes, which means that the memory of
1697 * <code>dsp</code> will be released once the information has been copied to
1698 * <code>sparsity_pattern</code>.
1699 *
1700 * @code
1701 *   {
1702 *   BlockDynamicSparsityPattern dsp(dofs_per_block, dofs_per_block);
1703 *  
1704 *   Table<2, DoFTools::Coupling> coupling(dim + 1, dim + 1);
1705 *   for (unsigned int c = 0; c < dim + 1; ++c)
1706 *   for (unsigned int d = 0; d < dim + 1; ++d)
1707 *   if (!((c == dim) && (d == dim)))
1708 *   coupling[c][d] = DoFTools::always;
1709 *   else
1710 *   coupling[c][d] = DoFTools::none;
1711 *  
1712 *   DoFTools::make_sparsity_pattern(
1713 *   dof_handler, coupling, dsp, constraints, false);
1714 *  
1715 *   sparsity_pattern.copy_from(dsp);
1716 *   }
1717 *  
1718 *   {
1719 *   BlockDynamicSparsityPattern preconditioner_dsp(dofs_per_block,
1720 *   dofs_per_block);
1721 *  
1722 *   Table<2, DoFTools::Coupling> preconditioner_coupling(dim + 1, dim + 1);
1723 *   for (unsigned int c = 0; c < dim + 1; ++c)
1724 *   for (unsigned int d = 0; d < dim + 1; ++d)
1725 *   if (((c == dim) && (d == dim)))
1726 *   preconditioner_coupling[c][d] = DoFTools::always;
1727 *   else
1728 *   preconditioner_coupling[c][d] = DoFTools::none;
1729 *  
1730 *   DoFTools::make_sparsity_pattern(dof_handler,
1731 *   preconditioner_coupling,
1732 *   preconditioner_dsp,
1733 *   constraints,
1734 *   false);
1735 *  
1736 *   preconditioner_sparsity_pattern.copy_from(preconditioner_dsp);
1737 *   }
1738 *  
1739 * @endcode
1740 *
1741 * Finally, the system matrix, the preconsitioner matrix, the solution and
1742 * the right hand side vector are created from the block structure similar
1743 * to the approach in @ref step_20 "step-20":
1744 *
1745 * @code
1746 *   system_matrix.reinit(sparsity_pattern);
1747 *   preconditioner_matrix.reinit(preconditioner_sparsity_pattern);
1748 *  
1749 *   solution.reinit(dofs_per_block);
1750 *   system_rhs.reinit(dofs_per_block);
1751 *   }
1752 *  
1753 *  
1754 * @endcode
1755 *
1756 *
1757 * <a name="StokesProblemassemble_system"></a>
1758 * <h4>StokesProblem::assemble_system</h4>
1759 *
1760
1761 *
1762 * The assembly process follows the discussion in @ref step_20 "step-20" and in the
1763 * introduction. We use the well-known abbreviations for the data structures
1764 * that hold the local matrices, right hand side, and global numbering of the
1765 * degrees of freedom for the present cell.
1766 *
1767 * @code
1768 *   template <int dim>
1769 *   void StokesProblem<dim>::assemble_system()
1770 *   {
1771 *   system_matrix = 0;
1772 *   system_rhs = 0;
1773 *   preconditioner_matrix = 0;
1774 *  
1775 *   QGauss<dim> quadrature_formula(degree + 2);
1776 *  
1777 *   FEValues<dim> fe_values(fe,
1778 *   quadrature_formula,
1779 *   update_values | update_quadrature_points |
1780 *   update_JxW_values | update_gradients);
1781 *  
1782 *   const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
1783 *  
1784 *   const unsigned int n_q_points = quadrature_formula.size();
1785 *  
1786 *   FullMatrix<double> local_matrix(dofs_per_cell, dofs_per_cell);
1787 *   FullMatrix<double> local_preconditioner_matrix(dofs_per_cell,
1788 *   dofs_per_cell);
1789 *   Vector<double> local_rhs(dofs_per_cell);
1790 *  
1791 *   std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
1792 *  
1793 *   const RightHandSide<dim> right_hand_side;
1794 *   std::vector<Tensor<1, dim>> rhs_values(n_q_points, Tensor<1, dim>());
1795 *  
1796 * @endcode
1797 *
1798 * Next, we need two objects that work as extractors for the FEValues
1799 * object. Their use is explained in detail in the report on @ref
1800 * vector_valued :
1801 *
1802 * @code
1803 *   const FEValuesExtractors::Vector velocities(0);
1804 *   const FEValuesExtractors::Scalar pressure(dim);
1805 *  
1806 * @endcode
1807 *
1808 * As an extension over @ref step_20 "step-20" and @ref step_21 "step-21", we include a few optimizations
1809 * that make assembly much faster for this particular problem. The
1810 * improvements are based on the observation that we do a few calculations
1811 * too many times when we do as in @ref step_20 "step-20": The symmetric gradient actually
1812 * has <code>dofs_per_cell</code> different values per quadrature point, but
1813 * we extract it <code>dofs_per_cell*dofs_per_cell</code> times from the
1814 * FEValues object - for both the loop over <code>i</code> and the inner
1815 * loop over <code>j</code>. In 3d, that means evaluating it @f$89^2=7921@f$
1816 * instead of @f$89@f$ times, a not insignificant difference.
1817 *
1818
1819 *
1820 * So what we're going to do here is to avoid such repeated calculations
1821 * by getting a vector of rank-2 tensors (and similarly for the divergence
1822 * and the basis function value on pressure) at the quadrature point prior
1823 * to starting the loop over the dofs on the cell. First, we create the
1824 * respective objects that will hold these values. Then, we start the loop
1825 * over all cells and the loop over the quadrature points, where we first
1826 * extract these values. There is one more optimization we implement here:
1827 * the local matrix (as well as the global one) is going to be symmetric,
1828 * since all the operations involved are symmetric with respect to @f$i@f$ and
1829 * @f$j@f$. This is implemented by simply running the inner loop not to
1830 * <code>dofs_per_cell</code>, but only up to <code>i</code>, the index of
1831 * the outer loop.
1832 *
1833 * @code
1834 *   std::vector<SymmetricTensor<2, dim>> symgrad_phi_u(dofs_per_cell);
1835 *   std::vector<double> div_phi_u(dofs_per_cell);
1836 *   std::vector<Tensor<1, dim>> phi_u(dofs_per_cell);
1837 *   std::vector<double> phi_p(dofs_per_cell);
1838 *  
1839 *   for (const auto &cell : dof_handler.active_cell_iterators())
1840 *   {
1841 *   fe_values.reinit(cell);
1842 *   local_matrix = 0;
1843 *   local_preconditioner_matrix = 0;
1844 *   local_rhs = 0;
1845 *  
1846 *   right_hand_side.value_list(fe_values.get_quadrature_points(),
1847 *   rhs_values);
1848 *  
1849 *   for (unsigned int q = 0; q < n_q_points; ++q)
1850 *   {
1851 *   for (unsigned int k = 0; k < dofs_per_cell; ++k)
1852 *   {
1853 *   symgrad_phi_u[k] =
1854 *   fe_values[velocities].symmetric_gradient(k, q);
1855 *   div_phi_u[k] = fe_values[velocities].divergence(k, q);
1856 *   phi_u[k] = fe_values[velocities].value(k, q);
1857 *   phi_p[k] = fe_values[pressure].value(k, q);
1858 *   }
1859 *  
1860 * @endcode
1861 *
1862 * Now finally for the bilinear forms of both the system matrix and
1863 * the matrix we use for the preconditioner. Recall that the
1864 * formulas for these two are
1865 * @f{align*}{
1866 * A_{ij} &= a(\varphi_i,\varphi_j)
1867 * \\ &= \underbrace{2(\varepsilon(\varphi_{i,\textbf{u}}),
1868 * \varepsilon(\varphi_{j,\textbf{u}}))_{\Omega}}
1869 * _{(1)}
1870 * \;
1871 * \underbrace{- (\textrm{div}\; \varphi_{i,\textbf{u}},
1872 * \varphi_{j,p})_{\Omega}}
1873 * _{(2)}
1874 * \;
1875 * \underbrace{- (\varphi_{i,p},
1876 * \textrm{div}\;
1877 * \varphi_{j,\textbf{u}})_{\Omega}}
1878 * _{(3)}
1879 * @f}
1880 * and
1881 * @f{align*}{
1882 * M_{ij} &= \underbrace{(\varphi_{i,p},
1883 * \varphi_{j,p})_{\Omega}}
1884 * _{(4)},
1885 * @f}
1886 * respectively, where @f$\varphi_{i,\textbf{u}}@f$ and @f$\varphi_{i,p}@f$
1887 * are the velocity and pressure components of the @f$i@f$th shape
1888 * function. The various terms above are then easily recognized in
1889 * the following implementation:
1890 *
1891 * @code
1892 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
1893 *   {
1894 *   for (unsigned int j = 0; j <= i; ++j)
1895 *   {
1896 *   local_matrix(i, j) +=
1897 *   (2 * (symgrad_phi_u[i] * symgrad_phi_u[j]) // (1)
1898 *   - div_phi_u[i] * phi_p[j] // (2)
1899 *   - phi_p[i] * div_phi_u[j]) // (3)
1900 *   * fe_values.JxW(q); // * dx
1901 *  
1902 *   local_preconditioner_matrix(i, j) +=
1903 *   (phi_p[i] * phi_p[j]) // (4)
1904 *   * fe_values.JxW(q); // * dx
1905 *   }
1906 * @endcode
1907 *
1908 * Note that in the implementation of (1) above, `operator*`
1909 * is overloaded for symmetric tensors, yielding the scalar
1910 * product between the two tensors.
1911 *
1912
1913 *
1914 * For the right-hand side, we need to multiply the (vector of)
1915 * velocity shape functions with the vector of body force
1916 * right-hand side components, both evaluated at the current
1917 * quadrature point. We have implemented the body forces as a
1918 * `TensorFunction<1,dim>`, so its values at quadrature points
1919 * are already tensors for which the application of `operator*`
1920 * against the velocity components of the shape function results
1921 * in the dot product, as intended.
1922 *
1923 * @code
1924 *   local_rhs(i) += phi_u[i] // phi_u_i(x_q)
1925 *   * rhs_values[q] // * f(x_q)
1926 *   * fe_values.JxW(q); // * dx
1927 *   }
1928 *   }
1929 *  
1930 * @endcode
1931 *
1932 * Before we can write the local data into the global matrix (and
1933 * simultaneously use the AffineConstraints object to apply
1934 * Dirichlet boundary conditions and eliminate hanging node constraints,
1935 * as we discussed in the introduction), we have to be careful about one
1936 * thing, though. We have only built half of the local matrices
1937 * because of symmetry, but we're going to save the full matrices
1938 * in order to use the standard functions for solving. This is done
1939 * by flipping the indices in case we are pointing into the empty part
1940 * of the local matrices.
1941 *
1942 * @code
1943 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
1944 *   for (unsigned int j = i + 1; j < dofs_per_cell; ++j)
1945 *   {
1946 *   local_matrix(i, j) = local_matrix(j, i);
1947 *   local_preconditioner_matrix(i, j) =
1948 *   local_preconditioner_matrix(j, i);
1949 *   }
1950 *  
1951 *   cell->get_dof_indices(local_dof_indices);
1952 *   constraints.distribute_local_to_global(local_matrix,
1953 *   local_rhs,
1954 *   local_dof_indices,
1955 *   system_matrix,
1956 *   system_rhs);
1957 *   constraints.distribute_local_to_global(local_preconditioner_matrix,
1958 *   local_dof_indices,
1959 *   preconditioner_matrix);
1960 *   }
1961 *  
1962 * @endcode
1963 *
1964 * Before we're going to solve this linear system, we generate a
1965 * preconditioner for the velocity-velocity matrix, i.e.,
1966 * <code>block(0,0)</code> in the system matrix. As mentioned above, this
1967 * depends on the spatial dimension. Since the two classes described by
1968 * the <code>InnerPreconditioner::type</code> alias have the same
1969 * interface, we do not have to do anything different whether we want to
1970 * use a sparse direct solver or an ILU:
1971 *
1972 * @code
1973 *   std::cout << " Computing preconditioner..." << std::endl << std::flush;
1974 *  
1975 *   A_preconditioner =
1976 *   std::make_shared<typename InnerPreconditioner<dim>::type>();
1977 *   A_preconditioner->initialize(
1978 *   system_matrix.block(0, 0),
1979 *   typename InnerPreconditioner<dim>::type::AdditionalData());
1980 *   }
1981 *  
1982 *  
1983 *  
1984 * @endcode
1985 *
1986 *
1987 * <a name="StokesProblemsolve"></a>
1988 * <h4>StokesProblem::solve</h4>
1989 *
1990
1991 *
1992 * After the discussion in the introduction and the definition of the
1993 * respective classes above, the implementation of the <code>solve</code>
1994 * function is rather straight-forward and done in a similar way as in
1995 * @ref step_20 "step-20". To start with, we need an object of the
1996 * <code>InverseMatrix</code> class that represents the inverse of the
1997 * matrix A. As described in the introduction, the inverse is generated with
1998 * the help of an inner preconditioner of type
1999 * <code>InnerPreconditioner::type</code>.
2000 *
2001 * @code
2002 *   template <int dim>
2003 *   void StokesProblem<dim>::solve()
2004 *   {
2005 *   const InverseMatrix<SparseMatrix<double>,
2006 *   typename InnerPreconditioner<dim>::type>
2007 *   A_inverse(system_matrix.block(0, 0), *A_preconditioner);
2008 *   Vector<double> tmp(solution.block(0).size());
2009 *  
2010 * @endcode
2011 *
2012 * This is as in @ref step_20 "step-20". We generate the right hand side @f$B A^{-1} F - G@f$
2013 * for the Schur complement and an object that represents the respective
2014 * linear operation @f$B A^{-1} B^T@f$, now with a template parameter
2015 * indicating the preconditioner - in accordance with the definition of
2016 * the class.
2017 *
2018 * @code
2019 *   {
2020 *   Vector<double> schur_rhs(solution.block(1).size());
2021 *   A_inverse.vmult(tmp, system_rhs.block(0));
2022 *   system_matrix.block(1, 0).vmult(schur_rhs, tmp);
2023 *   schur_rhs -= system_rhs.block(1);
2024 *  
2025 *   SchurComplement<typename InnerPreconditioner<dim>::type> schur_complement(
2026 *   system_matrix, A_inverse);
2027 *  
2028 * @endcode
2029 *
2030 * The usual control structures for the solver call are created...
2031 *
2032 * @code
2033 *   SolverControl solver_control(solution.block(1).size(),
2034 *   1e-6 * schur_rhs.l2_norm());
2035 *   SolverCG<Vector<double>> cg(solver_control);
2036 *  
2037 * @endcode
2038 *
2039 * Now to the preconditioner to the Schur complement. As explained in
2040 * the introduction, the preconditioning is done by a @ref GlossMassMatrix "mass matrix" in the
2041 * pressure variable.
2042 *
2043
2044 *
2045 * Actually, the solver needs to have the preconditioner in the form
2046 * @f$P^{-1}@f$, so we need to create an inverse operation. Once again, we
2047 * use an object of the class <code>InverseMatrix</code>, which
2048 * implements the <code>vmult</code> operation that is needed by the
2049 * solver. In this case, we have to invert the pressure mass matrix. As
2050 * it already turned out in earlier tutorial programs, the inversion of
2051 * a mass matrix is a rather cheap and straight-forward operation
2052 * (compared to, e.g., a Laplace matrix). The CG method with ILU
2053 * preconditioning converges in 5-10 steps, independently on the mesh
2054 * size. This is precisely what we do here: We choose another ILU
2055 * preconditioner and take it along to the InverseMatrix object via the
2056 * corresponding template parameter. A CG solver is then called within
2057 * the vmult operation of the inverse matrix.
2058 *
2059
2060 *
2061 * An alternative that is cheaper to build, but needs more iterations
2062 * afterwards, would be to choose a SSOR preconditioner with factor
2063 * 1.2. It needs about twice the number of iterations, but the costs for
2064 * its generation are almost negligible.
2065 *
2066 * @code
2067 *   SparseILU<double> preconditioner;
2068 *   preconditioner.initialize(preconditioner_matrix.block(1, 1),
2069 *   SparseILU<double>::AdditionalData());
2070 *  
2071 *   InverseMatrix<SparseMatrix<double>, SparseILU<double>> m_inverse(
2072 *   preconditioner_matrix.block(1, 1), preconditioner);
2073 *  
2074 * @endcode
2075 *
2076 * With the Schur complement and an efficient preconditioner at hand, we
2077 * can solve the respective equation for the pressure (i.e. block 0 in
2078 * the solution vector) in the usual way:
2079 *
2080 * @code
2081 *   cg.solve(schur_complement, solution.block(1), schur_rhs, m_inverse);
2082 *  
2083 * @endcode
2084 *
2085 * After this first solution step, the hanging node constraints have to
2086 * be distributed to the solution in order to achieve a consistent
2087 * pressure field.
2088 *
2089 * @code
2090 *   constraints.distribute(solution);
2091 *  
2092 *   std::cout << " " << solver_control.last_step()
2093 *   << " outer CG Schur complement iterations for pressure"
2094 *   << std::endl;
2095 *   }
2096 *  
2097 * @endcode
2098 *
2099 * As in @ref step_20 "step-20", we finally need to solve for the velocity equation where
2100 * we plug in the solution to the pressure equation. This involves only
2101 * objects we already know - so we simply multiply @f$p@f$ by @f$B^T@f$, subtract
2102 * the right hand side and multiply by the inverse of @f$A@f$. At the end, we
2103 * need to distribute the constraints from hanging nodes in order to
2104 * obtain a consistent flow field:
2105 *
2106 * @code
2107 *   {
2108 *   system_matrix.block(0, 1).vmult(tmp, solution.block(1));
2109 *   tmp *= -1;
2110 *   tmp += system_rhs.block(0);
2111 *  
2112 *   A_inverse.vmult(solution.block(0), tmp);
2113 *  
2114 *   constraints.distribute(solution);
2115 *   }
2116 *   }
2117 *  
2118 *  
2119 * @endcode
2120 *
2121 *
2122 * <a name="StokesProblemoutput_results"></a>
2123 * <h4>StokesProblem::output_results</h4>
2124 *
2125
2126 *
2127 * The next function generates graphical output. In this example, we are
2128 * going to use the VTK file format. We attach names to the individual
2129 * variables in the problem: <code>velocity</code> to the <code>dim</code>
2130 * components of velocity and <code>pressure</code> to the pressure.
2131 *
2132
2133 *
2134 * Not all visualization programs have the ability to group individual
2135 * vector components into a vector to provide vector plots; in particular,
2136 * this holds for some VTK-based visualization programs. In this case, the
2137 * logical grouping of components into vectors should already be described
2138 * in the file containing the data. In other words, what we need to do is
2139 * provide our output writers with a way to know which of the components of
2140 * the finite element logically form a vector (with @f$d@f$ components in @f$d@f$
2141 * space dimensions) rather than letting them assume that we simply have a
2142 * bunch of scalar fields. This is achieved using the members of the
2143 * <code>DataComponentInterpretation</code> namespace: as with the filename,
2144 * we create a vector in which the first <code>dim</code> components refer
2145 * to the velocities and are given the tag
2146 * DataComponentInterpretation::component_is_part_of_vector; we
2147 * finally push one tag
2148 * DataComponentInterpretation::component_is_scalar to describe
2149 * the grouping of the pressure variable.
2150 *
2151
2152 *
2153 * The rest of the function is then the same as in @ref step_20 "step-20".
2154 *
2155 * @code
2156 *   template <int dim>
2157 *   void
2158 *   StokesProblem<dim>::output_results(const unsigned int refinement_cycle) const
2159 *   {
2160 *   std::vector<std::string> solution_names(dim, "velocity");
2161 *   solution_names.emplace_back("pressure");
2162 *  
2163 *   std::vector<DataComponentInterpretation::DataComponentInterpretation>
2164 *   data_component_interpretation(
2165 *   dim, DataComponentInterpretation::component_is_part_of_vector);
2166 *   data_component_interpretation.push_back(
2167 *   DataComponentInterpretation::component_is_scalar);
2168 *  
2169 *   DataOut<dim> data_out;
2170 *   data_out.attach_dof_handler(dof_handler);
2171 *   data_out.add_data_vector(solution,
2172 *   solution_names,
2173 *   DataOut<dim>::type_dof_data,
2174 *   data_component_interpretation);
2175 *   data_out.build_patches();
2176 *  
2177 *   std::ofstream output(
2178 *   "solution-" + Utilities::int_to_string(refinement_cycle, 2) + ".vtk");
2179 *   data_out.write_vtk(output);
2180 *   }
2181 *  
2182 *  
2183 * @endcode
2184 *
2185 *
2186 * <a name="StokesProblemrefine_mesh"></a>
2187 * <h4>StokesProblem::refine_mesh</h4>
2188 *
2189
2190 *
2191 * This is the last interesting function of the <code>StokesProblem</code>
2192 * class. As indicated by its name, it takes the solution to the problem
2193 * and refines the mesh where this is needed. The procedure is the same as
2194 * in the respective step in @ref step_6 "step-6", with the exception that we base the
2195 * refinement only on the change in pressure, i.e., we call the Kelly error
2196 * estimator with a mask object of type ComponentMask that selects the
2197 * single scalar component for the pressure that we are interested in (we
2198 * get such a mask from the finite element class by specifying the component
2199 * we want). Additionally, we do not coarsen the grid again:
2200 *
2201 * @code
2202 *   template <int dim>
2203 *   void StokesProblem<dim>::refine_mesh()
2204 *   {
2205 *   Vector<float> estimated_error_per_cell(triangulation.n_active_cells());
2206 *  
2207 *   const FEValuesExtractors::Scalar pressure(dim);
2208 *   KellyErrorEstimator<dim>::estimate(
2209 *   dof_handler,
2210 *   QGauss<dim - 1>(degree + 1),
2211 *   std::map<types::boundary_id, const Function<dim> *>(),
2212 *   solution,
2213 *   estimated_error_per_cell,
2214 *   fe.component_mask(pressure));
2215 *  
2216 *   GridRefinement::refine_and_coarsen_fixed_number(triangulation,
2217 *   estimated_error_per_cell,
2218 *   0.3,
2219 *   0.0);
2220 *   triangulation.execute_coarsening_and_refinement();
2221 *   }
2222 *  
2223 *  
2224 * @endcode
2225 *
2226 *
2227 * <a name="StokesProblemrun"></a>
2228 * <h4>StokesProblem::run</h4>
2229 *
2230
2231 *
2232 * The last step in the Stokes class is, as usual, the function that
2233 * generates the initial grid and calls the other functions in the
2234 * respective order.
2235 *
2236
2237 *
2238 * We start off with a rectangle of size @f$4 \times 1@f$ (in 2d) or @f$4 \times 1
2239 * \times 1@f$ (in 3d), placed in @f$R^2/R^3@f$ as @f$(-2,2)\times(-1,0)@f$ or
2240 * @f$(-2,2)\times(0,1)\times(-1,0)@f$, respectively. It is natural to start
2241 * with equal mesh size in each direction, so we subdivide the initial
2242 * rectangle four times in the first coordinate direction. To limit the
2243 * scope of the variables involved in the creation of the mesh to the range
2244 * where we actually need them, we put the entire block between a pair of
2245 * braces:
2246 *
2247 * @code
2248 *   template <int dim>
2249 *   void StokesProblem<dim>::run()
2250 *   {
2251 *   {
2252 *   std::vector<unsigned int> subdivisions(dim, 1);
2253 *   subdivisions[0] = 4;
2254 *  
2255 *   const Point<dim> bottom_left = (dim == 2 ?
2256 *   Point<dim>(-2, -1) : // 2d case
2257 *   Point<dim>(-2, 0, -1)); // 3d case
2258 *  
2259 *   const Point<dim> top_right = (dim == 2 ?
2260 *   Point<dim>(2, 0) : // 2d case
2261 *   Point<dim>(2, 1, 0)); // 3d case
2262 *  
2263 *   GridGenerator::subdivided_hyper_rectangle(triangulation,
2264 *   subdivisions,
2265 *   bottom_left,
2266 *   top_right);
2267 *   }
2268 *  
2269 * @endcode
2270 *
2271 * A boundary indicator of 1 is set to all boundaries that are subject to
2272 * Dirichlet boundary conditions, i.e. to faces that are located at 0 in
2273 * the last coordinate direction. See the example description above for
2274 * details.
2275 *
2276 * @code
2277 *   for (const auto &cell : triangulation.active_cell_iterators())
2278 *   for (const auto &face : cell->face_iterators())
2279 *   if (face->center()[dim - 1] == 0)
2280 *   face->set_all_boundary_ids(1);
2281 *  
2282 *  
2283 * @endcode
2284 *
2285 * We then apply an initial refinement before solving for the first
2286 * time. In 3d, there are going to be more degrees of freedom, so we
2287 * refine less there:
2288 *
2289 * @code
2290 *   triangulation.refine_global(4 - dim);
2291 *  
2292 * @endcode
2293 *
2294 * As first seen in @ref step_6 "step-6", we cycle over the different refinement levels
2295 * and refine (except for the first cycle), setup the degrees of freedom
2296 * and matrices, assemble, solve and create output:
2297 *
2298 * @code
2299 *   for (unsigned int refinement_cycle = 0; refinement_cycle < 6;
2300 *   ++refinement_cycle)
2301 *   {
2302 *   std::cout << "Refinement cycle " << refinement_cycle << std::endl;
2303 *  
2304 *   if (refinement_cycle > 0)
2305 *   refine_mesh();
2306 *  
2307 *   setup_dofs();
2308 *  
2309 *   std::cout << " Assembling..." << std::endl << std::flush;
2310 *   assemble_system();
2311 *  
2312 *   std::cout << " Solving..." << std::flush;
2313 *   solve();
2314 *  
2315 *   output_results(refinement_cycle);
2316 *  
2317 *   std::cout << std::endl;
2318 *   }
2319 *   }
2320 *   } // namespace Step22
2321 *  
2322 *  
2323 * @endcode
2324 *
2325 *
2326 * <a name="Thecodemaincodefunction"></a>
2327 * <h3>The <code>main</code> function</h3>
2328 *
2329
2330 *
2331 * The main function is the same as in @ref step_20 "step-20". We pass the element degree as
2332 * a parameter and choose the space dimension at the well-known template slot.
2333 *
2334 * @code
2335 *   int main()
2336 *   {
2337 *   try
2338 *   {
2339 *   using namespace Step22;
2340 *  
2341 *   StokesProblem<2> flow_problem(1);
2342 *   flow_problem.run();
2343 *   }
2344 *   catch (std::exception &exc)
2345 *   {
2346 *   std::cerr << std::endl
2347 *   << std::endl
2348 *   << "----------------------------------------------------"
2349 *   << std::endl;
2350 *   std::cerr << "Exception on processing: " << std::endl
2351 *   << exc.what() << std::endl
2352 *   << "Aborting!" << std::endl
2353 *   << "----------------------------------------------------"
2354 *   << std::endl;
2355 *  
2356 *   return 1;
2357 *   }
2358 *   catch (...)
2359 *   {
2360 *   std::cerr << std::endl
2361 *   << std::endl
2362 *   << "----------------------------------------------------"
2363 *   << std::endl;
2364 *   std::cerr << "Unknown exception!" << std::endl
2365 *   << "Aborting!" << std::endl
2366 *   << "----------------------------------------------------"
2367 *   << std::endl;
2368 *   return 1;
2369 *   }
2370 *  
2371 *   return 0;
2372 *   }
2373 * @endcode
2374<a name="Results"></a>
2375<a name="Results"></a><h1>Results</h1>
2376
2377
2378<a name="Outputoftheprogramandgraphicalvisualization"></a><h3>Output of the program and graphical visualization</h3>
2379
2380
2381<a name="2Dcalculations"></a><h4>2D calculations</h4>
2382
2383
2384Running the program with the space dimension set to 2 in the <code>main</code>
2385function yields the following output (in "release mode",
2386See also <a href="http://www.math.colostate.edu/~bangerth/videos.676.18.html">video lecture 18</a>.):
2387@code
2388examples/step-22> make run
2389Refinement cycle 0
2390 Number of active cells: 64
2391 Number of degrees of freedom: 679 (594+85)
2392 Assembling...
2393 Computing preconditioner...
2394 Solving... 11 outer CG Schur complement iterations for pressure
2395
2396Refinement cycle 1
2397 Number of active cells: 160
2398 Number of degrees of freedom: 1683 (1482+201)
2399 Assembling...
2400 Computing preconditioner...
2401 Solving... 11 outer CG Schur complement iterations for pressure
2402
2403Refinement cycle 2
2404 Number of active cells: 376
2405 Number of degrees of freedom: 3813 (3370+443)
2406 Assembling...
2407 Computing preconditioner...
2408 Solving... 11 outer CG Schur complement iterations for pressure
2409
2410Refinement cycle 3
2411 Number of active cells: 880
2412 Number of degrees of freedom: 8723 (7722+1001)
2413 Assembling...
2414 Computing preconditioner...
2415 Solving... 11 outer CG Schur complement iterations for pressure
2416
2417Refinement cycle 4
2418 Number of active cells: 2008
2419 Number of degrees of freedom: 19383 (17186+2197)
2420 Assembling...
2421 Computing preconditioner...
2422 Solving... 11 outer CG Schur complement iterations for pressure
2423
2424Refinement cycle 5
2425 Number of active cells: 4288
2426 Number of degrees of freedom: 40855 (36250+4605)
2427 Assembling...
2428 Computing preconditioner...
2429 Solving... 11 outer CG Schur complement iterations for pressure
2430@endcode
2431
2432The entire computation above takes about 2 seconds on a reasonably
2433quick (for 2015 standards) machine.
2434
2435What we see immediately from this is that the number of (outer)
2436iterations does not increase as we refine the mesh. This confirms the
2437statement in the introduction that preconditioning the Schur
2438complement with the mass matrix indeed yields a matrix spectrally
2439equivalent to the identity matrix (i.e. with eigenvalues bounded above
2440and below independently of the mesh size or the relative sizes of
2441cells). In other words, the mass matrix and the Schur complement are
2442spectrally equivalent.
2443
2444In the images below, we show the grids for the first six refinement
2445steps in the program. Observe how the grid is refined in regions
2446where the solution rapidly changes: On the upper boundary, we have
2447Dirichlet boundary conditions that are -1 in the left half of the line
2448and 1 in the right one, so there is an abrupt change at @f$x=0@f$. Likewise,
2449there are changes from Dirichlet to Neumann data in the two upper
2450corners, so there is need for refinement there as well:
2451
2452<table width="60%" align="center">
2453 <tr>
2454 <td align="center">
2455 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-0.png" alt="">
2456 </td>
2457 <td align="center">
2458 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-1.png" alt="">
2459 </td>
2460 </tr>
2461 <tr>
2462 <td align="center">
2463 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-2.png" alt="">
2464 </td>
2465 <td align="center">
2466 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-3.png" alt="">
2467 </td>
2468 </tr>
2469 <tr>
2470 <td align="center">
2471 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-4.png" alt="">
2472 </td>
2473 <td align="center">
2474 <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-5.png" alt="">
2475 </td>
2476 </tr>
2477</table>
2478
2479Finally, following is a plot of the flow field. It shows fluid
2480transported along with the moving upper boundary and being replaced by
2481material coming from below:
2482
2483<img src="https://www.dealii.org/images/steps/developer/step-22.2d.solution.png" alt="">
2484
2485This plot uses the capability of VTK-based visualization programs (in
2486this case of VisIt) to show vector data; this is the result of us
2487declaring the velocity components of the finite element in use to be a
2488set of vector components, rather than independent scalar components in
2489the <code>StokesProblem@<dim@>::%output_results</code> function of this
2490tutorial program.
2491
2492
2493
2494<a name="3Dcalculations"></a><h4>3D calculations</h4>
2495
2496
2497In 3d, the screen output of the program looks like this:
2498
2499@code
2500Refinement cycle 0
2501 Number of active cells: 32
2502 Number of degrees of freedom: 1356 (1275+81)
2503 Assembling...
2504 Computing preconditioner...
2505 Solving... 13 outer CG Schur complement iterations for pressure.
2506
2507Refinement cycle 1
2508 Number of active cells: 144
2509 Number of degrees of freedom: 5088 (4827+261)
2510 Assembling...
2511 Computing preconditioner...
2512 Solving... 14 outer CG Schur complement iterations for pressure.
2513
2514Refinement cycle 2
2515 Number of active cells: 704
2516 Number of degrees of freedom: 22406 (21351+1055)
2517 Assembling...
2518 Computing preconditioner...
2519 Solving... 14 outer CG Schur complement iterations for pressure.
2520
2521Refinement cycle 3
2522 Number of active cells: 3168
2523 Number of degrees of freedom: 93176 (89043+4133)
2524 Assembling...
2525 Computing preconditioner...
2526 Solving... 15 outer CG Schur complement iterations for pressure.
2527
2528Refinement cycle 4
2529 Number of active cells: 11456
2530 Number of degrees of freedom: 327808 (313659+14149)
2531 Assembling...
2532 Computing preconditioner...
2533 Solving... 15 outer CG Schur complement iterations for pressure.
2534
2535Refinement cycle 5
2536 Number of active cells: 45056
2537 Number of degrees of freedom: 1254464 (1201371+53093)
2538 Assembling...
2539 Computing preconditioner...
2540 Solving... 14 outer CG Schur complement iterations for pressure.
2541@endcode
2542
2543Again, we see that the number of outer iterations does not increase as
2544we refine the mesh. Nevertheless, the compute time increases
2545significantly: for each of the iterations above separately, it takes about
25460.14 seconds, 0.63 seconds, 4.8 seconds, 35 seconds, 2 minutes and 33 seconds,
2547and 13 minutes and 12 seconds. This overall superlinear (in the number of
2548unknowns) increase in runtime is due to the fact that our inner solver is not
2549@f${\cal O}(N)@f$: a simple experiment shows that as we keep refining the mesh, the
2550average number of ILU-preconditioned CG iterations to invert the
2551velocity-velocity block @f$A@f$ increases.
2552
2553We will address the question of how possibly to improve our solver <a
2554href="#improved-solver">below</a>.
2555
2556As for the graphical output, the grids generated during the solution
2557look as follow:
2558
2559<table width="60%" align="center">
2560 <tr>
2561 <td align="center">
2562 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-0.png" alt="">
2563 </td>
2564 <td align="center">
2565 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-1.png" alt="">
2566 </td>
2567 </tr>
2568 <tr>
2569 <td align="center">
2570 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-2.png" alt="">
2571 </td>
2572 <td align="center">
2573 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-3.png" alt="">
2574 </td>
2575 </tr>
2576 <tr>
2577 <td align="center">
2578 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-4.png" alt="">
2579 </td>
2580 <td align="center">
2581 <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-5.png" alt="">
2582 </td>
2583 </tr>
2584</table>
2585
2586Again, they show essentially the location of singularities introduced
2587by boundary conditions. The vector field computed makes for an
2588interesting graph:
2589
2590<img src="https://www.dealii.org/images/steps/developer/step-22.3d.solution.png" alt="">
2591
2592The isocontours shown here as well are those of the pressure
2593variable, showing the singularity at the point of discontinuous
2594velocity boundary conditions.
2595
2596
2597
2598<a name="Sparsitypattern"></a><h3>Sparsity pattern</h3>
2599
2600
2601As explained during the generation of the sparsity pattern, it is
2602important to have the numbering of degrees of freedom in mind when
2603using preconditioners like incomplete LU decompositions. This is most
2604conveniently visualized using the distribution of nonzero elements in
2605the @ref GlossStiffnessMatrix "stiffness matrix".
2606
2607If we don't do anything special to renumber degrees of freedom (i.e.,
2608without using DoFRenumbering::Cuthill_McKee, but with using
2609DoFRenumbering::component_wise to ensure that degrees of freedom are
2610appropriately sorted into their corresponding blocks of the matrix and
2611vector), then we get the following image after the first adaptive
2612refinement in two dimensions:
2613
2614<img src="https://www.dealii.org/images/steps/developer/step-22.2d.sparsity-nor.png" alt="">
2615
2616In order to generate such a graph, you have to insert a piece of
2617code like the following to the end of the setup step.
2618@code
2619 {
2620 std::ofstream out ("sparsity_pattern.gpl");
2621 sparsity_pattern.print_gnuplot(out);
2622 }
2623@endcode
2624
2625It is clearly visible that the nonzero entries are spread over almost the
2626whole matrix. This makes preconditioning by ILU inefficient: ILU generates a
2627Gaussian elimination (LU decomposition) without fill-in elements, which means
2628that more tentative fill-ins left out will result in a worse approximation of
2629the complete decomposition.
2630
2631In this program, we have thus chosen a more advanced renumbering of
2632components. The renumbering with DoFRenumbering::Cuthill_McKee and grouping
2633the components into velocity and pressure yields the following output:
2634
2635<img src="https://www.dealii.org/images/steps/developer/step-22.2d.sparsity-ren.png" alt="">
2636
2637It is apparent that the situation has improved a lot. Most of the elements are
2638now concentrated around the diagonal in the (0,0) block in the matrix. Similar
2639effects are also visible for the other blocks. In this case, the ILU
2640decomposition will be much closer to the full LU decomposition, which improves
2641the quality of the preconditioner. (It may be interesting to note that the
2642sparse direct solver UMFPACK does some %internal renumbering of the equations
2643before actually generating a sparse LU decomposition; that procedure leads to
2644a very similar pattern to the one we got from the Cuthill-McKee algorithm.)
2645
2646Finally, we want to have a closer
2647look at a sparsity pattern in 3D. We show only the (0,0) block of the
2648matrix, again after one adaptive refinement. Apart from the fact that the matrix
2649size has increased, it is also visible that there are many more entries
2650in the matrix. Moreover, even for the optimized renumbering, there will be a
2651considerable amount of tentative fill-in elements. This illustrates why UMFPACK
2652is not a good choice in 3D - a full decomposition needs many new entries that
2653 eventually won't fit into the physical memory (RAM):
2654
2655<img src="https://www.dealii.org/images/steps/developer/step-22.3d.sparsity_uu-ren.png" alt="">
2656
2657
2658
2659<a name="Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
2660
2661
2662<a name="improved-solver">
2663<a name="Improvedlinearsolverin3D"></a><h4>Improved linear solver in 3D</h4>
2664
2665</a>
2666
2667We have seen in the section of computational results that the number of outer
2668iterations does not depend on the mesh size, which is optimal in a sense of
2669scalability. This does, however, not apply to the solver as a whole, as
2670mentioned above:
2671We did not look at the number of inner iterations when generating the inverse of
2672the matrix @f$A@f$ and the mass matrix @f$M_p@f$. Of course, this is unproblematic in
2673the 2D case where we precondition @f$A@f$ with a direct solver and the
2674<code>vmult</code> operation of the inverse matrix structure will converge in
2675one single CG step, but this changes in 3D where we only use an ILU
2676preconditioner. There, the number of required preconditioned CG steps to
2677invert @f$A@f$ increases as the mesh is refined, and each <code>vmult</code>
2678operation involves on average approximately 14, 23, 36, 59, 75 and 101 inner
2679CG iterations in the refinement steps shown above. (On the other hand,
2680the number of iterations for applying the inverse pressure mass matrix is
2681always around five, both in two and three dimensions.) To summarize, most work
2682is spent on solving linear systems with the same matrix @f$A@f$ over and over again.
2683What makes this look even worse is the fact that we
2684actually invert a matrix that is about 95 percent the size of the total system
2685matrix and stands for 85 percent of the non-zero entries in the sparsity
2686pattern. Hence, the natural question is whether it is reasonable to solve a
2687linear system with matrix @f$A@f$ for about 15 times when calculating the solution
2688to the block system.
2689
2690The answer is, of course, that we can do that in a few other (most of the time
2691better) ways.
2692Nevertheless, it has to be remarked that an indefinite system as the one
2693at hand puts indeed much higher
2694demands on the linear algebra than standard elliptic problems as we have seen
2695in the early tutorial programs. The improvements are still rather
2696unsatisfactory, if one compares with an elliptic problem of similar
2697size. Either way, we will introduce below a number of improvements to the
2698linear solver, a discussion that we will re-consider again with additional
2699options in the @ref step_31 "step-31" program.
2700
2701<a name="improved-ilu">
2702<a name="BetterILUdecompositionbysmartreordering"></a><h5>Better ILU decomposition by smart reordering</h5>
2703
2704</a>
2705A first attempt to improve the speed of the linear solution process is to choose
2706a dof reordering that makes the ILU being closer to a full LU decomposition, as
2707already mentioned in the in-code comments. The DoFRenumbering namespace compares
2708several choices for the renumbering of dofs for the Stokes equations. The best
2709result regarding the computing time was found for the King ordering, which is
2710accessed through the call DoFRenumbering::boost::king_ordering. With that
2711program, the inner solver needs considerably less operations, e.g. about 62
2712inner CG iterations for the inversion of @f$A@f$ at cycle 4 compared to about 75
2713iterations with the standard Cuthill-McKee-algorithm. Also, the computing time
2714at cycle 4 decreased from about 17 to 11 minutes for the <code>solve()</code>
2715call. However, the King ordering (and the orderings provided by the
2716DoFRenumbering::boost namespace in general) has a serious drawback - it uses
2717much more memory than the in-build deal versions, since it acts on abstract
2718graphs rather than the geometry provided by the triangulation. In the present
2719case, the renumbering takes about 5 times as much memory, which yields an
2720infeasible algorithm for the last cycle in 3D with 1.2 million
2721unknowns.
2722
2723<a name="BetterpreconditionerfortheinnerCGsolver"></a><h5>Better preconditioner for the inner CG solver</h5>
2724
2725Another idea to improve the situation even more would be to choose a
2726preconditioner that makes CG for the (0,0) matrix @f$A@f$ converge in a
2727mesh-independent number of iterations, say 10 to 30. We have seen such a
2728candidate in @ref step_16 "step-16": multigrid.
2729
2730<a name="BlockSchurcomplementpreconditioner"></a><h5>Block Schur complement preconditioner</h5>
2731
2732<a name="block-schur"></a>
2733Even with a good preconditioner for @f$A@f$, we still
2734need to solve of the same linear system repeatedly (with different
2735right hand sides, though) in order to make the Schur complement solve
2736converge. The approach we are going to discuss here is how inner iteration
2737and outer iteration can be combined. If we persist in calculating the Schur
2738complement, there is no other possibility.
2739
2740The alternative is to attack the block system at once and use an approximate
2741Schur complement as efficient preconditioner. The idea is as
2742follows: If we find a block preconditioner @f$P@f$ such that the matrix
2743@f{eqnarray*}
2744 P^{-1}\left(\begin{array}{cc}
2745 A & B^T \\ B & 0
2746 \end{array}\right)
2747@f}
2748is simple, then an iterative solver with that preconditioner will converge in a
2749few iterations. Using the Schur complement @f$S = B A^{-1} B^T@f$, one finds that
2750@f{eqnarray*}
2751 P^{-1}
2752 =
2753 \left(\begin{array}{cc}
2754 A^{-1} & 0 \\ S^{-1} B A^{-1} & -S^{-1}
2755 \end{array}\right)
2756@f}
2757would appear to be a good choice since
2758@f{eqnarray*}
2759 P^{-1}\left(\begin{array}{cc}
2760 A & B^T \\ B & 0
2761 \end{array}\right)
2762 =
2763 \left(\begin{array}{cc}
2764 A^{-1} & 0 \\ S^{-1} B A^{-1} & -S^{-1}
2765 \end{array}\right)\cdot \left(\begin{array}{cc}
2766 A & B^T \\ B & 0
2767 \end{array}\right)
2768 =
2769 \left(\begin{array}{cc}
2770 I & A^{-1} B^T \\ 0 & I
2771 \end{array}\right).
2772@f}
2773This is the approach taken by the paper by Silvester and Wathen referenced
2774to in the introduction (with the exception that Silvester and Wathen use
2775right preconditioning). In this case, a Krylov-based iterative method would
2776converge in one step only if exact inverses of @f$A@f$ and @f$S@f$ were applied,
2777since all the eigenvalues are one (and the number of iterations in such a
2778method is bounded by the number of distinct eigenvalues). Below, we will
2779discuss the choice of an adequate solver for this problem. First, we are
2780going to have a closer look at the implementation of the preconditioner.
2781
2782Since @f$P@f$ is aimed to be a preconditioner only, we shall use approximations to
2783the inverse of the Schur complement @f$S@f$ and the matrix @f$A@f$. Hence, the Schur
2784complement will be approximated by the pressure mass matrix @f$M_p@f$, and we use
2785a preconditioner to @f$A@f$ (without an InverseMatrix class around it) for
2786approximating @f$A^{-1}@f$.
2787
2788Here comes the class that implements the block Schur
2789complement preconditioner. The <code>vmult</code> operation for block vectors
2790according to the derivation above can be specified by three successive
2791operations:
2792@code
2793template <class PreconditionerA, class PreconditionerMp>
2794class BlockSchurPreconditioner : public Subscriptor
2795{
2796 public:
2797 BlockSchurPreconditioner (const BlockSparseMatrix<double> &S,
2798 const InverseMatrix<SparseMatrix<double>,PreconditionerMp> &Mpinv,
2799 const PreconditionerA &Apreconditioner);
2800
2801 void vmult (BlockVector<double> &dst,
2802 const BlockVector<double> &src) const;
2803
2804 private:
2807 PreconditionerMp > > m_inverse;
2808 const PreconditionerA &a_preconditioner;
2809
2810 mutable Vector<double> tmp;
2811
2812};
2813
2814template <class PreconditionerA, class PreconditionerMp>
2815BlockSchurPreconditioner<PreconditionerA, PreconditionerMp>::BlockSchurPreconditioner(
2817 const InverseMatrix<SparseMatrix<double>,PreconditionerMp> &Mpinv,
2818 const PreconditionerA &Apreconditioner
2819 )
2820 :
2821 system_matrix (&S),
2822 m_inverse (&Mpinv),
2823 a_preconditioner (Apreconditioner),
2824 tmp (S.block(1,1).m())
2825{}
2826
2827 // Now the interesting function, the multiplication of
2828 // the preconditioner with a BlockVector.
2829template <class PreconditionerA, class PreconditionerMp>
2830void BlockSchurPreconditioner<PreconditionerA, PreconditionerMp>::vmult (
2832 const BlockVector<double> &src) const
2833{
2834 // Form u_new = A^{-1} u
2835 a_preconditioner.vmult (dst.block(0), src.block(0));
2836 // Form tmp = - B u_new + p
2837 // (<code>SparseMatrix::residual</code>
2838 // does precisely this)
2839 system_matrix->block(1,0).residual(tmp, dst.block(0), src.block(1));
2840 // Change sign in tmp
2841 tmp *= -1;
2842 // Multiply by approximate Schur complement
2843 // (i.e. a pressure mass matrix)
2844 m_inverse->vmult (dst.block(1), tmp);
2845}
2846@endcode
2847
2848Since we act on the whole block system now, we have to live with one
2849disadvantage: we need to perform the solver iterations on
2850the full block system instead of the smaller pressure space.
2851
2852Now we turn to the question which solver we should use for the block
2853system. The first observation is that the resulting preconditioned matrix cannot
2854be solved with CG since it is neither positive definite nor symmetric.
2855
2856The deal.II libraries implement several solvers that are appropriate for the
2857problem at hand. One choice is the solver @ref SolverBicgstab "BiCGStab", which
2858was used for the solution of the unsymmetric advection problem in @ref step_9 "step-9". The
2859second option, the one we are going to choose, is @ref SolverGMRES "GMRES"
2860(generalized minimum residual). Both methods have their pros and cons - there
2861are problems where one of the two candidates clearly outperforms the other, and
2862vice versa.
2863<a href="http://en.wikipedia.org/wiki/GMRES#Comparison_with_other_solvers">Wikipedia</a>'s
2864article on the GMRES method gives a comparative presentation.
2865A more comprehensive and well-founded comparison can be read e.g. in the book by
2866J.W. Demmel (Applied Numerical Linear Algebra, SIAM, 1997, section 6.6.6).
2867
2868For our specific problem with the ILU preconditioner for @f$A@f$, we certainly need
2869to perform hundreds of iterations on the block system for large problem sizes
2870(we won't beat CG!). Actually, this disfavors GMRES: During the GMRES
2871iterations, a basis of Krylov vectors is successively built up and some
2872operations are performed on these vectors. The more vectors are in this basis,
2873the more operations and memory will be needed. The number of operations scales
2874as @f${\cal O}(n + k^2)@f$ and memory as @f${\cal O}(kn)@f$, where @f$k@f$ is the number of
2875vectors in the Krylov basis and @f$n@f$ the size of the (block) matrix.
2876To not let these demands grow excessively, deal.II limits the size @f$k@f$ of the
2877basis to 30 vectors by default.
2878Then, the basis is rebuilt. This implementation of the GMRES method is called
2879GMRES(k), with default @f$k=30@f$. What we have gained by this restriction,
2880namely a bound on operations and memory requirements, will be compensated by
2881the fact that we use an incomplete basis - this will increase the number of
2882required iterations.
2883
2884BiCGStab, on the other hand, won't get slower when many iterations are needed
2885(one iteration uses only results from one preceding step and
2886not all the steps as GMRES). Besides the fact the BiCGStab is more expensive per
2887step since two matrix-vector products are needed (compared to one for
2888CG or GMRES), there is one main reason which makes BiCGStab not appropriate for
2889this problem: The preconditioner applies the inverse of the pressure
2890mass matrix by using the InverseMatrix class. Since the application of the
2891inverse matrix to a vector is done only in approximative way (an exact inverse
2892is too expensive), this will also affect the solver. In the case of BiCGStab,
2893the Krylov vectors will not be orthogonal due to that perturbation. While
2894this is uncritical for a small number of steps (up to about 50), it ruins the
2895performance of the solver when these perturbations have grown to a significant
2896magnitude in the coarse of iterations.
2897
2898We did some experiments with BiCGStab and found it to
2899be faster than GMRES up to refinement cycle 3 (in 3D), but it became very slow
2900for cycles 4 and 5 (even slower than the original Schur complement), so the
2901solver is useless in this situation. Choosing a sharper tolerance for the
2902inverse matrix class (<code>1e-10*src.l2_norm()</code> instead of
2903<code>1e-6*src.l2_norm()</code>) made BiCGStab perform well also for cycle 4,
2904but did not change the failure on the very large problems.
2905
2906GMRES is of course also effected by the approximate inverses, but it is not as
2907sensitive to orthogonality and retains a relatively good performance also for
2908large sizes, see the results below.
2909
2910With this said, we turn to the realization of the solver call with GMRES with
2911@f$k=100@f$ temporary vectors:
2912
2913@code
2914 const SparseMatrix<double> &pressure_mass_matrix
2915 = preconditioner_matrix.block(1,1);
2916 SparseILU<double> pmass_preconditioner;
2917 pmass_preconditioner.initialize (pressure_mass_matrix,
2918 SparseILU<double>::AdditionalData());
2919
2920 InverseMatrix<SparseMatrix<double>,SparseILU<double> >
2921 m_inverse (pressure_mass_matrix, pmass_preconditioner);
2922
2923 BlockSchurPreconditioner<typename InnerPreconditioner<dim>::type,
2924 SparseILU<double> >
2925 preconditioner (system_matrix, m_inverse, *A_preconditioner);
2926
2927 SolverControl solver_control (system_matrix.m(),
2928 1e-6*system_rhs.l2_norm());
2929 GrowingVectorMemory<BlockVector<double> > vector_memory;
2930 SolverGMRES<BlockVector<double> >::AdditionalData gmres_data;
2931 gmres_data.max_n_tmp_vectors = 100;
2932
2933 SolverGMRES<BlockVector<double> > gmres(solver_control, vector_memory,
2934 gmres_data);
2935
2936 gmres.solve(system_matrix, solution, system_rhs,
2937 preconditioner);
2938
2939 constraints.distribute (solution);
2940
2941 std::cout << " "
2942 << solver_control.last_step()
2943 << " block GMRES iterations";
2944@endcode
2945
2946Obviously, one needs to add the include file @ref SolverGMRES
2947"<lac/solver_gmres.h>" in order to make this run.
2948We call the solver with a BlockVector template in order to enable
2949GMRES to operate on block vectors and matrices.
2950Note also that we need to set the (1,1) block in the system
2951matrix to zero (we saved the pressure mass matrix there which is not part of the
2952problem) after we copied the information to another matrix.
2953
2954Using the Timer class, we collect some statistics that compare the runtime
2955of the block solver with the one from the problem implementation above.
2956Besides the solution with the two options we also check if the solutions
2957of the two variants are close to each other (i.e. this solver gives indeed the
2958same solution as we had before) and calculate the infinity
2959norm of the vector difference.
2960
2961Let's first see the results in 2D:
2962@code
2963Refinement cycle 0
2964 Number of active cells: 64
2965 Number of degrees of freedom: 679 (594+85) [0.00162792 s]
2966 Assembling... [0.00108981 s]
2967 Computing preconditioner... [0.0025959 s]
2968 Solving...
2969 Schur complement: 11 outer CG iterations for p [0.00479603s ]
2970 Block Schur preconditioner: 12 GMRES iterations [0.00441718 s]
2971 l_infinity difference between solution vectors: 5.38258e-07
2972
2973Refinement cycle 1
2974 Number of active cells: 160
2975 Number of degrees of freedom: 1683 (1482+201) [0.00345707 s]
2976 Assembling... [0.00237417 s]
2977 Computing preconditioner... [0.00605702 s]
2978 Solving...
2979 Schur complement: 11 outer CG iterations for p [0.0123992s ]
2980 Block Schur preconditioner: 12 GMRES iterations [0.011909 s]
2981 l_infinity difference between solution vectors: 1.74658e-05
2982
2983Refinement cycle 2
2984 Number of active cells: 376
2985 Number of degrees of freedom: 3813 (3370+443) [0.00729299 s]
2986 Assembling... [0.00529909 s]
2987 Computing preconditioner... [0.0167508 s]
2988 Solving...
2989 Schur complement: 11 outer CG iterations for p [0.031672s ]
2990 Block Schur preconditioner: 12 GMRES iterations [0.029232 s]
2991 l_infinity difference between solution vectors: 7.81569e-06
2992
2993Refinement cycle 3
2994 Number of active cells: 880
2995 Number of degrees of freedom: 8723 (7722+1001) [0.017709 s]
2996 Assembling... [0.0126002 s]
2997 Computing preconditioner... [0.0435679 s]
2998 Solving...
2999 Schur complement: 11 outer CG iterations for p [0.0971651s ]
3000 Block Schur preconditioner: 12 GMRES iterations [0.0992041 s]
3001 l_infinity difference between solution vectors: 1.87249e-05
3002
3003Refinement cycle 4
3004 Number of active cells: 2008
3005 Number of degrees of freedom: 19383 (17186+2197) [0.039988 s]
3006 Assembling... [0.028281 s]
3007 Computing preconditioner... [0.118314 s]
3008 Solving...
3009 Schur complement: 11 outer CG iterations for p [0.252133s ]
3010 Block Schur preconditioner: 13 GMRES iterations [0.269125 s]
3011 l_infinity difference between solution vectors: 6.38657e-05
3012
3013Refinement cycle 5
3014 Number of active cells: 4288
3015 Number of degrees of freedom: 40855 (36250+4605) [0.0880702 s]
3016 Assembling... [0.0603511 s]
3017 Computing preconditioner... [0.278339 s]
3018 Solving...
3019 Schur complement: 11 outer CG iterations for p [0.53846s ]
3020 Block Schur preconditioner: 13 GMRES iterations [0.578667 s]
3021 l_infinity difference between solution vectors: 0.000173363
3022@endcode
3023
3024We see that there is no huge difference in the solution time between the
3025block Schur complement preconditioner solver and the Schur complement
3026itself. The reason is simple: we used a direct solve as preconditioner for
3027@f$A@f$ - so we cannot expect any gain by avoiding the inner iterations. We see
3028that the number of iterations has slightly increased for GMRES, but all in
3029all the two choices are fairly similar.
3030
3031The picture of course changes in 3D:
3032
3033@code
3034Refinement cycle 0
3035 Number of active cells: 32
3036 Number of degrees of freedom: 1356 (1275+81) [0.00845218 s]
3037 Assembling... [0.019372 s]
3038 Computing preconditioner... [0.00712395 s]
3039 Solving...
3040 Schur complement: 13 outer CG iterations for p [0.0320101s ]
3041 Block Schur preconditioner: 22 GMRES iterations [0.0048759 s]
3042 l_infinity difference between solution vectors: 2.15942e-05
3043
3044Refinement cycle 1
3045 Number of active cells: 144
3046 Number of degrees of freedom: 5088 (4827+261) [0.0346942 s]
3047 Assembling... [0.0857739 s]
3048 Computing preconditioner... [0.0465031 s]
3049 Solving...
3050 Schur complement: 14 outer CG iterations for p [0.349258s ]
3051 Block Schur preconditioner: 35 GMRES iterations [0.048759 s]
3052 l_infinity difference between solution vectors: 1.77657e-05
3053
3054Refinement cycle 2
3055 Number of active cells: 704
3056 Number of degrees of freedom: 22406 (21351+1055) [0.175669 s]
3057 Assembling... [0.437447 s]
3058 Computing preconditioner... [0.286435 s]
3059 Solving...
3060 Schur complement: 14 outer CG iterations for p [3.65519s ]
3061 Block Schur preconditioner: 63 GMRES iterations [0.497787 s]
3062 l_infinity difference between solution vectors: 5.08078e-05
3063
3064Refinement cycle 3
3065 Number of active cells: 3168
3066 Number of degrees of freedom: 93176 (89043+4133) [0.790985 s]
3067 Assembling... [1.97598 s]
3068 Computing preconditioner... [1.4325 s]
3069 Solving...
3070 Schur complement: 15 outer CG iterations for p [29.9666s ]
3071 Block Schur preconditioner: 128 GMRES iterations [5.02645 s]
3072 l_infinity difference between solution vectors: 0.000119671
3073
3074Refinement cycle 4
3075 Number of active cells: 11456
3076 Number of degrees of freedom: 327808 (313659+14149) [3.44995 s]
3077 Assembling... [7.54772 s]
3078 Computing preconditioner... [5.46306 s]
3079 Solving...
3080 Schur complement: 15 outer CG iterations for p [139.987s ]
3081 Block Schur preconditioner: 255 GMRES iterations [38.0946 s]
3082 l_infinity difference between solution vectors: 0.00020793
3083
3084Refinement cycle 5
3085 Number of active cells: 45056
3086 Number of degrees of freedom: 1254464 (1201371+53093) [19.6795 s]
3087 Assembling... [28.6586 s]
3088 Computing preconditioner... [22.401 s]
3089 Solving...
3090 Schur complement: 14 outer CG iterations for p [796.767s ]
3091 Block Schur preconditioner: 524 GMRES iterations [355.597 s]
3092 l_infinity difference between solution vectors: 0.000501219
3093@endcode
3094
3095Here, the block preconditioned solver is clearly superior to the Schur
3096complement, but the advantage gets less for more mesh points. This is
3097because GMRES(k) scales worse with the problem size than CG, as we discussed
3098above. Nonetheless, the improvement by a factor of 3-6 for moderate problem
3099sizes is quite impressive.
3100
3101
3102<a name="Combiningtheblockpreconditionerandmultigrid"></a><h5>Combining the block preconditioner and multigrid</h5>
3103
3104An ultimate linear solver for this problem could be imagined as a
3105combination of an optimal
3106preconditioner for @f$A@f$ (e.g. multigrid) and the block preconditioner
3107described above, which is the approach taken in the @ref step_31 "step-31"
3108and @ref step_32 "step-32" tutorial programs (where we use an algebraic multigrid
3109method) and @ref step_56 "step-56" (where we use a geometric multigrid method).
3110
3111
3112<a name="Noblockmatricesandvectors"></a><h5>No block matrices and vectors</h5>
3113
3114Another possibility that can be taken into account is to not set up a block
3115system, but rather solve the system of velocity and pressure all at once. The
3116options are direct solve with UMFPACK (2D) or GMRES with ILU
3117preconditioning (3D). It should be straightforward to try that.
3118
3119
3120
3121<a name="Moreinterestingtestcases"></a><h4>More interesting testcases</h4>
3122
3123
3124The program can of course also serve as a basis to compute the flow in more
3125interesting cases. The original motivation to write this program was for it to
3126be a starting point for some geophysical flow problems, such as the
3127movement of magma under places where continental plates drift apart (for
3128example mid-ocean ridges). Of course, in such places, the geometry is more
3129complicated than the examples shown above, but it is not hard to accommodate
3130for that.
3131
3132For example, by using the following modification of the boundary values
3133function
3134@code
3135template <int dim>
3136double
3137BoundaryValues<dim>::value (const Point<dim> &p,
3138 const unsigned int component) const
3139{
3140 Assert (component < this->n_components,
3141 ExcIndexRange (component, 0, this->n_components));
3142
3143 const double x_offset = std::atan(p[1]*4)/3;
3144
3145 if (component == 0)
3146 return (p[0] < x_offset ? -1 : (p[0] > x_offset ? 1 : 0));
3147 return 0;
3148}
3149@endcode
3150and the following way to generate the mesh as the domain
3151@f$[-2,2]\times[-2,2]\times[-1,0]@f$
3152@code
3153 std::vector<unsigned int> subdivisions (dim, 1);
3154 subdivisions[0] = 4;
3155 if (dim>2)
3156 subdivisions[1] = 4;
3157
3158 const Point<dim> bottom_left = (dim == 2 ?
3159 Point<dim>(-2,-1) :
3160 Point<dim>(-2,-2,-1));
3161 const Point<dim> top_right = (dim == 2 ?
3162 Point<dim>(2,0) :
3163 Point<dim>(2,2,0));
3164
3166 subdivisions,
3167 bottom_left,
3168 top_right);
3169@endcode
3170then we get images where the fault line is curved:
3171<table width="60%" align="center">
3172 <tr>
3173 <td align="center">
3174 <img src="https://www.dealii.org/images/steps/developer/step-22.3d-extension.png" alt="">
3175 </td>
3176 <td align="center">
3177 <img src="https://www.dealii.org/images/steps/developer/step-22.3d-grid-extension.png" alt="">
3178 </td>
3179 </tr>
3180</table>
3181 *
3182 *
3183<a name="PlainProg"></a>
3184<h1> The plain program</h1>
3185@include "step-22.cc"
3186*/
BlockType & block(const unsigned int i)
Definition point.h:112
Point< 2 > second
Definition grid_out.cc:4616
Point< 2 > first
Definition grid_out.cc:4615
__global__ void set(Number *val, const Number s, const size_type N)
#define Assert(cond, exc)
void loop(ITERATOR begin, std_cxx20::type_identity_t< ITERATOR > 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())
Definition loop.h:439
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern, const AffineConstraints< number > &constraints=AffineConstraints< number >(), const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
const Event initial
Definition event.cc:65
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 >())
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 >())
std::vector< types::global_dof_index > count_dofs_per_fe_component(const DoFHandler< dim, spacedim > &dof_handler, const bool vector_valued_once=false, const std::vector< unsigned int > &target_component={})
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)
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ diagonal
Matrix is diagonal.
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition utilities.cc:189
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)
VectorType::value_type * end(VectorType &V)
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())
int(&) functions(const void *v1, const void *v2)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
Definition types.h:33
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
DEAL_II_HOST constexpr SymmetricTensor< 2, dim, Number > invert(const SymmetricTensor< 2, dim, Number > &)
std::array< Number, 1 > eigenvalues(const SymmetricTensor< 2, 1, Number > &T)