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// $Id: matrices.h 20602 2010-02-13 17:44:17Z bangerth $
// Version: $Name$
//
// Copyright (C) 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
// to the file deal.II/doc/license.html for the text and
// further information on this license.
//
//---------------------------------------------------------------------------
#ifndef __deal2__matrices_h
#define __deal2__matrices_h
#include <base/config.h>
#include <base/exceptions.h>
#include <base/thread_management.h>
#include <dofs/function_map.h>
#include <map>
DEAL_II_NAMESPACE_OPEN
// forward declarations
template <int dim> class Quadrature;
template<typename number> class Vector;
template<typename number> class FullMatrix;
template<typename number> class SparseMatrix;
template <typename number> class BlockSparseMatrix;
template <typename Number> class BlockVector;
template <int dim, int spacedim> class Mapping;
template <int dim, int spacedim> class DoFHandler;
template <int dim, int spacedim> class MGDoFHandler;
template <int dim, int spacedim> class FEValues;
namespace hp
{
template <int> class QCollection;
template <int, int> class MappingCollection;
template <int, int> class DoFHandler;
}
#ifdef DEAL_II_USE_PETSC
namespace PETScWrappers
{
class SparseMatrix;
class Vector;
namespace MPI
{
class SparseMatrix;
class BlockSparseMatrix;
class Vector;
class BlockVector;
}
}
#endif
#ifdef DEAL_II_USE_TRILINOS
namespace TrilinosWrappers
{
class SparseMatrix;
class Vector;
class BlockSparseMatrix;
class BlockVector;
namespace MPI
{
class Vector;
class BlockVector;
}
}
#endif
/**
* This class provides functions that assemble certain standard matrices for a
* given triangulation, using a given finite element, a given mapping and a
* quadrature formula. All functions are static, so it is not necessary to
* create an object of this type.
*
*
* <h3>Conventions for all functions</h3>
*
* There exist two versions of each function. One with a Mapping
* argument and one without. If a code uses a mapping different from
* MappingQ1 the functions <em>with</em> mapping argument should
* be used. Code that uses only MappingQ1 may also use the
* functions <em>without</em> Mapping argument. Each of these
* latter functions create a MappingQ1 object and just call the
* respective functions with that object as mapping argument.
*
* All functions take a sparse matrix object to hold the matrix to be
* created. The functions assume that the matrix is initialized with a
* sparsity pattern (SparsityPattern) corresponding to the given degree
* of freedom handler, i.e. the sparsity structure is already as needed.
* You can do this by calling the DoFTools::make_sparsity_pattern()
* function.
*
* Furthermore it is assumed that no relevant data is in the matrix. Some
* entries will be overwritten and some others will contain invalid data if
* the matrix wasn't empty before. Therefore you may want to clear the matrix
* before assemblage.
*
* All created matrices are `raw': they are not condensed,
* i.e. hanging nodes are not eliminated. The reason is that you may
* want to add several matrices and could then condense afterwards
* only once, instead of for every matrix. To actually do computations
* with these matrices, you have to condense the matrix using the
* ConstraintMatrix@p ::condense function; you also have to
* condense the right hand side accordingly and distribute the
* solution afterwards.
*
* If you want to use boundary conditions with the matrices generated
* by the functions of this class, you have to use a function like
* <tt>ProblemBase<>::apply_dirichlet_bc</tt> to matrix and right hand
* side.
*
*
* <h3>Supported matrices</h3>
*
* At present there are functions to create the following matrices:
* <ul>
* <li> @p create_mass_matrix: create the matrix with entries
* $m_{ij} = \int_\Omega \phi_i(x) \phi_j(x) dx$ by numerical
* quadrature. Here, the $\phi_i$ are the basis functions of the
* finite element space given.
*
* A coefficient may be given to evaluate
* $m_{ij} = \int_\Omega a(x) \phi_i(x) \phi_j(x) dx$ instead.
*
* <li> @p create_laplace_matrix: create the matrix with entries
* $a_{ij} = \int_\Omega \nabla\phi_i(x) \nabla\phi_j(x) dx$ by
* numerical quadrature.
*
* Again, a coefficient may be given to evaluate
* $a_{ij} = \int_\Omega a(x) \nabla\phi_i(x) \nabla\phi_j(x) dx$ instead.
* </ul>
*
* Make sure that the order of the Quadrature formula given to these
* functions is sufficiently high to compute the matrices with the
* required accuracy. For the choice of this quadrature rule you need
* to take into account the polynomial degree of the FiniteElement
* basis functions, the roughness of the coefficient @p a, as well as
* the degree of the given @p Mapping (if any).
*
* Note, that for system elements the mass matrix and the laplace matrix is
* implemented such that each components couple only with itself, i.e. there
* is no coupling of shape functions belonging to different components. If the
* degrees of freedom have been sorted according to their vector component
* (e.g., using DoFRenumbering::component_wise()), then the resulting matrices
* will be block diagonal.
*
* If the finite element for which the mass matrix or the laplace
* matrix is to be built has more than one component, this function
* accepts a single coefficient as well as a vector valued coefficient
* function. For the latter case make sure that the number of
* components coincides with the number of components of the system
* finite element.
*
*
* <h3>Matrices on the boundary</h3>
*
* The create_boundary_mass_matrix() creates the matrix with entries $m_{ij} =
* \int_{\Gamma} \phi_i \phi_j dx$, where $\Gamma$ is the union of boundary
* parts with indicators contained in a FunctionMap passed to the function
* (i.e. if you want to set up the mass matrix for the parts of the boundary
* with indicators zero and 2, you pass the function a map of <tt>unsigned
* char</tt>s as parameter @p boundary_functions containing the keys zero and
* 2). The size of the matrix is equal to the number of degrees of freedom
* that have support on the boundary, i.e. it is <em>not</em> a matrix on all
* degrees of freedom, but only a subset. (The $\phi_i$ in the formula are
* this subsect of basis functions which have at least part of their support
* on $\Gamma$.) In order to determine which shape functions are to be
* considered, and in order to determine in which order, the function takes a
* @p dof_to_boundary_mapping; this object maps global DoF numbers to a
* numbering of the degrees of freedom located on the boundary, and can be
* obtained using the function DoFTools::map_dof_to_boundary_indices().
*
* In order to work, the function needs a matrix of the correct size, built on
* top of a corresponding sparsity pattern. Since we only work on a subset of
* the degrees of freedom, we can't use the matrices and sparsity patterns
* that are created for the entire set of degrees of freedom. Rather, you
* should use the DoFHandler@p ::make_boundary_sparsity_pattern() function to
* create the correct sparsity pattern, and build a matrix on top of it.
*
* Note that at present there is no function that computes the mass matrix for
* <em>all</em> shape functions, though such a function would be trivial to
* implement.
*
*
* <h3>Right hand sides</h3>
*
* In many cases, you will not only want to build the matrix, but also
* a right hand side, which will give a vector with
* $f_i = \int_\Omega f(x) \phi_i(x) dx$. For this purpose, each function
* exists in two versions, one only building the matrix and one also
* building the right hand side vector. If you want to create a right
* hand side vector without creating a matrix, you can use the
* VectorTools::create_right_hand_side() function. The use of the
* latter may be useful if you want to create many right hand side
* vectors.
*
* @ingroup numerics
* @author Wolfgang Bangerth, 1998, Ralf Hartmann, 2001
*/
class MatrixCreator
{
public:
/**
* Assemble the mass matrix. If no
* coefficient is given, it is assumed
* to be unity.
*
* If the library is configured
* to use multithreading, this
* function works in parallel.
*
* See the general doc of this class
* for more information.
*/
template <int dim, typename number, int spacedim>
static void create_mass_matrix (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
SparseMatrix<number> &matrix,
const Function<spacedim> * const a = 0);
/**
* Calls the create_mass_matrix()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, typename number, int spacedim>
static void create_mass_matrix (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
SparseMatrix<number> &matrix,
const Function<spacedim> * const a = 0);
/**
* Assemble the mass matrix and a
* right hand side vector. If no
* coefficient is given, it is
* assumed to be unity.
*
* If the library is configured
* to use multithreading, this
* function works in parallel.
*
* See the general doc of this
* class for more information.
*/
template <int dim, typename number, int spacedim>
static void create_mass_matrix (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
SparseMatrix<number> &matrix,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const Function<spacedim> * const a = 0);
/**
* Calls the create_mass_matrix()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, typename number, int spacedim>
static void create_mass_matrix (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
SparseMatrix<number> &matrix,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const Function<spacedim> * const a = 0);
/**
* Same function as above, but for hp
* objects.
*/
template <int dim, typename number, int spacedim>
static void create_mass_matrix (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
SparseMatrix<number> &matrix,
const Function<spacedim> * const a = 0);
/**
* Same function as above, but for hp
* objects.
*/
template <int dim, typename number, int spacedim>
static void create_mass_matrix (const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
SparseMatrix<number> &matrix,
const Function<spacedim> * const a = 0);
/**
* Same function as above, but for hp
* objects.
*/
template <int dim, typename number, int spacedim>
static void create_mass_matrix (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
SparseMatrix<number> &matrix,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const Function<spacedim> * const a = 0);
/**
* Same function as above, but for hp
* objects.
*/
template <int dim, typename number, int spacedim>
static void create_mass_matrix (const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
SparseMatrix<number> &matrix,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const Function<spacedim> * const a = 0);
/**
* Assemble the mass matrix and a
* right hand side vector along
* the boundary.
*
* The matrix is assumed to
* already be initialized with a
* suiting sparsity pattern (the
* DoFHandler provides an
* appropriate function).
*
* If the library is configured
* to use multithreading, this
* function works in parallel.
*
* @arg @p weight: an optional
* weight for the computation of
* the mass matrix. If no weight
* is given, it is set to one.
*
* @arg @p component_mapping: if
* the components in @p
* boundary_functions and @p dof
* do not coincide, this vector
* allows them to be remapped. If
* the vector is not empty, it
* has to have one entry for each
* component in @p dof. This
* entry is the component number
* in @p boundary_functions that
* should be used for this
* component in @p dof. By
* default, no remapping is
* applied.
*
* @todo This function does not
* work for finite elements with
* cell-dependent shape functions.
*/
template <int dim, int spacedim>
static
void create_boundary_mass_matrix (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim-1> &q,
SparseMatrix<double> &matrix,
const typename FunctionMap<spacedim>::type &boundary_functions,
Vector<double> &rhs_vector,
std::vector<unsigned int>&dof_to_boundary_mapping,
const Function<spacedim> * const weight = 0,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
// * Same function, but for 1d.
// */
// static
// void create_boundary_mass_matrix (const Mapping<1,1> &mapping,
// const DoFHandler<1,1> &dof,
// const Quadrature<0> &q,
// SparseMatrix<double> &matrix,
// const FunctionMap<1>::type &boundary_functions,
// Vector<double> &rhs_vector,
// std::vector<unsigned int>&dof_to_boundary_mapping,
// const Function<1> * const a = 0);
// //codimension 1
// static
// void create_boundary_mass_matrix (const Mapping<1,2> &mapping,
// const DoFHandler<1,2> &dof,
// const Quadrature<0> &q,
// SparseMatrix<double> &matrix,
// const FunctionMap<2>::type &boundary_functions,
// Vector<double> &rhs_vector,
// std::vector<unsigned int>&dof_to_boundary_mapping,
// const Function<2> * const a = 0);
/**
* Calls the
* create_boundary_mass_matrix()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, int spacedim>
static
void create_boundary_mass_matrix (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim-1> &q,
SparseMatrix<double> &matrix,
const typename FunctionMap<spacedim>::type &boundary_functions,
Vector<double> &rhs_vector,
std::vector<unsigned int>&dof_to_boundary_mapping,
const Function<spacedim> * const a = 0,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
/**
* Same function as above, but for hp
* objects.
*/
template <int dim, int spacedim>
static
void create_boundary_mass_matrix (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim-1> &q,
SparseMatrix<double> &matrix,
const typename FunctionMap<spacedim>::type &boundary_functions,
Vector<double> &rhs_vector,
std::vector<unsigned int>&dof_to_boundary_mapping,
const Function<spacedim> * const a = 0,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
/**
* Same function as above, but for hp
* objects.
*/
// static
// void create_boundary_mass_matrix (const hp::MappingCollection<1,1> &mapping,
// const hp::DoFHandler<1,1> &dof,
// const hp::QCollection<0> &q,
// SparseMatrix<double> &matrix,
// const FunctionMap<1>::type &boundary_functions,
// Vector<double> &rhs_vector,
// std::vector<unsigned int>&dof_to_boundary_mapping,
// const Function<1> * const a = 0);
/**
* Same function as above, but for hp
* objects.
*/
template <int dim, int spacedim>
static
void create_boundary_mass_matrix (const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim-1> &q,
SparseMatrix<double> &matrix,
const typename FunctionMap<spacedim>::type &boundary_functions,
Vector<double> &rhs_vector,
std::vector<unsigned int>&dof_to_boundary_mapping,
const Function<spacedim> * const a = 0,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
/**
* Assemble the Laplace
* matrix. If no coefficient is
* given, it is assumed to be
* constant one.
*
* If the library is configured
* to use multithreading, this
* function works in parallel.
*
* See the general doc of this
* class for more information.
*/
template <int dim, int spacedim>
static void create_laplace_matrix (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
SparseMatrix<double> &matrix,
const Function<spacedim> * const a = 0);
/**
* Calls the
* create_laplace_matrix()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, int spacedim>
static void create_laplace_matrix (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
SparseMatrix<double> &matrix,
const Function<spacedim> * const a = 0);
/**
* Assemble the Laplace matrix
* and a right hand side
* vector. If no coefficient is
* given, it is assumed to be
* constant one.
*
* If the library is configured
* to use multithreading, this
* function works in parallel.
*
* See the general doc of this
* class for more information.
*/
template <int dim, int spacedim>
static void create_laplace_matrix (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
SparseMatrix<double> &matrix,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const Function<spacedim> * const a = 0);
/**
* Calls the
* create_laplace_matrix()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, int spacedim>
static void create_laplace_matrix (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
SparseMatrix<double> &matrix,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const Function<spacedim> * const a = 0);
/**
* Like the functions above, but for hp
* dof handlers, mappings, and quadrature
* collections.
*/
template <int dim, int spacedim>
static void create_laplace_matrix (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
SparseMatrix<double> &matrix,
const Function<spacedim> * const a = 0);
/**
* Like the functions above, but for hp
* dof handlers, mappings, and quadrature
* collections.
*/
template <int dim, int spacedim>
static void create_laplace_matrix (const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
SparseMatrix<double> &matrix,
const Function<spacedim> * const a = 0);
/**
* Like the functions above, but for hp
* dof handlers, mappings, and quadrature
* collections.
*/
template <int dim, int spacedim>
static void create_laplace_matrix (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
SparseMatrix<double> &matrix,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const Function<spacedim> * const a = 0);
/**
* Like the functions above, but for hp
* dof handlers, mappings, and quadrature
* collections.
*/
template <int dim, int spacedim>
static void create_laplace_matrix (const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
SparseMatrix<double> &matrix,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const Function<spacedim> * const a = 0);
/**
* Exception
*/
DeclException0 (ExcComponentMismatch);
private:
/**
* Convenience abbreviation for
* pairs of DoF handler cell
* iterators. This type works
* just like a
* <tt>std::pair<iterator,iterator></tt>
* but is templatized on the
* dof handler that shouls be used.
*/
template <typename DH>
struct IteratorRange
{
/**
* Typedef for the iterator type.
*/
typedef typename DH::active_cell_iterator active_cell_iterator;
/**
* Abbreviation for a pair of
* iterators.
*/
typedef std::pair<active_cell_iterator,active_cell_iterator> iterator_pair;
/**
* Constructor. Initialize
* the two values by the
* given values.
*/
IteratorRange (const active_cell_iterator &first,
const active_cell_iterator &second);
/**
* Constructor taking a pair
* of values for
* initialization.
*/
IteratorRange (const iterator_pair &ip);
/**
* Pair of iterators denoting
* a half-open range.
*/
active_cell_iterator first, second;
};
/**
* Version of the same function
* (without suffix @p _1) with
* the same argument list that
* operates only on an interval
* of iterators. Used for
* parallelization. The mutex is
* used to synchronise access to
* the matrix.
*/
template <int dim, int spacedim>
static
void create_boundary_mass_matrix_1 (std_cxx1x::tuple<const Mapping<dim,spacedim> &,
const DoFHandler<dim,spacedim> &,
const Quadrature<dim-1> & > commons,
SparseMatrix<double> &matrix,
const typename FunctionMap<spacedim>::type &boundary_functions,
Vector<double> &rhs_vector,
std::vector<unsigned int>&dof_to_boundary_mapping,
const Function<spacedim> * const coefficient,
const std::vector<unsigned int>& component_mapping,
const IteratorRange<DoFHandler<dim,spacedim> > range,
Threads::ThreadMutex &mutex);
/**
* Same as above, but for hp.
*/
template <int dim, int spacedim>
static
void create_boundary_mass_matrix_1 (std_cxx1x::tuple<const hp::MappingCollection<dim,spacedim> &,
const hp::DoFHandler<dim,spacedim>&,
const hp::QCollection<dim-1> & > commons,
SparseMatrix<double> &matrix,
const typename FunctionMap<spacedim>::type &boundary_functions,
Vector<double> &rhs_vector,
std::vector<unsigned int>&dof_to_boundary_mapping,
const Function<spacedim> * const coefficient,
const std::vector<unsigned int>& component_mapping,
const IteratorRange<hp::DoFHandler<dim,spacedim> > range,
Threads::ThreadMutex &mutex);
};
/**
* Provide a collection of functions operating on matrices. These include
* the application of boundary conditions to a linear system of equations
* and others.
*
*
* <h3>Boundary conditions</h3>
*
* The apply_boundary_values() function inserts boundary conditions
* into a system of equations. To actually do this you have to
* specify a list of degree of freedom indices along with the values
* these degrees of freedom shall assume. To see how to get such a
* list, see the discussion of the
* VectorTools::interpolate_boundary_values function.
*
* There are two ways to incorporate fixed degrees of freedom such as boundary
* nodes into a linear system, as discussed below.
*
*
* <h3>Global elimination</h3>
*
* In the first method, we first assemble the global linear system without
* respect for fixed degrees of freedom, and in a second step eliminate them
* again from the linear system. The inclusion into the assembly process is as
* follows: when the matrix and vectors are set up, a list of nodes subject to
* dirichlet bc is made and matrix and vectors are modified accordingly. This
* is done by deleting all entries in the matrix in the line of this degree of
* freedom, setting the main diagonal entry to a suitable positive value and
* the right hand side element to a value so that the solution of the linear
* system will have the boundary value at this node. To decouple the remaining
* linear system of equations and to make the system symmetric again (at least
* if it was before), one Gauss elimination step is performed with this line,
* by adding this (now almost empty) line to all other lines which couple with
* the given degree of freedom and thus eliminating all coupling between this
* degree of freedom and others. Now the respective column also consists only
* of zeroes, apart from the main diagonal entry. Alternatively, the functions
* in this class take a boolean parameter that allows to omit this last step,
* if symmetry of the resulting linear system is not required. Note that
* usually even CG can cope with a non-symmetric linear system with this
* particular structure.
*
* Finding which rows contain an entry in the column for which we are
* presently performing a Gauss elimination step is either difficult
* or very simple, depending on the circumstances. If the sparsity
* pattern is symmetric (whether the matrix is symmetric is irrelevant
* here), then we can infer the rows which have a nonzero entry in the
* present column by looking at which columns in the present row are
* nonempty. In this case, we only need to look into a fixed number of
* rows and need not search all rows. On the other hand, if the
* sparsity pattern is nonsymmetric, then we need to use an iterative
* solver which can handle nonsymmetric matrices in any case, so there
* may be no need to do the Gauss elimination anyway. In fact, this is
* the way the function works: it takes a parameter
* (@p elininate_columns) that specifies whether the sparsity pattern
* is symmetric; if so, then the column is eliminated and the right
* hand side is also modified accordingly. If not, then only the row
* is deleted and the column is not touched at all, and all right hand
* side values apart from the one corresponding to the present row
* remain unchanged.
*
* If the sparsity pattern for your matrix is non-symmetric, you must
* set the value of this parameter to @p false in any case, since then
* we can't eliminate the column without searching all rows, which
* would be too expensive (if @p N be the number of rows, and @p m the
* number of nonzero elements per row, then eliminating one column is
* an <tt>O(N*log(m))</tt> operation, since searching in each row takes
* <tt>log(m)</tt> operations). If your sparsity pattern is symmetric, but
* your matrix is not, then you might specify @p false as well. If your
* sparsity pattern and matrix are both symmetric, you might want to
* specify @p true (the complexity of eliminating one row is then
* <tt>O(m*log(m))</tt>, since we only have to search @p m rows for the
* respective element of the column). Given the fact that @p m is
* roughly constant, irrespective of the discretization, and that the
* number of boundary nodes is <tt>sqrt(N)</tt> in 2d, the algorithm for
* symmetric sparsity patterns is <tt>O(sqrt(N)*m*log(m))</tt>, while it
* would be <tt>O(N*sqrt(N)*log(m))</tt> for the general case; the latter
* is too expensive to be performed.
*
* It seems as if we had to make clear not to overwrite the lines of
* other boundary nodes when doing the Gauss elimination
* step. However, since we reset the right hand side when passing such
* a node, it is not a problem to change the right hand side values of
* other boundary nodes not yet processed. It would be a problem to
* change those entries of nodes already processed, but since the
* matrix entry of the present column on the row of an already
* processed node is zero, the Gauss step does not change the right
* hand side. We need therefore not take special care of other
* boundary nodes.
*
* To make solving faster, we preset the solution vector with the right
* boundary values (as to why this is necessary, see the discussion below in
* the description of local elimination). It it not clear whether the deletion
* of coupling between the boundary degree of freedom and other dofs really
* forces the corresponding entry in the solution vector to have the right
* value when using iterative solvers, since their search directions may
* contain components in the direction of the boundary node. For this reason,
* we perform a very simple line balancing by not setting the main diagonal
* entry to unity, but rather to the value it had before deleting this line,
* or to the first nonzero main diagonal entry if it is zero for some reason.
* Of course we have to change the right hand side appropriately. This is not
* a very good strategy, but it at least should give the main diagonal entry a
* value in the right order of dimension, which makes the solvution process a
* bit more stable. A refined algorithm would set the entry to the mean of the
* other diagonal entries, but this seems to be too expensive.
*
* In some cases, it might be interesting to solve several times with
* the same matrix, but for different right hand sides or boundary
* values. However, since the modification for boundary values of the
* right hand side vector depends on the original matrix, this is not
* possible without storing the original matrix somewhere and applying
* the @p apply_boundary_conditions function to a copy of it each
* time we want to solve. In that case, you can use the
* FilteredMatrix class in the @p LAC sublibrary. There you can
* also find a formal (mathematical) description of the process of
* modifying the matrix and right hand side vectors for boundary
* values.
*
*
* <h3>Local elimination</h3>
*
* The second way of handling boundary values is to modify the local
* matrix and vector contributions appropriately before transferring
* them into the global sparse matrix and vector. This is what
* local_apply_boundary_values() does. The advantage is that we save
* the call to the apply_boundary_values function (which is expensive
* because it has to work on sparse data structures). On the other
* hand, the local_apply_boundary_values() function is called many
* times, even if we only have a very small number of fixed boundary
* nodes, and the main drawback is that this function doesn't work as
* expected if there are hanging nodes that also need to be
* treated. The reason that this function doesn't work is that it is
* meant to be run before distribution into the global matrix,
* i.e. before hanging nodes are distributed; since hanging nodes can
* be constrained to a boundary node, the treatment of hanging nodes
* can add entries again to rows and columns corresponding to boundary
* values and that we have already vacated in the local elimination
* step. To make things worse, in 3d constrained nodes can even lie on
* the boundary. Thus, it is imperative that boundary node elimination
* happens @em after hanging node elimination, but this can't be
* achieved with local elimination of boundary nodes unless there are
* no hanging node constraints at all.
*
* Local elimination has one additional drawback: we don't have access
* to the solution vector, only to the local contributions to the
* matrix and right hand side. The problem with this is subtle, but
* can lead to very hard to find difficulties: when we eliminate a
* degree of freedom, we delete the row and column of this unknown,
* and set the diagonal entry to some positive value. To make the
* problem more or less well-conditioned, we set this diagonal entry
* to the absolute value of its prior value if that was non-zero, or
* to the average magnitude of all other nonzero diagonal
* elements. Then we set the right hand side value such that the
* resulting solution entry has the right value as given by the
* boundary values. Since we add these contributions up over all local
* contributions, the diagonal entry and the respective value in the
* right hand side are added up correspondingly, so that the entry in
* the solution of the linear system is still valid.
*
* A problem arises, however, if the diagonal entries so chosen are not
* appropriate for the linear system. Consider, for example, a mixed Laplace
* problem with matrix <tt>[[A B][C^T 0]]</tt>, where we only specify boundary
* values for the second component of the solution. In the mixed formulation,
* the stress-strain tensor only appears in either the matrix @p B or @p C, so
* one of them may be significantly larger or smaller than the other one. Now,
* if we eliminate boundary values, we delete some rows and columns, but we
* also introduce a few entries on the diagonal of the lower right block, so
* that we get the system <tt>[[A' B'][C'^T X]]</tt>. The diagonal entries in
* the matrix @p X will be of the same order of magnitude as those in @p
* A. Now, if we solve this system in the Schur complement formulation, we
* have to invert the matrix <tt>X-C'^TA'^{-1}B'</tt>. Deleting rows and
* columns above makes sure that boundary nodes indeed have empty rows and
* columns in the Schur complement as well, except for the entries in @p
* X. However, the entries in @p X may be of significantly different orders of
* magnitude than those in <tt>C'^TA'^{-1}B'</tt>! If this is the case, we may
* run into trouble with iterative solvers. For example, assume that we start
* with zero entries in the solution vector and that the entries in @p X are
* several orders of magnitude too small; in this case, iterative solvers will
* compute the residual vector in each step and form correction vectors, but
* since the entries in @p X are so small, the residual contributions for
* boundary nodes are really small, despite the fact that the boundary nodes
* are still at values close to zero and not in accordance with the prescribed
* boundary values. Since the residual is so small, the corrections the
* iterative solver computes are very small, and in the end the solver will
* indicate convergence to a small total residual with the boundary values
* still being significantly wrong.
*
* We avoid this problem in the global elimination process described above by
* 'priming' the solution vector with the correct values for boundary
* nodes. However, we can't do this for the local elimination
* process. Therefore, if you experience a problem like the one above, you
* need to either increase the diagonal entries in @p X to a size that matches
* those in the other part of the Schur complement, or, simpler, prime the
* solution vector before you start the solver.
*
* In conclusion, local elimination of boundary nodes only works if
* there are no hanging nodes and even then doesn't always work fully
* satisfactorily.
*
* @ingroup numerics
* @author Wolfgang Bangerth, 1998, 2000, 2004, 2005
*/
class MatrixTools : public MatrixCreator
{
public:
/**
* Apply dirichlet boundary conditions
* to the system matrix and vectors
* as described in the general
* documentation.
*/
template <typename number>
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
SparseMatrix<number> &matrix,
Vector<number> &solution,
Vector<number> &right_hand_side,
const bool eliminate_columns = true);
/**
* Apply dirichlet boundary
* conditions to the system
* matrix and vectors as
* described in the general
* documentation. This function
* works for block sparse
* matrices and block vectors
*/
template <typename number>
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
BlockSparseMatrix<number> &matrix,
BlockVector<number> &solution,
BlockVector<number> &right_hand_side,
const bool eliminate_columns = true);
#ifdef DEAL_II_USE_PETSC
/**
* Apply dirichlet boundary conditions to
* the system matrix and vectors as
* described in the general
* documentation. This function works on
* the classes that are used to wrap
* PETSc objects.
*
* Note that this function is not very
* efficient: it needs to alternatingly
* read and write into the matrix, a
* situation that PETSc does not handle
* too well. In addition, we only get rid
* of rows corresponding to boundary
* nodes, but the corresponding case of
* deleting the respective columns
* (i.e. if @p eliminate_columns is @p
* true) is not presently implemented,
* and probably will never because it is
* too expensive without direct access to
* the PETSc data structures. (This leads
* to the situation where the action
* indicates by the default value of the
* last argument is actually not
* implemented; that argument has
* <code>true</code> as its default value
* to stay consistent with the other
* functions of same name in this class.)
* A third reason against this function
* is that it doesn't handle the case
* where the matrix is distributed across
* an MPI system.
*
* This function is used in
* step-17 and
* step-18.
*/
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
PETScWrappers::SparseMatrix &matrix,
PETScWrappers::Vector &solution,
PETScWrappers::Vector &right_hand_side,
const bool eliminate_columns = true);
/**
* Same function, but for parallel PETSc
* matrices.
*/
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
PETScWrappers::MPI::SparseMatrix &matrix,
PETScWrappers::MPI::Vector &solution,
PETScWrappers::MPI::Vector &right_hand_side,
const bool eliminate_columns = true);
/**
* Same function, but for
* parallel PETSc matrices. Note
* that this function only
* operates on the local range of
* the parallel matrix, i.e. it
* only eliminates rows
* corresponding to degrees of
* freedom for which the row is
* stored on the present
* processor. All other boundary
* nodes are ignored, and it
* doesn't matter whether they
* are present in the first
* argument to this function or
* not. A consequence of this,
* however, is that this function
* has to be called from all
* processors that participate in
* sharing the contents of the
* given matrices and vectors. It
* is also implied that the local
* range for all objects passed
* to this function is the same.
*/
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
PETScWrappers::MPI::SparseMatrix &matrix,
PETScWrappers::Vector &solution,
PETScWrappers::MPI::Vector &right_hand_side,
const bool eliminate_columns = true);
/**
* Same as above but for BlockSparseMatrix.
*/
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
PETScWrappers::MPI::BlockSparseMatrix &matrix,
PETScWrappers::MPI::BlockVector &solution,
PETScWrappers::MPI::BlockVector &right_hand_side,
const bool eliminate_columns = true);
#endif
#ifdef DEAL_II_USE_TRILINOS
/**
* Apply dirichlet boundary
* conditions to the system matrix
* and vectors as described in the
* general documentation. This
* function works on the classes
* that are used to wrap Trilinos
* objects.
*
* Note that this function is not
* very efficient: it needs to
* alternatingly read and write
* into the matrix, a situation
* that Trilinos does not handle
* too well. In addition, we only
* get rid of rows corresponding to
* boundary nodes, but the
* corresponding case of deleting
* the respective columns (i.e. if
* @p eliminate_columns is @p true)
* is not presently implemented,
* and probably will never because
* it is too expensive without
* direct access to the Trilinos
* data structures. (This leads to
* the situation where the action
* indicates by the default value
* of the last argument is actually
* not implemented; that argument
* has <code>true</code> as its
* default value to stay consistent
* with the other functions of same
* name in this class.) A third
* reason against this function is
* that it doesn't handle the case
* where the matrix is distributed
* across an MPI system.
*/
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
TrilinosWrappers::SparseMatrix &matrix,
TrilinosWrappers::Vector &solution,
TrilinosWrappers::Vector &right_hand_side,
const bool eliminate_columns = true);
/**
* This function does the same as
* the one above, except now
* working on block structures.
*/
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
TrilinosWrappers::BlockSparseMatrix &matrix,
TrilinosWrappers::BlockVector &solution,
TrilinosWrappers::BlockVector &right_hand_side,
const bool eliminate_columns = true);
/**
* Apply dirichlet boundary
* conditions to the system matrix
* and vectors as described in the
* general documentation. This
* function works on the classes
* that are used to wrap Trilinos
* objects.
*
* Note that this function is not
* very efficient: it needs to
* alternatingly read and write
* into the matrix, a situation
* that Trilinos does not handle
* too well. In addition, we only
* get rid of rows corresponding to
* boundary nodes, but the
* corresponding case of deleting
* the respective columns (i.e. if
* @p eliminate_columns is @p true)
* is not presently implemented,
* and probably will never because
* it is too expensive without
* direct access to the Trilinos
* data structures. (This leads to
* the situation where the action
* indicates by the default value
* of the last argument is actually
* not implemented; that argument
* has <code>true</code> as its
* default value to stay consistent
* with the other functions of same
* name in this class.) This
* function does work on MPI vector
* types.
*/
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
TrilinosWrappers::SparseMatrix &matrix,
TrilinosWrappers::MPI::Vector &solution,
TrilinosWrappers::MPI::Vector &right_hand_side,
const bool eliminate_columns = true);
/**
* This function does the same as
* the one above, except now working
* on block structures.
*/
static void
apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
TrilinosWrappers::BlockSparseMatrix &matrix,
TrilinosWrappers::MPI::BlockVector &solution,
TrilinosWrappers::MPI::BlockVector &right_hand_side,
const bool eliminate_columns = true);
#endif
/**
* Rather than applying boundary
* values to the global matrix
* and vector after creating the
* global matrix, this function
* does so during assembly, by
* modifying the local matrix and
* vector contributions. If you
* call this function on all
* local contributions, the
* resulting matrix will have the
* same entries, and the final
* call to
* apply_boundary_values() on the
* global system will not be
* necessary.
*
* Since this function does not
* have to work on the
* complicated data structures of
* sparse matrices, it is
* relatively cheap. It may
* therefore be a win if you have
* many fixed degrees of freedom
* (e.g. boundary nodes), or if
* access to the sparse matrix is
* expensive (e.g. for block
* sparse matrices, or for PETSc
* or trilinos
* matrices). However, it doesn't
* work as expected if there are
* also hanging nodes to be
* considered. More caveats are
* listed in the general
* documentation of this class.
*/
static void
local_apply_boundary_values (const std::map<unsigned int,double> &boundary_values,
const std::vector<unsigned int> &local_dof_indices,
FullMatrix<double> &local_matrix,
Vector<double> &local_rhs,
const bool eliminate_columns);
/**
* Exception
*/
DeclException0 (ExcBlocksDontMatch);
};
DEAL_II_NAMESPACE_CLOSE
#endif
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