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// $Id: vector_tools.h 31932 2013-12-08 02:15:54Z heister $
//
// Copyright (C) 1998 - 2013 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef __deal2__vector_tools_h
#define __deal2__vector_tools_h
#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/base/point.h>
#include <deal.II/dofs/function_map.h>
#include <deal.II/fe/mapping_q.h>
#include <deal.II/hp/mapping_collection.h>
#include <map>
#include <vector>
#include <set>
DEAL_II_NAMESPACE_OPEN
template <int dim> class Function;
template <int dim> struct FunctionMap;
template <int dim> class Quadrature;
template <int dim> class QGauss;
template <typename number> class Vector;
template <typename number> class FullMatrix;
template <int dim, int spacedim> class Mapping;
template <int dim, int spacedim> class DoFHandler;
template <typename gridtype> class InterGridMap;
namespace hp
{
template <int dim, int spacedim> class DoFHandler;
template <int dim, int spacedim> class MappingCollection;
template <int dim> class QCollection;
}
class ConstraintMatrix;
//TODO: Move documentation of functions to the functions!
/**
* Provide a namespace which offers some operations on vectors. Among
* these are assembling of standard vectors, integration of the
* difference of a finite element solution and a continuous function,
* interpolations and projections of continuous functions to the
* finite element space and other operations.
*
* @note There exist two versions of almost each function. One with a
* Mapping argument and one without. If a code uses a mapping
* different from MappingQ1 the functions <b>with</b> mapping argument
* should be used. Code that uses only MappingQ1 may also use the
* functions without Mapping argument. Each of these latter functions
* create a MappingQ1 object and just call the respective functions
* with that object as mapping argument. The functions without Mapping
* argument still exist to ensure backward compatibility. Nevertheless
* it is advised to change the user's codes to store a specific
* Mapping object and to use the functions that take this Mapping
* object as argument. This gives the possibility to easily extend the
* user codes to work also on mappings of higher degree, this just by
* exchanging MappingQ1 by, for example, a MappingQ or another Mapping
* object of interest.
*
* <h3>Description of operations</h3>
*
* This collection of methods offers the following operations:
* <ul>
* <li> Interpolation: assign each degree of freedom in the vector to be
* the value of the function given as argument. This is identical to
* saying that the resulting finite element function (which is
* isomorphic to the output vector) has exact function values in all
* support points of trial functions. The support point of a trial
* function is the point where its value equals one, e.g. for linear
* trial functions the support points are four corners of an
* element. This function therefore relies on the assumption that a
* finite element is used for which the degrees of freedom are
* function values (Lagrange elements) rather than gradients, normal
* derivatives, second derivatives, etc (Hermite elements, quintic
* Argyris element, etc.).
*
* It seems inevitable that some values of the vector to be created are set
* twice or even more than that. The reason is that we have to loop over
* all cells and get the function values for each of the trial functions
* located thereon. This applies also to the functions located on faces and
* corners which we thus visit more than once. While setting the value
* in the vector is not an expensive operation, the evaluation of the
* given function may be, taking into account that a virtual function has
* to be called.
*
* <li> Projection: compute the <i>L</i><sup>2</sup>-projection of the
* given function onto the finite element space, i.e. if <i>f</i> is
* the function to be projected, compute <i>f<sub>h</sub></i> in
* <i>V<sub>h</sub></i> such that
* (<i>f<sub>h</sub></i>,<i>v<sub>h</sub></i>)=(<i>f</i>,<i>v<sub>h</sub></i>)
* for all discrete test functions <i>v<sub>h</sub></i>. This is done
* through the solution of the linear system of equations <i> M v =
* f</i> where <i>M</i> is the mass matrix $m_{ij} = \int_\Omega
* \phi_i(x) \phi_j(x) dx$ and $f_i = \int_\Omega f(x) \phi_i(x)
* dx$. The solution vector $v$ then is the nodal representation of
* the projection <i>f<sub>h</sub></i>. The project() functions are
* used in the step-21 and step-23
* tutorial programs.
*
* In order to get proper results, it be may necessary to treat
* boundary conditions right. Below are listed some cases where this
* may be needed. If needed, this is done by <i>L</i><sup>2</sup>-projection of
* the trace of the given function onto the finite element space
* restricted to the boundary of the domain, then taking this
* information and using it to eliminate the boundary nodes from the
* mass matrix of the whole domain, using the
* MatrixTools::apply_boundary_values() function. The projection of
* the trace of the function to the boundary is done with the
* VectorTools::project_boundary_values() (see below) function,
* which is called with a map of boundary functions FunctioMap in
* which all boundary indicators from zero to numbers::internal_face_boundary_id-1
* (numbers::internal_face_boundary_id is used for other purposes,
* see the Triangulation class documentation) point
* to the function to be projected. The projection to the boundary
* takes place using a second quadrature formula on the boundary
* given to the project() function. The first quadrature formula is
* used to compute the right hand side and for numerical quadrature
* of the mass matrix.
*
* The projection of the boundary values first, then eliminating
* them from the global system of equations is not needed
* usually. It may be necessary if you want to enforce special
* restrictions on the boundary values of the projected function,
* for example in time dependent problems: you may want to project
* the initial values but need consistency with the boundary values
* for later times. Since the latter are projected onto the boundary
* in each time step, it is necessary that we also project the
* boundary values of the initial values, before projecting them to
* the whole domain.
*
* Obviously, the results of the two schemes for projection are
* different. Usually, when projecting to the boundary first, the
* <i>L</i><sup>2</sup>-norm of the difference between original
* function and projection over the whole domain will be larger
* (factors of five have been observed) while the
* <i>L</i><sup>2</sup>-norm of the error integrated over the
* boundary should of course be less. The reverse should also hold
* if no projection to the boundary is performed.
*
* The selection whether the projection to the boundary first is
* needed is done with the <tt>project_to_boundary_first</tt> flag
* passed to the function. If @p false is given, the additional
* quadrature formula for faces is ignored.
*
* You should be aware of the fact that if no projection to the boundary
* is requested, a function with zero boundary values may not have zero
* boundary values after projection. There is a flag for this especially
* important case, which tells the function to enforce zero boundary values
* on the respective boundary parts. Since enforced zero boundary values
* could also have been reached through projection, but are more economically
* obtain using other methods, the @p project_to_boundary_first flag is
* ignored if the @p enforce_zero_boundary flag is set.
*
* The solution of the linear system is presently done using a simple CG
* method without preconditioning and without multigrid. This is clearly not
* too efficient, but sufficient in many cases and simple to implement. This
* detail may change in the future.
*
* <li> Creation of right hand side vectors:
* The create_right_hand_side() function computes the vector
* $f_i = \int_\Omega f(x) \phi_i(x) dx$. This is the same as what the
* <tt>MatrixCreator::create_*</tt> functions which take a right hand side do,
* but without assembling a matrix.
*
* <li> Creation of right hand side vectors for point sources:
* The create_point_source_vector() function computes the vector
* $f_i = \int_\Omega \delta(x-x_0) \phi_i(x) dx$.
*
* <li> Creation of boundary right hand side vectors: The
* create_boundary_right_hand_side() function computes the vector
* $f_i = \int_{\partial\Omega} g(x) \phi_i(x) dx$. This is the
* right hand side contribution of boundary forces when having
* inhomogeneous Neumann boundary values in Laplace's equation or
* other second order operators. This function also takes an
* optional argument denoting over which parts of the boundary the
* integration shall extend. If the default argument is used, it is applied
* to all boundaries.
*
* <li> Interpolation of boundary values:
* The MatrixTools::apply_boundary_values() function takes a list
* of boundary nodes and their values. You can get such a list by interpolation
* of a boundary function using the interpolate_boundary_values() function.
* To use it, you have to
* specify a list of pairs of boundary indicators (of type <tt>types::boundary_id</tt>;
* see the section in the documentation of the Triangulation class for more
* details) and the according functions denoting the dirichlet boundary values
* of the nodes on boundary faces with this boundary indicator.
*
* Usually, all other boundary conditions, such as inhomogeneous Neumann values
* or mixed boundary conditions are handled in the weak formulation. No attempt
* is made to include these into the process of matrix and vector assembly therefore.
*
* Within this function, boundary values are interpolated, i.e. a node is given
* the point value of the boundary function. In some cases, it may be necessary
* to use the L2-projection of the boundary function or any other method. For
* this purpose we refer to the project_boundary_values()
* function below.
*
* You should be aware that the boundary function may be evaluated at nodes
* on the interior of faces. These, however, need not be on the true
* boundary, but rather are on the approximation of the boundary represented
* by the mapping of the unit cell to the real cell. Since this mapping will
* in most cases not be the exact one at the face, the boundary function is
* evaluated at points which are not on the boundary and you should make
* sure that the returned values are reasonable in some sense anyway.
*
* In 1d the situation is a bit different since there faces (i.e. vertices) have
* no boundary indicator. It is assumed that if the boundary indicator zero
* is given in the list of boundary functions, the left boundary point is to be
* interpolated while the right boundary point is associated with the boundary
* index 1 in the map. The respective boundary functions are then evaluated at
* the place of the respective boundary point.
*
* <li> Projection of boundary values:
* The project_boundary_values() function acts similar to the
* interpolate_boundary_values() function, apart from the fact that it does
* not get the nodal values of boundary nodes by interpolation but rather
* through the <i>L</i><sup>2</sup>-projection of the trace of the function to the boundary.
*
* The projection takes place on all boundary parts with boundary
* indicators listed in the map (FunctioMap::FunctionMap)
* of boundary functions. These boundary parts may or may not be
* continuous. For these boundary parts, the mass matrix is
* assembled using the
* MatrixTools::create_boundary_mass_matrix() function, as
* well as the appropriate right hand side. Then the resulting
* system of equations is solved using a simple CG method (without
* preconditioning), which is in most cases sufficient for the
* present purpose.
*
* <li> Computing errors:
* The function integrate_difference() performs the calculation of
* the error between a given (continuous) reference function and the
* finite element solution in different norms. The integration is
* performed using a given quadrature formula and assumes that the
* given finite element objects equals that used for the computation
* of the solution.
*
* The result is stored in a vector (named @p difference), where each entry
* equals the given norm of the difference on a cell. The order of entries
* is the same as a @p cell_iterator takes when started with @p begin_active and
* promoted with the <tt>++</tt> operator.
*
* This data, one number per active cell, can be used to generate
* graphical output by directly passing it to the DataOut class
* through the DataOut::add_data_vector function. Alternatively, it
* can be interpolated to the nodal points of a finite element field
* using the DoFTools::distribute_cell_to_dof_vector function.
*
* Presently, there is the possibility to compute the following values from the
* difference, on each cell: @p mean, @p L1_norm, @p L2_norm, @p Linfty_norm,
* @p H1_seminorm and @p H1_norm, see VectorTools::NormType.
* For the mean difference value, the reference function minus the numerical
* solution is computed, not the other way round.
*
* The infinity norm of the difference on a given cell returns the maximum
* absolute value of the difference at the quadrature points given by the
* quadrature formula parameter. This will in some cases not be too good
* an approximation, since for example the Gauss quadrature formulae do
* not evaluate the difference at the end or corner points of the cells.
* You may want to choose a quadrature formula with more quadrature points
* or one with another distribution of the quadrature points in this case.
* You should also take into account the superconvergence properties of finite
* elements in some points: for example in 1D, the standard finite element
* method is a collocation method and should return the exact value at nodal
* points. Therefore, the trapezoidal rule should always return a vanishing
* L-infinity error. Conversely, in 2D the maximum L-infinity error should
* be located at the vertices or at the center of the cell, which would make
* it plausible to use the Simpson quadrature rule. On the other hand, there
* may be superconvergence at Gauss integration points. These examples are not
* intended as a rule of thumb, rather they are thought to illustrate that the
* use of the wrong quadrature formula may show a significantly wrong result
* and care should be taken to chose the right formula.
*
* The <i>H</i><sup>1</sup> seminorm is the <i>L</i><sup>2</sup>
* norm of the gradient of the difference. The square of the full
* <i>H</i><sup>1</sup> norm is the sum of the square of seminorm
* and the square of the <i>L</i><sup>2</sup> norm.
*
* To get the global <i>L<sup>1</sup></i> error, you have to sum up the
* entries in @p difference, e.g. using
* Vector::l1_norm() function. For the global <i>L</i><sup>2</sup>
* difference, you have to sum up the squares of the entries and
* take the root of the sum, e.g. using
* Vector::l2_norm(). These two operations
* represent the <i>l</i><sub>1</sub> and <i>l</i><sub>2</sub> norms of the vectors, but you need
* not take the absolute value of each entry, since the cellwise
* norms are already positive.
*
* To get the global mean difference, simply sum up the elements as above.
* To get the $L_\infty$ norm, take the maximum of the vector elements, e.g.
* using the Vector::linfty_norm() function.
*
* For the global <i>H</i><sup>1</sup> norm and seminorm, the same rule applies as for the
* <i>L</i><sup>2</sup> norm: compute the <i>l</i><sub>2</sub> norm
* of the cell error vector.
*
* Note that, in the codimension one case, if you ask for a norm
* that requires the computation of a gradient, then the provided
* function is automatically projected along the curve, and the
* difference is only computed on the tangential part of the
* gradient, since no information is available on the normal
* component of the gradient anyway.
* </ul>
*
* All functions use the finite element given to the DoFHandler object the last
* time that the degrees of freedom were distributed over the triangulation. Also,
* if access to an object describing the exact form of the boundary is needed, the
* pointer stored within the triangulation object is accessed.
*
* @note Instantiations for this template are provided for some vector types,
* in particular <code>Vector<float>, Vector<double>,
* BlockVector<float>, BlockVector<double></code>; others can be
* generated in application code (see the section on @ref Instantiations in
* the manual).
*
* @ingroup numerics
* @author Wolfgang Bangerth, Ralf Hartmann, Guido Kanschat, 1998, 1999, 2000, 2001
*/
namespace VectorTools
{
/**
* Denote which norm/integral is
* to be computed by the
* integrate_difference()
* function of this class. The
* following possibilities are
* implemented:
*/
enum NormType
{
/**
* The function or
* difference of functions
* is integrated on each
* cell.
*/
mean,
/**
* The absolute value of
* the function is
* integrated.
*/
L1_norm,
/**
* The square of the
* function is integrated
* and the the square root
* of the result is
* computed on each cell.
*/
L2_norm,
/**
* The absolute value to
* the <i>p</i>th power is
* integrated and the pth
* root is computed on each
* cell. The exponent
* <i>p</i> is the last
* parameter of the
* function.
*/
Lp_norm,
/**
* The maximum absolute
* value of the function.
*/
Linfty_norm,
/**
* #L2_norm of the gradient.
*/
H1_seminorm,
/**
* The square of this norm
* is the square of the
* #L2_norm plus the square
* of the #H1_seminorm.
*/
H1_norm,
/**
* #Lp_norm of the gradient.
*/
W1p_seminorm,
/**
* same as #H1_norm for
* <i>L<sup>p</sup></i>.
*/
W1p_norm,
/**
* #Linfty_norm of the gradient.
*/
W1infty_seminorm,
/**
* same as #H1_norm for
* <i>L<sup>infty</sup></i>.
*/
W1infty_norm
};
/**
* @name Interpolation and projection
*/
//@{
/**
* Compute the interpolation of
* @p function at the support
* points to the finite element
* space described by the Triangulation
* and FiniteElement object with which
* the given DoFHandler argument is
* initialized. It is assumed that the
* number of components of
* @p function matches that of
* the finite element used by
* @p dof.
*
* Note that you may have to call
* <tt>hanging_nodes.distribute(vec)</tt>
* with the hanging nodes from
* space @p dof afterwards, to
* make the result continuous
* again.
*
* The template argument <code>DH</code>
* may either be of type DoFHandler or
* hp::DoFHandler.
*
* See the general documentation
* of this class for further
* information.
*
* @todo The @p mapping argument should be
* replaced by a hp::MappingCollection in
* case of a hp::DoFHandler.
*/
template <class VECTOR, int dim, int spacedim, template <int,int> class DH>
void interpolate (const Mapping<dim,spacedim> &mapping,
const DH<dim,spacedim> &dof,
const Function<spacedim> &function,
VECTOR &vec);
/**
* Calls the @p interpolate()
* function above with
* <tt>mapping=MappingQ1@<dim>@()</tt>.
*/
template <class VECTOR, class DH>
void interpolate (const DH &dof,
const Function<DH::space_dimension> &function,
VECTOR &vec);
/**
* Interpolate different finite
* element spaces. The
* interpolation of vector
* @p data_1 is executed from the
* FE space represented by
* @p dof_1 to the vector @p data_2
* on FE space @p dof_2. The
* interpolation on each cell is
* represented by the matrix
* @p transfer. Curved boundaries
* are neglected so far.
*
* Note that you may have to call
* <tt>hanging_nodes.distribute(data_2)</tt>
* with the hanging nodes from
* space @p dof_2 afterwards, to
* make the result continuous
* again.
*
* @note Instantiations for this template
* are provided for some vector types
* (see the general documentation of the
* class), but only the same vector for
* InVector and OutVector. Other
* combinations must be instantiated by
* hand.
*/
template <int dim, class InVector, class OutVector, int spacedim>
void interpolate (const DoFHandler<dim,spacedim> &dof_1,
const DoFHandler<dim,spacedim> &dof_2,
const FullMatrix<double> &transfer,
const InVector &data_1,
OutVector &data_2);
/**
* Gives the interpolation of a
* @p dof1-function @p u1 to a
* @p dof2-function @p u2, where @p
* dof1 and @p dof2 represent
* different triangulations with a
* common coarse grid.
*
* dof1 and dof2 need to have the
* same finite element
* discretization.
*
* Note that for continuous
* elements on grids with hanging
* nodes (i.e. locally refined
* grids) this function does not
* give the expected output.
* Indeed, the resulting output
* vector does not necessarily
* respect continuity
* requirements at hanging nodes,
* due to local cellwise
* interpolation.
*
* For this case (continuous
* elements on grids with hanging
* nodes), please use the
* interpolate_to_different_mesh
* function with an additional
* ConstraintMatrix argument,
* see below, or make the field
* conforming yourself by calling the
* @p ConstraintsMatrix::distribute
* function of your hanging node
* constraints object.
*/
template <int dim, int spacedim,
template <int,int> class DH,
class VECTOR>
void
interpolate_to_different_mesh (const DH<dim, spacedim> &dof1,
const VECTOR &u1,
const DH<dim, spacedim> &dof2,
VECTOR &u2);
/**
* Gives the interpolation of a
* @p dof1-function @p u1 to a
* @p dof2-function @p u2, where @p
* dof1 and @p dof2 represent
* different triangulations with a
* common coarse grid.
*
* dof1 and dof2 need to have the
* same finite element
* discretization.
*
* @p constraints is a hanging node
* constraints object corresponding
* to @p dof2. This object is
* particularly important when
* interpolating onto continuous
* elements on grids with hanging
* nodes (locally refined grids):
* Without it - due to cellwise
* interpolation - the resulting
* output vector does not necessarily
* respect continuity requirements
* at hanging nodes.
*/
template <int dim, int spacedim,
template <int,int> class DH,
class VECTOR>
void
interpolate_to_different_mesh (const DH<dim, spacedim> &dof1,
const VECTOR &u1,
const DH<dim, spacedim> &dof2,
const ConstraintMatrix &constraints,
VECTOR &u2);
/**
* The same function as above, but
* takes an InterGridMap object
* directly as a parameter. Useful
* for interpolating several vectors
* at the same time.
*
* @p intergridmap
* has to be initialized via
* InterGridMap::make_mapping pointing
* from a source DoFHandler to a
* destination DoFHandler.
*/
template <int dim, int spacedim,
template <int,int> class DH,
class VECTOR>
void
interpolate_to_different_mesh (const InterGridMap<DH<dim, spacedim> > &intergridmap,
const VECTOR &u1,
const ConstraintMatrix &constraints,
VECTOR &u2);
/**
* Compute the projection of
* @p function to the finite element space.
*
* By default, projection to the boundary
* and enforcement of zero boundary values
* are disabled. The ordering of arguments
* to this function is such that you need
* not give a second quadrature formula if
* you don't want to project to the
* boundary first, but that you must if you
* want to do so.
*
* This function needs the mass
* matrix of the finite element
* space on the present grid. To
* this end, the mass matrix is
* assembled exactly using
* MatrixTools::create_mass_matrix. This
* function performs numerical
* quadrature using the given
* quadrature rule; you should
* therefore make sure that the
* given quadrature formula is
* also sufficient for the
* integration of the mass
* matrix.
*
* See the general documentation of this
* class for further information.
*
* In 1d, the default value of
* the boundary quadrature
* formula is an invalid object
* since integration on the
* boundary doesn't happen in
* 1d.
*/
template <int dim, class VECTOR, int spacedim>
void project (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const ConstraintMatrix &constraints,
const Quadrature<dim> &quadrature,
const Function<spacedim> &function,
VECTOR &vec,
const bool enforce_zero_boundary = false,
const Quadrature<dim-1> &q_boundary = (dim > 1 ?
QGauss<dim-1>(2) :
Quadrature<dim-1>(0)),
const bool project_to_boundary_first = false);
/**
* Calls the project()
* function above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, class VECTOR, int spacedim>
void project (const DoFHandler<dim,spacedim> &dof,
const ConstraintMatrix &constraints,
const Quadrature<dim> &quadrature,
const Function<spacedim> &function,
VECTOR &vec,
const bool enforce_zero_boundary = false,
const Quadrature<dim-1> &q_boundary = (dim > 1 ?
QGauss<dim-1>(2) :
Quadrature<dim-1>(0)),
const bool project_to_boundary_first = false);
/**
* Same as above, but for arguments of type hp::DoFHandler,
* hp::QuadratureCollection, hp::MappingCollection
*/
template <int dim, class VECTOR, int spacedim>
void project (const hp::MappingCollection<dim, spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const ConstraintMatrix &constraints,
const hp::QCollection<dim> &quadrature,
const Function<spacedim> &function,
VECTOR &vec,
const bool enforce_zero_boundary = false,
const hp::QCollection<dim-1> &q_boundary = hp::QCollection<dim-1>(dim > 1 ?
QGauss<dim-1>(2) :
Quadrature<dim-1>(0)),
const bool project_to_boundary_first = false);
/**
* Calls the project()
* function above, with a collection of
* MappingQ1@<dim@>() objects.
*/
template <int dim, class VECTOR, int spacedim>
void project (const hp::DoFHandler<dim,spacedim> &dof,
const ConstraintMatrix &constraints,
const hp::QCollection<dim> &quadrature,
const Function<spacedim> &function,
VECTOR &vec,
const bool enforce_zero_boundary = false,
const hp::QCollection<dim-1> &q_boundary = hp::QCollection<dim-1>(dim > 1 ?
QGauss<dim-1>(2) :
Quadrature<dim-1>(0)),
const bool project_to_boundary_first = false);
/**
* Compute Dirichlet boundary
* conditions. This function makes up a map
* of degrees of freedom subject
* to Dirichlet boundary
* conditions and the corresponding values to
* be assigned to them, by
* interpolation around the
* boundary. If the
* @p boundary_values object contained
* values before, the new ones
* are added, or the old ones
* overwritten if a node of the
* boundary part to be used
* was already in the
* map of boundary values.
*
* The parameter
* @p function_map
* provides a list of boundary
* indicators to be handled by
* this function and corresponding
* boundary value functions. The
* keys of this map
* correspond to the number
* @p boundary_indicator of the
* face. numbers::internal_face_boundary_id
* is an illegal value for this key since
* it is reserved for interior faces.
*
* The flags in the last
* parameter, @p component_mask
* denote which components of the
* finite element space shall be
* interpolated. If it is left as
* specified by the default value
* (i.e. an empty array), all
* components are
* interpolated. If it is
* different from the default
* value, it is assumed that the
* number of entries equals the
* number of components in the
* boundary functions and the
* finite element, and those
* components in the given
* boundary function will be used
* for which the respective flag
* was set in the component mask.
* See also @ref GlossComponentMask. As an example, assume that you are
* solving the Stokes equations in 2d, with variables $(u,v,p)$ and that
* you only want to interpolate boundary values for the pressure, then
* the component mask should correspond to <code>(true,true,false)</code>.
*
* @note Whether a component mask has been specified or not, the number
* of components of the functions
* in @p function_map must match that
* of the finite element used by
* @p dof. In other words, for the example above, you need to provide a
* Function object that has 3 components (the two velocities and the
* pressure), even though you are only
* interested in the first two of them. interpolate_boundary_values()
* will then call this function to obtain a vector of 3 values at each
* interpolation point but only take the first two and discard the third.
* In other words, you are free to return whatever you like in the third
* component of the vector returned by Function::vector_value, but the
* Function object must state that it has 3 components.
*
* If the finite element used has
* shape functions that are
* non-zero in more than one
* component (in deal.II speak:
* they are non-primitive), then
* these components can presently
* not be used for interpolating
* boundary values. Thus, the
* elements in the component mask
* corresponding to the
* components of these
* non-primitive shape functions
* must be @p false.
*
* See the general documentation of this class for more
* information.
*/
template <class DH>
void
interpolate_boundary_values (const Mapping<DH::dimension,DH::space_dimension> &mapping,
const DH &dof,
const typename FunctionMap<DH::space_dimension>::type &function_map,
std::map<types::global_dof_index,double> &boundary_values,
const ComponentMask &component_mask = ComponentMask());
/**
* Like the previous function, but take a mapping collection to go with
* the hp::DoFHandler object.
*/
template <int dim, int spacedim>
void
interpolate_boundary_values (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const typename FunctionMap<spacedim>::type &function_map,
std::map<types::global_dof_index,double> &boundary_values,
const ComponentMask &component_mask = ComponentMask());
/**
* Same function as above, but
* taking only one pair of
* boundary indicator and
* corresponding boundary
* function. The same comments apply as for the previous function, in particular
* about the use of the component mask and the requires size of the function
* object.
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <class DH>
void
interpolate_boundary_values (const Mapping<DH::dimension,DH::space_dimension> &mapping,
const DH &dof,
const types::boundary_id boundary_component,
const Function<DH::space_dimension> &boundary_function,
std::map<types::global_dof_index,double> &boundary_values,
const ComponentMask &component_mask = ComponentMask());
/**
* Calls the other
* interpolate_boundary_values()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
* The same comments apply as for the previous function, in particular
* about the use of the component mask and the requires size of the function
* object.
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <class DH>
void
interpolate_boundary_values (const DH &dof,
const types::boundary_id boundary_component,
const Function<DH::space_dimension> &boundary_function,
std::map<types::global_dof_index,double> &boundary_values,
const ComponentMask &component_mask = ComponentMask());
/**
* Calls the other
* interpolate_boundary_values()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
* The same comments apply as for the previous function, in particular
* about the use of the component mask and the requires size of the function
* object.
*/
template <class DH>
void
interpolate_boundary_values (const DH &dof,
const typename FunctionMap<DH::space_dimension>::type &function_map,
std::map<types::global_dof_index,double> &boundary_values,
const ComponentMask &component_mask = ComponentMask());
/**
* Insert the (algebraic) constraints due
* to Dirichlet boundary conditions into
* a ConstraintMatrix @p
* constraints. This function identifies
* the degrees of freedom subject to
* Dirichlet boundary conditions, adds
* them to the list of constrained DoFs
* in @p constraints and sets the
* respective inhomogeneity to the value
* interpolated around the boundary. If
* this routine encounters a DoF that
* already is constrained (for instance
* by a hanging node constraint, see
* below, or any other type of
* constraint, e.g. from periodic
* boundary conditions), the old setting
* of the constraint (dofs the entry is
* constrained to, inhomogeneities) is
* kept and nothing happens.
*
* @note When combining adaptively
* refined meshes with hanging node
* constraints and boundary conditions
* like from the current function within
* one ConstraintMatrix object, the
* hanging node constraints should always
* be set first, and then the boundary
* conditions since boundary conditions
* are not set in the second operation on
* degrees of freedom that are already
* constrained. This makes sure that the
* discretization remains conforming as
* is needed. See the discussion on
* conflicting constraints in the module
* on @ref constraints .
*
* The parameter @p boundary_component
* corresponds to the number @p
* boundary_indicator of the face.
*
* The flags in the last
* parameter, @p component_mask
* denote which components of the
* finite element space shall be
* interpolated. If it is left as
* specified by the default value
* (i.e. an empty array), all
* components are
* interpolated. If it is
* different from the default
* value, it is assumed that the
* number of entries equals the
* number of components in the
* boundary functions and the
* finite element, and those
* components in the given
* boundary function will be used
* for which the respective flag
* was set in the component mask.
* See also @ref GlossComponentMask. As an example, assume that you are
* solving the Stokes equations in 2d, with variables $(u,v,p)$ and that
* you only want to interpolate boundary values for the pressure, then
* the component mask should correspond to <code>(true,true,false)</code>.
*
* @note Whether a component mask has been specified or not, the number
* of components of the functions
* in @p function_map must match that
* of the finite element used by
* @p dof. In other words, for the example above, you need to provide a
* Function object that has 3 components (the two velocities and the
* pressure), even though you are only
* interested in the first two of them. interpolate_boundary_values()
* will then call this function to obtain a vector of 3 values at each
* interpolation point but only take the first two and discard the third.
* In other words, you are free to return whatever you like in the third
* component of the vector returned by Function::vector_value, but the
* Function object must state that it has 3 components.
*
* If the finite element used has shape
* functions that are non-zero in more
* than one component (in deal.II
* speak: they are non-primitive), then
* these components can presently not
* be used for interpolating boundary
* values. Thus, the elements in the
* component mask corresponding to the
* components of these non-primitive
* shape functions must be @p false.
*
* See the general documentation of this class for more
* information.
*
* @ingroup constraints
*/
template <class DH>
void
interpolate_boundary_values (const Mapping<DH::dimension,DH::space_dimension> &mapping,
const DH &dof,
const typename FunctionMap<DH::space_dimension>::type &function_map,
ConstraintMatrix &constraints,
const ComponentMask &component_mask = ComponentMask());
/**
* Same function as above, but taking
* only one pair of boundary indicator
* and corresponding boundary
* function.
* The same comments apply as for the previous function, in particular
* about the use of the component mask and the requires size of the function
* object.
*
* @ingroup constraints
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <class DH>
void
interpolate_boundary_values (const Mapping<DH::dimension,DH::space_dimension> &mapping,
const DH &dof,
const types::boundary_id boundary_component,
const Function<DH::space_dimension> &boundary_function,
ConstraintMatrix &constraints,
const ComponentMask &component_mask = ComponentMask());
/**
* Calls the other
* interpolate_boundary_values()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
* The same comments apply as for the previous function, in particular
* about the use of the component mask and the requires size of the function
* object.
*
* @ingroup constraints
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <class DH>
void
interpolate_boundary_values (const DH &dof,
const types::boundary_id boundary_component,
const Function<DH::space_dimension> &boundary_function,
ConstraintMatrix &constraints,
const ComponentMask &component_mask = ComponentMask());
/**
* Calls the other
* interpolate_boundary_values()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
* The same comments apply as for the previous function, in particular
* about the use of the component mask and the requires size of the function
* object.
*
* @ingroup constraints
*/
template <class DH>
void
interpolate_boundary_values (const DH &dof,
const typename FunctionMap<DH::space_dimension>::type &function_map,
ConstraintMatrix &constraints,
const ComponentMask &component_mask = ComponentMask());
/**
* Project a function to the boundary
* of the domain, using the given
* quadrature formula for the faces. If
* the @p boundary_values contained
* values before, the new ones are
* added, or the old one overwritten if
* a node of the boundary part to be
* projected on already was in the
* variable.
*
* If @p component_mapping is empty, it
* is assumed that the number of
* components of @p boundary_function
* matches that of the finite element
* used by @p dof.
*
* In 1d, projection equals
* interpolation. Therefore,
* interpolate_boundary_values is
* called.
*
* @arg @p boundary_values: the result
* of this function, a map containing
* all indices of degrees of freedom at
* the boundary (as covered by the
* boundary parts in @p
* boundary_functions) and the computed
* dof value for this degree of
* freedom.
*
* @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.
*/
template <int dim, int spacedim>
void project_boundary_values (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const typename FunctionMap<spacedim>::type &boundary_functions,
const Quadrature<dim-1> &q,
std::map<types::global_dof_index,double> &boundary_values,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
/**
* Calls the project_boundary_values()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, int spacedim>
void project_boundary_values (const DoFHandler<dim,spacedim> &dof,
const typename FunctionMap<spacedim>::type &boundary_function,
const Quadrature<dim-1> &q,
std::map<types::global_dof_index,double> &boundary_values,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
/**
* Same as above, but for objects of type hp::DoFHandler
*/
template <int dim, int spacedim>
void project_boundary_values (const hp::MappingCollection<dim, spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const typename FunctionMap<spacedim>::type &boundary_functions,
const hp::QCollection<dim-1> &q,
std::map<types::global_dof_index,double> &boundary_values,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
/**
* Calls the project_boundary_values()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, int spacedim>
void project_boundary_values (const hp::DoFHandler<dim,spacedim> &dof,
const typename FunctionMap<spacedim>::type &boundary_function,
const hp::QCollection<dim-1> &q,
std::map<types::global_dof_index,double> &boundary_values,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
/**
* Project a function to the boundary of
* the domain, using the given quadrature
* formula for the faces. This function
* identifies the degrees of freedom
* subject to Dirichlet boundary
* conditions, adds them to the list of
* constrained DoFs in @p constraints and
* sets the respective inhomogeneity to
* the value resulting from the
* projection operation. If this routine
* encounters a DoF that already is
* constrained (for instance by a hanging
* node constraint, see below, or any
* other type of constraint, e.g. from
* periodic boundary conditions), the old
* setting of the constraint (dofs the
* entry is constrained to,
* inhomogeneities) is kept and nothing
* happens.
*
* @note When combining adaptively
* refined meshes with hanging node
* constraints and boundary conditions
* like from the current function within
* one ConstraintMatrix object, the
* hanging node constraints should always
* be set first, and then the boundary
* conditions since boundary conditions
* are not set in the second operation on
* degrees of freedom that are already
* constrained. This makes sure that the
* discretization remains conforming as
* is needed. See the discussion on
* conflicting constraints in the module
* on @ref constraints .
*
* If @p component_mapping is empty, it
* is assumed that the number of
* components of @p boundary_function
* matches that of the finite element
* used by @p dof.
*
* In 1d, projection equals
* interpolation. Therefore,
* interpolate_boundary_values is
* called.
*
* @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.
*
* @ingroup constraints
*/
template <int dim, int spacedim>
void project_boundary_values (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const typename FunctionMap<spacedim>::type &boundary_functions,
const Quadrature<dim-1> &q,
ConstraintMatrix &constraints,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
/**
* Calls the project_boundary_values()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*
* @ingroup constraints
*/
template <int dim, int spacedim>
void project_boundary_values (const DoFHandler<dim,spacedim> &dof,
const typename FunctionMap<spacedim>::type &boundary_function,
const Quadrature<dim-1> &q,
ConstraintMatrix &constraints,
std::vector<unsigned int> component_mapping = std::vector<unsigned int>());
/**
* Compute constraints that correspond to
* boundary conditions of the form
* $\vec{n}\times\vec{u}=\vec{n}\times\vec{f}$,
* i.e. the tangential components of $u$
* and $f$ shall coincide.
*
* If the ConstraintMatrix @p constraints
* contained values or other
* constraints before, the new ones are
* added or the old ones overwritten,
* if a node of the boundary part to be
* used was already in the list of
* constraints. This is handled by
* using inhomogeneous constraints. Please
* note that when combining adaptive meshes
* and this kind of constraints, the
* Dirichlet conditions should be set
* first, and then completed by hanging
* node constraints, in order to make sure
* that the discretization remains
* consistent. See the discussion on
* conflicting constraints in the
* module on @ref constraints .
*
* This function is explecitly written to
* use with the FE_Nedelec elements. Thus
* it throws an exception, if it is
* called with other finite elements.
*
* The second argument of this function
* denotes the first vector component in
* the finite element that corresponds to
* the vector function that you want to
* constrain. For example, if we want to
* solve Maxwell's equations in 3d and the
* finite element has components
* $(E_x,E_y,E_z,B_x,B_y,B_z)$ and we want
* the boundary conditions
* $\vec{n}\times\vec{B}=\vec{n}\times\vec{f}$,
* then @p first_vector_component would
* be 3. Vectors are implicitly assumed to
* have exactly <code>dim</code> components
* that are ordered in the same way as we
* usually order the coordinate directions,
* i.e. $x$-, $y$-, and finally
* $z$-component.
*
* The parameter @p boundary_component
* corresponds to the number
* @p boundary_indicator of the face.
* numbers::internal_face_boundary_id
* is an illegal value, since it is
* reserved for interior faces.
*
* The last argument is denoted to compute
* the normal vector $\vec{n}$ at the
* boundary points.
*
* <h4>Computing constraints</h4>
*
* To compute the constraints we use
* projection-based interpolation as proposed
* in Solin, Segeth and Dolezel
* (Higher order finite elements, Chapman&Hall,
* 2004) on every face located at the
* boundary.
*
* First one projects $\vec{f}$ on the
* lowest-order edge shape functions. Then the
* remaining part $(I-P_0)\vec{f}$ of the
* function is projected on the remaining
* higher-order edge shape functions. In the
* last step we project $(I-P_0-P_e)\vec{f}$
* on the bubble shape functions defined on
* the face.
*
* @ingroup constraints
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim>
void project_boundary_values_curl_conforming (const DoFHandler<dim> &dof_handler,
const unsigned int first_vector_component,
const Function<dim> &boundary_function,
const types::boundary_id boundary_component,
ConstraintMatrix &constraints,
const Mapping<dim> &mapping = StaticMappingQ1<dim>::mapping);
/**
* Same as above for the hp-namespace.
*
* @ingroup constraints
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim>
void project_boundary_values_curl_conforming (const hp::DoFHandler<dim> &dof_handler,
const unsigned int first_vector_component,
const Function<dim> &boundary_function,
const types::boundary_id boundary_component,
ConstraintMatrix &constraints,
const hp::MappingCollection<dim, dim> &mapping_collection = hp::StaticMappingQ1<dim>::mapping_collection);
/**
* Compute constraints that correspond to
* boundary conditions of the form
* $\vec{n}^T\vec{u}=\vec{n}^T\vec{f}$,
* i.e. the normal components of $u$
* and $f$ shall coincide.
*
* If the ConstraintMatrix @p constraints
* contained values or other
* constraints before, the new ones are
* added or the old ones overwritten,
* if a node of the boundary part to be
* used was already in the list of
* constraints. This is handled by
* using inhomogeneous constraints. Please
* note that when combining adaptive meshes
* and this kind of constraints, the
* Dirichlet conditions should be set
* first, and then completed by hanging
* node constraints, in order to make sure
* that the discretization remains
* consistent. See the discussion on
* conflicting constraints in the
* module on @ref constraints .
*
* This function is explecitly written to
* use with the FE_RaviartThomas elements.
* Thus it throws an exception, if it is
* called with other finite elements.
*
* The second argument of this function
* denotes the first vector component in
* the finite element that corresponds to
* the vector function that you want to
* constrain. Vectors are implicitly
* assumed to have exactly
* <code>dim</code> components that are
* ordered in the same way as we
* usually order the coordinate directions,
* i.e. $x$-, $y$-, and finally
* $z$-component.
*
* The parameter @p boundary_component
* corresponds to the number
* @p boundary_indicator of the face.
* numbers::internal_face_boundary_id
* is an illegal value, since it is
* reserved for interior faces.
*
* The last argument is denoted to compute
* the normal vector $\vec{n}$ at the
* boundary points.
*
* <h4>Computing constraints</h4>
*
* To compute the constraints we use
* interpolation operator proposed
* in Brezzi, Fortin (Mixed and Hybrid
* (Finite Element Methods, Springer,
* 1991) on every face located at the
* boundary.
*
* @ingroup constraints
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template<int dim>
void project_boundary_values_div_conforming (const DoFHandler<dim> &dof_handler,
const unsigned int first_vector_component,
const Function<dim> &boundary_function,
const types::boundary_id boundary_component,
ConstraintMatrix &constraints,
const Mapping<dim> &mapping = StaticMappingQ1<dim>::mapping);
/**
* Same as above for the hp-namespace.
*
* @ingroup constraints
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template<int dim>
void project_boundary_values_div_conforming (const hp::DoFHandler<dim> &dof_handler,
const unsigned int first_vector_component,
const Function<dim> &boundary_function,
const types::boundary_id boundary_component,
ConstraintMatrix &constraints,
const hp::MappingCollection<dim, dim> &mapping_collection = hp::StaticMappingQ1<dim>::mapping_collection);
/**
* This function computes the constraints that correspond to boundary conditions of the
* form $\vec n \cdot \vec u=0$, i.e. no normal flux if $\vec u$ is a
* vector-valued quantity. These conditions have exactly the form handled by
* the ConstraintMatrix class, so instead of creating a map between boundary
* degrees of freedom and corresponding value, we here create a list of
* constraints that are written into a ConstraintMatrix. This object may
* already have some content, for example from hanging node constraints,
* that remains untouched. These constraints have to be applied to the
* linear system like any other such constraints, i.e. you have to condense
* the linear system with the constraints before solving, and you have to
* distribute the solution vector afterwards.
*
* The use of this function is explained in more detail in step-31. It
* doesn't make much sense in 1d, so the function throws an exception in
* that case.
*
* The second argument of this function denotes the first vector component
* in the finite element that corresponds to the vector function that you
* want to constrain. For example, if we were solving a Stokes equation in
* 2d and the finite element had components $(u,v,p)$, then @p
* first_vector_component would be zero. On the other hand, if we solved the
* Maxwell equations in 3d and the finite element has components
* $(E_x,E_y,E_z,B_x,B_y,B_z)$ and we want the boundary condition $\vec
* n\cdot \vec B=0$, then @p first_vector_component would be 3. Vectors are
* implicitly assumed to have exactly <code>dim</code> components that are
* ordered in the same way as we usually order the coordinate directions,
* i.e. $x$-, $y$-, and finally $z$-component. The function assumes, but
* can't check, that the vector components in the range
* <code>[first_vector_component,first_vector_component+dim)</code> come
* from the same base finite element. For example, in the Stokes example
* above, it would not make sense to use a
* <code>FESystem@<dim@>(FE_Q@<dim@>(2), 1, FE_Q@<dim@>(1), dim)</code>
* (note that the first velocity vector component is a $Q_2$ element,
* whereas all the other ones are $Q_1$ elements) as there would be points
* on the boundary where the $x$-velocity is defined but no corresponding
* $y$- or $z$-velocities.
*
* The third argument denotes the set of boundary indicators on which the
* boundary condition is to be enforced. Note that, as explained below, this
* is one of the few functions where it makes a difference where we call the
* function multiple times with only one boundary indicator, or whether we
* call the function onces with the whole set of boundary indicators at
* once.
*
* The mapping argument is used to compute the boundary points where the
* function needs to request the normal vector $\vec n$ from the boundary
* description.
*
* @note When combining adaptively refined meshes with hanging node
* constraints and boundary conditions like from the current function within
* one ConstraintMatrix object, the hanging node constraints should always
* be set first, and then the boundary conditions since boundary conditions
* are not set in the second operation on degrees of freedom that are
* already constrained. This makes sure that the discretization remains
* conforming as is needed. See the discussion on conflicting constraints in
* the module on @ref constraints .
*
*
* <h4>Computing constraints in 2d</h4>
*
* Computing these constraints requires some smarts. The main question
* revolves around the question what the normal vector is. Consider the
* following situation: <p ALIGN="center"> @image html no_normal_flux_1.png
* </p>
*
* Here, we have two cells that use a bilinear mapping
* (i.e. MappingQ1). Consequently, for each of the cells, the normal vector
* is perpendicular to the straight edge. If the two edges at the top and
* right are meant to approximate a curved boundary (as indicated by the
* dashed line), then neither of the two computed normal vectors are equal
* to the exact normal vector (though they approximate it as the mesh is
* refined further). What is worse, if we constrain $\vec n \cdot \vec u=0$
* at the common vertex with the normal vector from both cells, then we
* constrain the vector $\vec u$ with respect to two linearly independent
* vectors; consequently, the constraint would be $\vec u=0$ at this point
* (i.e. <i>all</i> components of the vector), which is not what we wanted.
*
* To deal with this situation, the algorithm works in the following way: at
* each point where we want to constrain $\vec u$, we first collect all
* normal vectors that adjacent cells might compute at this point. We then
* do not constrain $\vec n \cdot \vec u=0$ for <i>each</i> of these normal
* vectors but only for the <i>average</i> of the normal vectors. In the
* example above, we therefore record only a single constraint $\vec n \cdot
* \vec {\bar u}=0$, where $\vec {\bar u}$ is the average of the two
* indicated normal vectors.
*
* Unfortunately, this is not quite enough. Consider the situation here:
*
* <p ALIGN="center">
* @image html no_normal_flux_2.png
* </p>
*
* If again the top and right edges approximate a curved boundary, and the
* left boundary a separate boundary (for example straight) so that the
* exact boundary has indeed a corner at the top left vertex, then the above
* construction would not work: here, we indeed want the constraint that
* $\vec u$ at this point (because the normal velocities with respect to
* both the left normal as well as the top normal vector should be zero),
* not that the velocity in the direction of the average normal vector is
* zero.
*
* Consequently, we use the following heuristic to determine whether all
* normal vectors computed at one point are to be averaged: if two normal
* vectors for the same point are computed on <i>different</i> cells, then
* they are to be averaged. This covers the first example above. If they are
* computed from the same cell, then the fact that they are different is
* considered indication that they come from different parts of the boundary
* that might be joined by a real corner, and must not be averaged.
*
* There is one problem with this scheme. If, for example, the same domain
* we have considered above, is discretized with the following mesh, then we
* get into trouble:
*
* <p ALIGN="center">
* @image html no_normal_flux_3.png
* </p>
*
* Here, the algorithm assumes that the boundary does not have a corner at
* the point where faces $F1$ and $F2$ join because at that point there are
* two different normal vectors computed from different cells. If you intend
* for there to be a corner of the exact boundary at this point, the only
* way to deal with this is to assign the two parts of the boundary
* different boundary indicators and call this function twice, once for each
* boundary indicators; doing so will yield only one normal vector at this
* point per invocation (because we consider only one boundary part at a
* time), with the result that the normal vectors will not be averaged. This
* situation also needs to be taken into account when using this function
* around reentrant corners on Cartesian meshes. If no-normal-flux boundary
* conditions are to be enforced on non-Cartesian meshes around reentrant
* corners, one may even get cycles in the constraints as one will in
* general constrain different components from the two sides. In that case,
* set a no-slip constraint on the reentrant vertex first.
*
*
* <h4>Computing constraints in 3d</h4>
*
* The situation is more complicated in 3d. Consider the following case
* where we want to compute the constraints at the marked vertex:
*
* <p ALIGN="center">
* @image html no_normal_flux_4.png
* </p>
*
* Here, we get four different normal vectors, one from each of the four
* faces that meet at the vertex. Even though they may form a complete set
* of vectors, it is not our intent to constrain all components of the
* vector field at this point. Rather, we would like to still allow
* tangential flow, where the term "tangential" has to be suitably defined.
*
* In a case like this, the algorithm proceeds as follows: for each cell
* that has computed two tangential vectors at this point, we compute the
* unconstrained direction as the outer product of the two tangential
* vectors (if necessary multiplied by minus one). We then average these
* tangential vectors. Finally, we compute constraints for the two
* directions perpendicular to this averaged tangential direction.
*
* There are cases where one cell contributes two tangential directions and
* another one only one; for example, this would happen if both top and
* front faces of the left cell belong to the boundary selected whereas only
* the top face of the right cell belongs to it, maybe indicating the the entire
* front part of the domain is a smooth manifold whereas the top really forms
* two separate manifolds that meet in a ridge, and that no-flux boundary
* conditions are only desired on the front manifold and the right one on top.
* In cases like these, it's difficult to define what should happen. The
* current implementation simply ignores the one contribution from the
* cell that only contributes one normal vector. In the example shown, this
* is acceptable because the normal vector for the front face of the left
* cell is the same as the normal vector provided by the front face of
* the right cell (the surface is planar) but it would be a problem if the
* front manifold would be curved. Regardless, it is unclear how one would
* proceed in this case and ignoring the single cell is likely the best
* one can do.
*
*
* <h4>Results</h4>
*
* Because it makes for good pictures, here are two images of vector fields
* on a circle and on a sphere to which the constraints computed by this
* function have been applied:
*
* <p ALIGN="center">
* @image html no_normal_flux_5.png
* @image html no_normal_flux_6.png
* </p>
*
* The vectors fields are not physically reasonable but the tangentiality
* constraint is clearly enforced. The fact that the vector fields are zero
* at some points on the boundary is an artifact of the way it is created,
* it is not constrained to be zero at these points.
*
* @ingroup constraints
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim, template <int, int> class DH, int spacedim>
void
compute_no_normal_flux_constraints (const DH<dim,spacedim> &dof_handler,
const unsigned int first_vector_component,
const std::set<types::boundary_id> &boundary_ids,
ConstraintMatrix &constraints,
const Mapping<dim, spacedim> &mapping = StaticMappingQ1<dim>::mapping);
/**
* Compute the constraints that correspond to boundary conditions of the
* form $\vec n \times \vec u=0$, i.e. flow normal to the boundary if $\vec
* u$ is a vector-valued quantity. This function constrains exactly those
* vector-valued components that are left unconstrained by
* compute_no_normal_flux_constraints, and leaves the one component
* unconstrained that is constrained by compute_no_normal_flux_constraints.
*/
template <int dim, template <int, int> class DH, int spacedim>
void
compute_normal_flux_constraints (const DH<dim,spacedim> &dof_handler,
const unsigned int first_vector_component,
const std::set<types::boundary_id> &boundary_ids,
ConstraintMatrix &constraints,
const Mapping<dim, spacedim> &mapping = StaticMappingQ1<dim>::mapping);
//@}
/**
* @name Assembling of right hand sides
*/
//@{
/**
* Create a right hand side
* vector. Prior content of the
* given @p rhs_vector vector is
* deleted.
*
* See the general documentation of this
* class for further information.
*/
template <int dim, int spacedim>
void create_right_hand_side (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector);
/**
* Calls the create_right_hand_side()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, int spacedim>
void create_right_hand_side (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector);
/**
* Like the previous set of functions,
* but for hp objects.
*/
template <int dim, int spacedim>
void create_right_hand_side (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector);
/**
* Like the previous set of functions,
* but for hp objects.
*/
template <int dim, int spacedim>
void create_right_hand_side (const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim> &q,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector);
/**
* Create a right hand side
* vector for a point source at point @p p. In other words, it creates
* a vector $F$ so that
* $F_i = \int_\Omega \delta(x-p) \phi_i(x) dx$.
* Prior content of the
* given @p rhs_vector vector is
* deleted.
*
* See the general documentation of this
* class for further information.
*/
template <int dim, int spacedim>
void create_point_source_vector(const Mapping<dim,spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Point<spacedim> &p,
Vector<double> &rhs_vector);
/**
* Calls the create_point_source_vector()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, int spacedim>
void create_point_source_vector(const DoFHandler<dim,spacedim> &dof,
const Point<spacedim> &p,
Vector<double> &rhs_vector);
/**
* Like the previous set of functions,
* but for hp objects.
*/
template <int dim, int spacedim>
void create_point_source_vector(const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const Point<spacedim> &p,
Vector<double> &rhs_vector);
/**
* Like the previous set of functions,
* but for hp objects. The function uses
* the default Q1 mapping object. Note
* that if your hp::DoFHandler uses any
* active fe index other than zero, then
* you need to call the function above
* that provides a mapping object for
* each active fe index.
*/
template <int dim, int spacedim>
void create_point_source_vector(const hp::DoFHandler<dim,spacedim> &dof,
const Point<spacedim> &p,
Vector<double> &rhs_vector);
/**
* Create a right hand side
* vector for a point source at point @p p
* for vector-valued finite elements.
* Prior content of the
* given @p rhs_vector vector is
* deleted.
*
* See the general documentation of this
* class for further information.
*/
template <int dim, int spacedim>
void create_point_source_vector(const Mapping<dim,spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Point<spacedim> &p,
const Point<dim> &orientation,
Vector<double> &rhs_vector);
/**
* Calls the create_point_source_vector()
* function for vector-valued finite elements,
* see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, int spacedim>
void create_point_source_vector(const DoFHandler<dim,spacedim> &dof,
const Point<spacedim> &p,
const Point<dim> &orientation,
Vector<double> &rhs_vector);
/**
* Like the previous set of functions,
* but for hp objects.
*/
template <int dim, int spacedim>
void create_point_source_vector(const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const Point<spacedim> &p,
const Point<dim> &orientation,
Vector<double> &rhs_vector);
/**
* Like the previous set of functions,
* but for hp objects. The function uses
* the default Q1 mapping object. Note
* that if your hp::DoFHandler uses any
* active fe index other than zero, then
* you need to call the function above
* that provides a mapping object for
* each active fe index.
*/
template <int dim, int spacedim>
void create_point_source_vector(const hp::DoFHandler<dim,spacedim> &dof,
const Point<spacedim> &p,
const Point<dim> &orientation,
Vector<double> &rhs_vector);
/**
* Create a right hand side
* vector from boundary
* forces. Prior content of the
* given @p rhs_vector vector is
* deleted.
*
* See the general documentation of this
* class for further information.
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim, int spacedim>
void create_boundary_right_hand_side (const Mapping<dim,spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim-1> &q,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const std::set<types::boundary_id> &boundary_indicators = std::set<types::boundary_id>());
/**
* Calls the
* create_boundary_right_hand_side()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim, int spacedim>
void create_boundary_right_hand_side (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim-1> &q,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const std::set<types::boundary_id> &boundary_indicators = std::set<types::boundary_id>());
/**
* Same as the set of functions above,
* but for hp objects.
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim, int spacedim>
void create_boundary_right_hand_side (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim-1> &q,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const std::set<types::boundary_id> &boundary_indicators = std::set<types::boundary_id>());
/**
* Calls the
* create_boundary_right_hand_side()
* function, see above, with a
* single Q1 mapping as
* collection. This function
* therefore will only work if
* the only active fe index in
* use is zero.
*
* @see @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim, int spacedim>
void create_boundary_right_hand_side (const hp::DoFHandler<dim,spacedim> &dof,
const hp::QCollection<dim-1> &q,
const Function<spacedim> &rhs,
Vector<double> &rhs_vector,
const std::set<types::boundary_id> &boundary_indicators = std::set<types::boundary_id>());
//@}
/**
* @name Evaluation of functions
* and errors
*/
//@{
/**
* Compute the error of the finite element solution. Integrate the
* difference between a reference function which is given as a
* continuous function object, and a finite element function. The
* result of this function is the vector @p difference that contains
* one value per active cell $K$ of the triangulation. Each of the values
* of this vector $d$ equals
* @f{align*}
* d_K = \| u-u_h \|_X
* @f}
* where $X$ denotes the norm chosen and $u$ represents the exact solution.
*
* It is assumed that the number of components of the function
* @p exact_solution matches that of the finite element used by @p dof.
*
* @param[in] mapping The mapping that is used when integrating the
* difference $u-u_h$.
* @param[in] dof The DoFHandler object that describes the finite
* element space in which the solution vector lives.
* @param[in] fe_function A vector with nodal values representing the
* numerical approximation $u_h$. This vector needs to correspond
* to the finite element space represented by @p dof
* @param[out] difference The vector of values $d_K$ computed as above.
* @param[in] q The quadrature formula used to approximate the integral
* shown above. Note that some quadrature formulas are more useful
* than other in integrating $u-u_h$. For example, it is known that
* the $Q_1$ approximation $u_h$ to the exact solution $u$ of a Laplace
* equation is particularly accurate (in fact, superconvergent, i.e.
* accurate to higher order) at the 4 Gauss points of a cell in 2d
* (or 8 points in 3d) that correspond to a QGauss(2) object. Consequently,
* because a QGauss(2) formula only evaluates the two solutions at these
* particular points, choosing this quadrature formula may indicate an error
* far smaller than it actually is.
* @param[in] norm The norm $X$ shown above that should be computed.
* @param[in] weight The additional argument @p weight allows to evaluate weighted
* norms. The weight function may be scalar, establishing a weight
* in the domain for all components equally. This may be used, for
* instance, to only integrate over parts of the domain. The weight function
* may also be vector-valued, with as many components as the finite element:
* Then, different components get different weights. A typical application is when
* the error with respect to only one or a subset of the solution
* variables is to be computed, in which the other components would
* have weight values equal to zero. The ComponentSelectFunction
* class is particularly useful for this purpose as it provides
* such as "mask" weight..
* The weight function is expected to be positive, but negative
* values are not filtered. By default, no weighting function is
* given, i.e. weight=1 in the whole domain for all vector
* components uniformly.
* @param[in] exponent This value denotes the $p$ used in computing
* $L^p$-norms and $W^{1,p}$-norms. The value is ignores if a @p norm
* other than NormType::Lp_norm or NormType::W1p_norm is chosen.
*
*
* See the general documentation of this
* class for more information.
*
* @note If the integration here happens over the cells of a
* parallel::distribute::Triangulation object, then this function
* computes the vector elements $d_K$ for an output vector with as
* many cells as there are active cells of the triangulation object
* of the current processor. However, not all active cells are in
* fact locally owned: some may be ghost or artificial cells (see
* @ref GlossGhostCell "here" and @ref GlossArtificialCell
* "here"). The vector computed will, in the case of a distributed
* triangulation, contain zeros for cells that are not locally
* owned. As a consequence, in order to compute the <i>global</i>
* $L_2$ error (for example), the errors from different processors
* need to be combined, but this is simple because every processor
* only computes contributions for those cells of the global
* triangulation it locally owns (and these sets are, by definition,
* mutually disjoint). Consequently, the following piece of code
* computes the global $L_2$ error across multiple processors
* sharing a parallel::distribute::Triangulation:
* @code
* Vector<double> local_errors (tria.n_active_cells());
* VectorTools::integrate_difference (mapping, dof,
* solution, exact_solution,
* local_errors,
* QGauss<dim>(fe.degree+2),
* NormType::L2_norm);
* const double total_local_error = local_errors.l2_norm();
* const double total_global_error
* = std::sqrt (Utilities::MPI::sum (total_local_error * total_local_error, MPI_COMM_WORLD));
* @endcode
* The squaring and taking the square root is necessary in order to
* compute the sum of squares of norms over all all cells in the definition
* of the $L_2$ norm:
* @f{align*}
* \textrm{error} = \sqrt{\sum_K \|u-u_h\|_{L_2(K)}^2}
* @f}
* Obviously, if you are interested in computing the $L_1$ norm of the
* error, the correct form of the last two lines would have been
* @code
* const double total_local_error = local_errors.l1_norm();
* const double total_global_error
* = Utilities::MPI::sum (total_local_error, MPI_COMM_WORLD);
* @endcode
* instead, and similar considerations hold when computing the $L_\infty$
* norm of the error.
*
* Instantiations for this template
* are provided for some vector types
* (see the general documentation of the
* class), but only for InVectors as in
* the documentation of the class,
* OutVector only Vector<double> and
* Vector<float>.
*/
template <int dim, class InVector, class OutVector, int spacedim>
void integrate_difference (const Mapping<dim,spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Function<spacedim> &exact_solution,
OutVector &difference,
const Quadrature<dim> &q,
const NormType &norm,
const Function<spacedim> *weight = 0,
const double exponent = 2.);
/**
* Calls the integrate_difference()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, class InVector, class OutVector, int spacedim>
void integrate_difference (const DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Function<spacedim> &exact_solution,
OutVector &difference,
const Quadrature<dim> &q,
const NormType &norm,
const Function<spacedim> *weight = 0,
const double exponent = 2.);
/**
* Same as above for hp.
*/
template <int dim, class InVector, class OutVector, int spacedim>
void integrate_difference (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Function<spacedim> &exact_solution,
OutVector &difference,
const hp::QCollection<dim> &q,
const NormType &norm,
const Function<spacedim> *weight = 0,
const double exponent = 2.);
/**
* Calls the integrate_difference()
* function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, class InVector, class OutVector, int spacedim>
void integrate_difference (const hp::DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Function<spacedim> &exact_solution,
OutVector &difference,
const hp::QCollection<dim> &q,
const NormType &norm,
const Function<spacedim> *weight = 0,
const double exponent = 2.);
/**
* Point error evaluation. Find
* the first cell containing the
* given point and compute the
* difference of a (possibly
* vector-valued) finite element
* function and a continuous
* function (with as many vector
* components as the finite
* element) at this point.
*
* This is a wrapper function
* using a Q1-mapping for cell
* boundaries to call the other
* point_difference() function.
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
void point_difference (const DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Function<spacedim> &exact_solution,
Vector<double> &difference,
const Point<spacedim> &point);
/**
* Point error evaluation. Find
* the first cell containing the
* given point and compute the
* difference of a (possibly
* vector-valued) finite element
* function and a continuous
* function (with as many vector
* components as the finite
* element) at this point.
*
* Compared with the other
* function of the same name,
* this function uses an
* arbitrary mapping to evaluate
* the difference.
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
void point_difference (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Function<spacedim> &exact_solution,
Vector<double> &difference,
const Point<spacedim> &point);
/**
* Evaluate a possibly
* vector-valued finite element
* function defined by the given
* DoFHandler and nodal vector at
* the given point, and return
* the (vector) value of this
* function through the last
* argument.
*
* This is a wrapper function
* using a Q1-mapping for cell
* boundaries to call the other
* point_difference() function.
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
void
point_value (const DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Point<spacedim> &point,
Vector<double> &value);
/**
* Same as above for hp.
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
void
point_value (const hp::DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Point<spacedim> &point,
Vector<double> &value);
/**
* Evaluate a scalar finite
* element function defined by
* the given DoFHandler and nodal
* vector at the given point, and
* return the value of this
* function.
*
* Compared with the other
* function of the same name,
* this is a wrapper function using
* a Q1-mapping for cells.
*
* This function is used in the
* "Possibilities for extensions" part of
* the results section of @ref step_3
* "step-3".
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
double
point_value (const DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Point<spacedim> &point);
/**
* Same as above for hp.
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
double
point_value (const hp::DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Point<spacedim> &point);
/**
* Evaluate a possibly
* vector-valued finite element
* function defined by the given
* DoFHandler and nodal vector at
* the given point, and return
* the (vector) value of this
* function through the last
* argument.
*
* Compared with the other
* function of the same name,
* this function uses an arbitrary
* mapping to evaluate the difference.
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
void
point_value (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Point<spacedim> &point,
Vector<double> &value);
/**
* Same as above for hp.
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
void
point_value (const hp::MappingCollection<dim, spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Point<spacedim> &point,
Vector<double> &value);
/**
* Evaluate a scalar finite
* element function defined by
* the given DoFHandler and nodal
* vector at the given point, and
* return the value of this
* function.
*
* Compared with the other
* function of the same name,
* this function uses an arbitrary
* mapping to evaluate the difference.
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
double
point_value (const Mapping<dim,spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Point<spacedim> &point);
/**
* Same as above for hp.
*
* @note If the cell in which the point is found
* is not locally owned, an exception of type
* VectorTools::ExcPointNotAvailableHere
* is thrown.
*/
template <int dim, class InVector, int spacedim>
double
point_value (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const InVector &fe_function,
const Point<spacedim> &point);
//@}
/**
* Mean value operations
*/
//@{
/**
* Subtract the (algebraic) mean value from a vector.
*
* This function is most frequently used as a mean-value filter for
* Stokes: The pressure in Stokes' equations with only Dirichlet
* boundaries for the velocities is only determined up to a constant.
* This function allows to subtract the mean value of the pressure. It is
* usually called in a preconditioner and generates updates with mean
* value zero. The mean value is computed as the mean value of the
* degrees of freedom values as given by the input vector; they are not
* weighted by the area of cells, i.e. the mean is computed as $\sum_i
* v_i$, rather than as $\int_\Omega v(x) = \int_\Omega \sum_i v_i
* \phi_i(x)$. The latter can be obtained from the
* VectorTools::compute_mean_function, however.
*
* Apart from the vector @p v to operate on, this function takes a
* boolean mask @p p_select that has a true entry for every element of
* the vector for which the mean value shall be computed and later
* subtracted. The argument is used to denote which components of the
* solution vector correspond to the pressure, and avoid touching all
* other components of the vector, such as the velocity components.
* (Note, however, that the mask is not a @ref GlossComponentMask
* operating on the vector components of the finite element the solution
* vector @p v may be associated with; rather, it is a mask on the entire
* vector, without reference to what the vector elements mean.)
*
* The boolean mask @p p_select has an empty vector as default value,
* which will be interpreted as selecting all vector elements, hence,
* subtracting the algebraic mean value on the whole vector. This allows
* to call this function without a boolean mask if the whole vector
* should be processed.
*
* @note In the context of using this function to filter out the kernel
* of an operator (such as the null space of the Stokes operator that
* consists of the constant pressures), this function only makes sense
* for finite elements for which the null space indeed consists of the
* vector $(1,1,\ldots,1)^T$. This is the case for example for the usual
* Lagrange elements where the sum of all shape functions equals the
* function that is constant one. However, it is not true for some other
* functions: for example, for the FE_DGP element (another valid choice
* for the pressure in Stokes discretizations), the first shape function
* on each cell is constant while further elements are $L_2$ orthogonal
* to it (on the reference cell); consequently, the sum of all shape
* functions is not equal to one, and the vector that is associated with
* the constant mode is not equal to $(1,1,\ldots,1)^T$. For such
* elements, a different procedure has to be used when subtracting the
* mean value.
*/
template <class VECTOR>
void subtract_mean_value(VECTOR &v,
const std::vector<bool> &p_select = std::vector<bool>());
/**
* Compute the mean value of one component of the solution.
*
* This function integrates the chosen component over the whole domain
* and returns the result, i.e. it computes $\int_\Omega [u_h(x)]_c \;
* dx$ where $c$ is the vector component and $u_h$ is the function
* representation of the nodal vector given as fourth argument. The
* integral is evaluated numerically using the quadrature formula given
* as third argument.
*
* This function is used in the "Possibilities for extensions" part of
* the results section of @ref step_3 "step-3".
*
* @note The function is most often used when solving a problem whose
* solution is only defined up to a constant, for example a pure Neumann
* problem or the pressure in a Stokes or Navier-Stokes problem. In both
* cases, subtracting the mean value as computed by the current function,
* from the nodal vector does not generally yield the desired result of a
* finite element function with mean value zero. In fact, it only works
* for Lagrangian elements. For all other elements, you will need to
* compute the mean value and subtract it right inside the evaluation
* routine.
*/
template <int dim, class InVector, int spacedim>
double compute_mean_value (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &quadrature,
const InVector &v,
const unsigned int component);
/**
* Calls the other compute_mean_value() function, see above, with
* <tt>mapping=MappingQ1@<dim@>()</tt>.
*/
template <int dim, class InVector, int spacedim>
double compute_mean_value (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &quadrature,
const InVector &v,
const unsigned int component);
//@}
/**
* Exception
*/
DeclException0 (ExcInvalidBoundaryIndicator);
/**
* Exception
*/
DeclException0 (ExcNonInterpolatingFE);
/**
* Exception
*/
DeclException0 (ExcPointNotAvailableHere);
}
DEAL_II_NAMESPACE_CLOSE
#endif
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