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2422 2423 | // ---------------------------------------------------------------------
//
// Copyright (C) 1998 - 2016 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 dealii__vector_tools_h
#define dealii__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, typename Number> class Function;
template <int dim, typename Number> 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 all functions, one that takes an
* explicit Mapping argument and one that does not. The second one generally
* calls the first with an implicit $Q_1$ argument (i.e., with an argument of
* kind MappingQGeneric(1)). If your intend your code to use a different
* mapping than a (bi-/tri-)linear one, then you need to call the functions
* <b>with</b> mapping argument should be used.
*
*
* <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 FunctionMap 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 namespace. 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,
/**
* #L2_norm of the divergence of a vector field
*/
Hdiv_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>DoFHandlerType</code> may either be of type
* DoFHandler or hp::DoFHandler.
*
* See the general documentation of this namespace for further information.
*
* @todo The @p mapping argument should be replaced by a
* hp::MappingCollection in case of a hp::DoFHandler.
*/
template <typename VectorType, int dim, int spacedim, template <int, int> class DoFHandlerType>
void interpolate (const Mapping<dim,spacedim> &mapping,
const DoFHandlerType<dim,spacedim> &dof,
const Function<spacedim,double> &function,
VectorType &vec);
/**
* Calls the @p interpolate() function above with
* <tt>mapping=MappingQGeneric1@<dim>@()</tt>.
*/
template <typename VectorType, typename DoFHandlerType>
void interpolate (const DoFHandlerType &dof,
const Function<DoFHandlerType::space_dimension,double> &function,
VectorType &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 namespace), 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);
/**
* This function is a kind of generalization or modification of the very
* first interpolate() function in the series. It interpolations a set of
* functions onto the finite element space given by the DoFHandler argument
* where the determination which function to use is made based on the
* material id (see
* @ref GlossMaterialId)
* of each cell.
*
* @param mapping - The mapping to use to determine the location of
* support points at which the functions are to be evaluated.
* @param dof_handler - DoFHandler initialized with Triangulation and
* FiniteElement objects,
* @param function_map - std::map reflecting the correspondence between
* material ids and functions,
* @param dst - global FE vector at the support points,
* @param component_mask - mask of components that shall be interpolated
*
* @note If a material id of some group of cells is missed in @p
* function_map, then @p dst will not be updated in the respective degrees
* of freedom of the output vector For example, if @p dst was successfully
* initialized to capture the degrees of freedom of the @p dof_handler of
* the problem with all zeros in it, then those zeros which correspond to
* the missed material ids will still remain in @p dst even after calling
* this function.
*
* @note Degrees of freedom located on faces between cells of different
* material ids will get their value by that cell which was called last in
* the respective loop over cells implemented in this function. Since this
* process is kind of arbitrary, you cannot control it. However, if you want
* to have control over the order in which cells are visited, let us take a
* look at the following example: Let @p u be a variable of interest which
* is approximated by some CG finite element. Let @p 0, @p 1 and @p 2 be
* material ids of cells on the triangulation. Let 0: 0.0, 1: 1.0, 2: 2.0 be
* the whole @p function_map that you want to pass to this function, where
* @p key is a material id and @p value is a value of @p u. By using the
* whole @p function_map you do not really know which values will be
* assigned to the face DoFs. On the other hand, if you split the whole @p
* function_map into three smaller independent objects 0: 0.0 and 1: 1.0 and
* 2: 2.0 and make three distinct calls of this function passing each of
* these objects separately (the order depends on what you want to get
* between cells), then each subsequent call will rewrite the intercell @p
* dofs of the previous one.
*
* @author Valentin Zingan, 2013
*/
template <typename VectorType, typename DoFHandlerType>
void
interpolate_based_on_material_id
(const Mapping<DoFHandlerType::dimension, DoFHandlerType::space_dimension> &mapping,
const DoFHandlerType &dof_handler,
const std::map<types::material_id, const Function<DoFHandlerType::space_dimension, double> *> &function_map,
VectorType &dst,
const ComponentMask &component_mask = ComponentMask());
/**
* 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.
*
* @note: This function works with parallel::distributed::Triangulation, but
* only if the parallel partitioning is the same for both meshes (see the
* parallel::distributed::Triangulation<dim>::no_automatic_repartitioning
* flag).
*/
template <int dim, int spacedim,
template <int, int> class DoFHandlerType,
typename VectorType>
void
interpolate_to_different_mesh (const DoFHandlerType<dim, spacedim> &dof1,
const VectorType &u1,
const DoFHandlerType<dim, spacedim> &dof2,
VectorType &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 DoFHandlerType,
typename VectorType>
void
interpolate_to_different_mesh (const DoFHandlerType<dim, spacedim> &dof1,
const VectorType &u1,
const DoFHandlerType<dim, spacedim> &dof2,
const ConstraintMatrix &constraints,
VectorType &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 DoFHandlerType,
typename VectorType>
void
interpolate_to_different_mesh
(const InterGridMap<DoFHandlerType<dim, spacedim> > &intergridmap,
const VectorType &u1,
const ConstraintMatrix &constraints,
VectorType &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 namespace 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, typename VectorType, int spacedim>
void project (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const ConstraintMatrix &constraints,
const Quadrature<dim> &quadrature,
const Function<spacedim,double> &function,
VectorType &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=MappingQGeneric@<dim@>(1)</tt>.
*/
template <int dim, typename VectorType, int spacedim>
void project (const DoFHandler<dim,spacedim> &dof,
const ConstraintMatrix &constraints,
const Quadrature<dim> &quadrature,
const Function<spacedim,double> &function,
VectorType &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, typename VectorType, 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,double> &function,
VectorType &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 $Q_1$ mapping
* objects, i.e., with hp::StaticMappingQ1::mapping_collection.
*/
template <int dim, typename VectorType, int spacedim>
void project (const hp::DoFHandler<dim,spacedim> &dof,
const ConstraintMatrix &constraints,
const hp::QCollection<dim> &quadrature,
const Function<spacedim,double> &function,
VectorType &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. For each degree of freedom at the boundary, if its index
* already exists in @p boundary_values then its boundary value will be
* overwritten, otherwise a new entry with proper index and boundary value
* for this degree of freedom will be inserted into @p 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_id 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 velocity, 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 namespace for more information.
*/
template <typename DoFHandlerType>
void
interpolate_boundary_values
(const Mapping<DoFHandlerType::dimension,DoFHandlerType::space_dimension> &mapping,
const DoFHandlerType &dof,
const typename FunctionMap<DoFHandlerType::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 <typename DoFHandlerType>
void
interpolate_boundary_values
(const Mapping<DoFHandlerType::dimension,DoFHandlerType::space_dimension> &mapping,
const DoFHandlerType &dof,
const types::boundary_id boundary_component,
const Function<DoFHandlerType::space_dimension,double> &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=MappingQGeneric@<dim,spacedim@>(1)</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 <typename DoFHandlerType>
void
interpolate_boundary_values
(const DoFHandlerType &dof,
const types::boundary_id boundary_component,
const Function<DoFHandlerType::space_dimension,double> &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=MappingQGeneric@<dim,spacedim@>(1)</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 <typename DoFHandlerType>
void
interpolate_boundary_values
(const DoFHandlerType &dof,
const typename FunctionMap<DoFHandlerType::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_id 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 namespace for more information.
*
* @ingroup constraints
*/
template <typename DoFHandlerType>
void
interpolate_boundary_values
(const Mapping<DoFHandlerType::dimension,DoFHandlerType::space_dimension> &mapping,
const DoFHandlerType &dof,
const typename FunctionMap<DoFHandlerType::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 <typename DoFHandlerType>
void
interpolate_boundary_values
(const Mapping<DoFHandlerType::dimension,DoFHandlerType::space_dimension> &mapping,
const DoFHandlerType &dof,
const types::boundary_id boundary_component,
const Function<DoFHandlerType::space_dimension,double> &boundary_function,
ConstraintMatrix &constraints,
const ComponentMask &component_mask = ComponentMask());
/**
* Calls the other interpolate_boundary_values() function, see above, with
* <tt>mapping=MappingQGeneric@<dim,spacedim@>(1)</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 <typename DoFHandlerType>
void
interpolate_boundary_values
(const DoFHandlerType &dof,
const types::boundary_id boundary_component,
const Function<DoFHandlerType::space_dimension,double> &boundary_function,
ConstraintMatrix &constraints,
const ComponentMask &component_mask = ComponentMask());
/**
* Calls the other interpolate_boundary_values() function, see above, with
* <tt>mapping=MappingQGeneric@<dim,spacedim@>(1)</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 <typename DoFHandlerType>
void
interpolate_boundary_values
(const DoFHandlerType &dof,
const typename FunctionMap<DoFHandlerType::space_dimension>::type &function_map,
ConstraintMatrix &constraints,
const ComponentMask &component_mask = ComponentMask());
/**
* Project a function or a set of functions to the boundary of the domain.
* In other words, compute the solution of the following problem: Find $u_h
* \in V_h$ (where $V_h$ is the finite element space represented by the
* DoFHandler argument of this function) so that
* @f{align*}{
* \int_{\Gamma} \varphi_i u_h
* = \sum_{k \in {\cal K}} \int_{\Gamma_k} \varphi_i f_k,
* \qquad \forall \varphi_i \in V_h
* @f}
* where $\Gamma = \bigcup_{k \in {\cal K}} \Gamma_k$, $\Gamma_k \subset
* \partial\Omega$, $\cal K$ is the set of indices and $f_k$ the
* corresponding boundary functions represented in the function map argument
* @p boundary_values to this function, and the integrals are evaluated by
* quadrature. This problem has a non-unique solution in the interior, but
* it is well defined for the degrees of freedom on the part of the
* boundary, $\Gamma$, for which we do the integration. The values of
* $u_h|_\Gamma$, i.e., the nodal values of the degrees of freedom of this
* function along the boundary, are then what is computed by this function.
*
* @param[in] mapping The mapping that will be used in the transformations
* necessary to integrate along the boundary.
* @param[in] dof The DoFHandler that describes the finite element space and
* the numbering of degrees of freedom.
* @param[in] boundary_functions A map from boundary indicators to pointers
* to functions that describe the desired values on those parts of the
* boundary marked with this boundary indicator (see
* @ref GlossBoundaryIndicator "Boundary indicator").
* The projection happens on only those parts of the boundary whose
* indicators are represented in this map.
* @param[in] q The face quadrature used in the integration necessary to
* compute the mass matrix and right hand side of the projection.
* @param[out] boundary_values The result of this function. It is 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. For each degree of freedom at the
* boundary, if its index already exists in @p boundary_values then its
* boundary value will be overwritten, otherwise a new entry with proper
* index and boundary value for this degree of freedom will be inserted into
* @p boundary_values.
* @param[in] component_mapping It is sometimes convenient to project a
* vector-valued function onto only parts of a finite element space (for
* example, to project a function with <code>dim</code> components onto the
* velocity components of a <code>dim+1</code> component DoFHandler for a
* Stokes problem). To allow for this, this argument allows components to be
* remapped. If the vector is not empty, it has to have one entry for each
* vector component of the finite element used 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=MappingQGeneric@<dim,spacedim@>(1)</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=MappingQGeneric@<dim,spacedim@>(1)</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=MappingQGeneric@<dim,spacedim@>(1)</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 explicitly 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_id 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,double> &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,double> &boundary_function,
const types::boundary_id boundary_component,
ConstraintMatrix &constraints,
const hp::MappingCollection<dim, dim> &mapping_collection = hp::StaticMappingQ1<dim>::mapping_collection);
/**
* This function is an updated version of the
* project_boundary_values_curl_conforming function. The intention is to fix
* a problem when using the previous function in conjunction with non-
* rectangular geometries (i.e. elements with non-rectangular faces). The
* L2-projection method used has been taken from the paper "Electromagnetic
* scattering simulation using an H (curl) conforming hp finite element
* method in three dimensions" by PD Ledger, K Morgan and O Hassan ( Int. J.
* Num. Meth. Fluids, Volume 53, Issue 8, pages 1267–1296).
*
* This function will compute constraints that correspond to Dirichlet
* boundary conditions of the form
* $\vec{n}\times\vec{E}=\vec{n}\times\vec{F}$ i.e. the tangential
* components of $\vec{E}$ and $f$ shall coincide.
*
* <h4>Computing constraints</h4>
*
* To compute the constraints we use a projection method based upon the
* paper mentioned above. In 2D this is done in a single stage for the edge-
* based shape functions, regardless of the order of the finite element. In
* 3D this is done in two stages, edges first and then faces.
*
* For each cell, each edge, $e$, is projected by solving the linear system
* $Ax=b$ where $x$ is the vector of contraints on degrees of freedom on the
* edge and
*
* $A_{ij} = \int_{e} (\vec{s}_{i}\cdot\vec{t})(\vec{s}_{j}\cdot\vec{t}) dS$
*
* $b_{i} = \int_{e} (\vec{s}_{i}\cdot\vec{t})(\vec{F}\cdot\vec{t}) dS$
*
* with $\vec{s}_{i}$ the $i^{th}$ shape function and $\vec{t}$ the tangent
* vector.
*
* Once all edge constraints, $x$, have been computed, we may compute the
* face constraints in a similar fashion, taking into account the residuals
* from the edges.
*
* For each face on the cell, $f$, we solve the linear system $By=c$ where
* $y$ is the vector of constraints on degrees of freedom on the face and
*
* $B_{ij} = \int_{f} (\vec{n} \times \vec{s}_{i}) \cdot (\vec{n} \times
* \vec{s}_{j}) dS$
*
* $c_{i} = \int_{f} (\vec{n} \times \vec{r}) \cdot (\vec{n} \times
* \vec{s}_i) dS$
*
* and $\vec{r} = \vec{F} - \sum_{e \in f} \sum{i \in e} x_{i}\vec{s}_i$,
* the edge residual.
*
* The resulting constraints are then given in the solutions $x$ and $y$.
*
* 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.
*
* <h4>Arguments to this function></h4>
*
* This function is explicitly for use with FE_Nedelec elements, or with
* FESystem elements which contain FE_Nedelec elements. It will throw an
* exception if called with any other finite element. The user must ensure
* that FESystem elements are correctly setup when using this function as
* this check not possible in this case.
*
* The second argument of this function denotes the first vector component
* of the finite element which corresponds to the vector function that you
* wish to constrain. For example, if we are solving Maxwell's equations in
* 3D and have 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. The @p boundary_function must return 6
* components in this example, with the first 3 corresponding to $\vec{E}$
* and the second 3 corresponding to $\vec{B}$. 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_id 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.
*
*
* @ingroup constraints
*
* @see
* @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim>
void project_boundary_values_curl_conforming_l2 (const DoFHandler<dim> &dof_handler,
const unsigned int first_vector_component,
const Function<dim,double> &boundary_function,
const types::boundary_id boundary_component,
ConstraintMatrix &constraints,
const Mapping<dim> &mapping = StaticMappingQ1<dim>::mapping);
/**
* hp-namespace version of project_boundary_values_curl_conforming_l2
* (above).
*
* @ingroup constraints
*/
template <int dim>
void project_boundary_values_curl_conforming_l2 (const hp::DoFHandler<dim> &dof_handler,
const unsigned int first_vector_component,
const Function<dim,double> &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 the
* solution $u$ and a given $f$ shall coincide. The function $f$ is given by
* @p boundary_function and the resulting constraints are added to @p
* constraints for faces with boundary indicator @p boundary_component.
*
* This function is explicitly written to use with the FE_RaviartThomas
* elements. Thus it throws an exception, if it is called with other finite
* elements.
*
* 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.
*
* The argument @p first_vector_component denotes the first vector component
* in the finite element that corresponds to the vector function $\vec{u}$
* 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 @p boundary_id of
* the faces where the boundary conditions are applied.
* numbers::internal_face_boundary_id is an illegal value, since it is
* reserved for interior faces. The @p mapping is used 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,double> &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,double> &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 $\vec u \cdot \vec n=\vec u_\Gamma \cdot \vec n$,
* i.e. normal flux constraints 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 $\vec
* B\cdot \vec n=\vec B_\Gamma\cdot \vec n$, 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 once with the whole set of boundary indicators at once.
*
* The forth parameter describes the boundary function that is used for
* computing these constraints.
*
* 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.,
* MappingQGeneric(1)). 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 $\vec u \cdot \vec n=
* \vec u_\Gamma \cdot \vec n$ 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
* $\vec u=\vec u_\Gamma$ 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 $\vec u \cdot \vec n=\vec u_\Gamma \cdot \vec n$ 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 $\vec u \cdot \vec {\bar n}=\vec u_\Gamma \cdot \vec
* {\bar n}$, where $\vec {\bar n}$ 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 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
* normal-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 DoFHandlerType, int spacedim>
void
compute_nonzero_normal_flux_constraints
(const DoFHandlerType<dim,spacedim> &dof_handler,
const unsigned int first_vector_component,
const std::set<types::boundary_id> &boundary_ids,
typename FunctionMap<spacedim>::type &function_map,
ConstraintMatrix &constraints,
const Mapping<dim, spacedim> &mapping = StaticMappingQ1<dim>::mapping);
/**
* Same as above for homogeneous normal-flux constraints.
*
* @ingroup constraints
*
* @see
* @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim, template <int, int> class DoFHandlerType, int spacedim>
void
compute_no_normal_flux_constraints
(const DoFHandlerType<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 $\vec u \times \vec n=\vec u_\Gamma \times \vec n$, i.e. tangential
* flow constraints 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.
*
* @ingroup constraints
*
* @see
* @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim, template <int, int> class DoFHandlerType, int spacedim>
void
compute_nonzero_tangential_flux_constraints
(const DoFHandlerType<dim,spacedim> &dof_handler,
const unsigned int first_vector_component,
const std::set<types::boundary_id> &boundary_ids,
typename FunctionMap<spacedim>::type &function_map,
ConstraintMatrix &constraints,
const Mapping<dim, spacedim> &mapping = StaticMappingQ1<dim>::mapping);
/**
* Same as above for homogeneous tangential-flux constraints.
*
* @ingroup constraints
*
* @see
* @ref GlossBoundaryIndicator "Glossary entry on boundary indicators"
*/
template <int dim, template <int, int> class DoFHandlerType, int spacedim>
void
compute_normal_flux_constraints
(const DoFHandlerType<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 namespace 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,double> &rhs,
Vector<double> &rhs_vector);
/**
* Calls the create_right_hand_side() function, see above, with
* <tt>mapping=MappingQGeneric@<dim@>(1)</tt>.
*/
template <int dim, int spacedim>
void create_right_hand_side (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &q,
const Function<spacedim,double> &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,double> &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,double> &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 namespace 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=MappingQGeneric@<dim@>(1)</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. This
* variation of the function is meant for vector-valued problems with
* exactly dim components (it will also work for problems with more than dim
* components, and in this case simply consider only the first dim
* components of the shape functions). It computes a right hand side that
* corresponds to a forcing function that is equal to a delta function times
* a given direction. In other words, it creates a vector $F$ so that $F_i =
* \int_\Omega [\mathbf d \delta(x-p)] \cdot \phi_i(x) dx$. Note here that
* $\phi_i$ is a vector-valued function. $\mathbf d$ is the given direction
* of the source term $\mathbf d \delta(x-p)$ and corresponds to the @p
* direction argument to be passed to this function.
*
* Prior content of the given @p rhs_vector vector is deleted.
*
* See the general documentation of this namespace 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> &direction,
Vector<double> &rhs_vector);
/**
* Calls the create_point_source_vector() function for vector-valued finite
* elements, see above, with <tt>mapping=MappingQGeneric@<dim@>(1)</tt>.
*/
template <int dim, int spacedim>
void create_point_source_vector(const DoFHandler<dim,spacedim> &dof,
const Point<spacedim> &p,
const Point<dim> &direction,
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> &direction,
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> &direction,
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 namespace 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,double> &rhs,
Vector<double> &rhs_vector,
const std::set<types::boundary_id> &boundary_ids = std::set<types::boundary_id>());
/**
* Calls the create_boundary_right_hand_side() function, see above, with
* <tt>mapping=MappingQGeneric@<dim@>(1)</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,double> &rhs,
Vector<double> &rhs_vector,
const std::set<types::boundary_id> &boundary_ids = 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,double> &rhs,
Vector<double> &rhs_vector,
const std::set<types::boundary_id> &boundary_ids = 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,double> &rhs,
Vector<double> &rhs_vector,
const std::set<types::boundary_id> &boundary_ids = 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[in] exact_solution The exact solution that is used to compute the
* error.
* @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. If the
* norm is NormType::Hdiv_seminorm, then the finite element on which this
* function is called needs to have at least dim vector components, and the
* divergence will be computed on the first div components. This works, for
* example, on the finite elements used for the mixed Laplace (step-20) and
* the Stokes equations (step-22).
* @param[in] weight The additional argument @p weight allows to evaluate
* weighted norms. The weight function may be scalar, establishing a
* spatially variable 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 case the
* other components would have weight values equal to zero. The
* ComponentSelectFunction class is particularly useful for this purpose as
* it provides such a "mask" weight. The weight function is expected to be
* positive, but negative values are not filtered. The default value of this
* function, a null pointer, is interpreted as "no weighting function",
* 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 namespace 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),
* VectorTools::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 namespace), but only for InVectors as in
* the documentation of the namespace, 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,double> &exact_solution,
OutVector &difference,
const Quadrature<dim> &q,
const NormType &norm,
const Function<spacedim,double> *weight = 0,
const double exponent = 2.);
/**
* Calls the integrate_difference() function, see above, with
* <tt>mapping=MappingQGeneric@<dim@>(1)</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,double> &exact_solution,
OutVector &difference,
const Quadrature<dim> &q,
const NormType &norm,
const Function<spacedim,double> *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,double> &exact_solution,
OutVector &difference,
const hp::QCollection<dim> &q,
const NormType &norm,
const Function<spacedim,double> *weight = 0,
const double exponent = 2.);
/**
* Calls the integrate_difference() function, see above, with
* <tt>mapping=MappingQGeneric@<dim@>(1)</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,double> &exact_solution,
OutVector &difference,
const hp::QCollection<dim> &q,
const NormType &norm,
const Function<spacedim,double> *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, typename VectorType, int spacedim>
void point_difference (const DoFHandler<dim,spacedim> &dof,
const VectorType &fe_function,
const Function<spacedim,double> &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, typename VectorType, int spacedim>
void point_difference (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const VectorType &fe_function,
const Function<spacedim,double> &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, typename VectorType, int spacedim>
void
point_value (const DoFHandler<dim,spacedim> &dof,
const VectorType &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, typename VectorType, int spacedim>
void
point_value (const hp::DoFHandler<dim,spacedim> &dof,
const VectorType &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, typename VectorType, int spacedim>
double
point_value (const DoFHandler<dim,spacedim> &dof,
const VectorType &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, typename VectorType, int spacedim>
double
point_value (const hp::DoFHandler<dim,spacedim> &dof,
const VectorType &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, typename VectorType, int spacedim>
void
point_value (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const VectorType &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, typename VectorType, int spacedim>
void
point_value (const hp::MappingCollection<dim, spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const VectorType &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, typename VectorType, int spacedim>
double
point_value (const Mapping<dim,spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const VectorType &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, typename VectorType, int spacedim>
double
point_value (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const VectorType &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) gradient of this function through the last argument.
*
* This is a wrapper function using a Q1-mapping for cell boundaries to call
* the other point_gradient() 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, typename VectorType, int spacedim>
void
point_gradient (const DoFHandler<dim,spacedim> &dof,
const VectorType &fe_function,
const Point<spacedim> &point,
std::vector<Tensor<1, spacedim, typename VectorType::value_type> > &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, typename VectorType, int spacedim>
void
point_gradient (const hp::DoFHandler<dim,spacedim> &dof,
const VectorType &fe_function,
const Point<spacedim> &point,
std::vector<Tensor<1, spacedim, typename VectorType::value_type> > &value);
/**
* Evaluate a scalar finite element function defined by the given DoFHandler
* and nodal vector at the given point, and return the gradient of this
* function.
*
* Compared with the other function of the same name, this is a wrapper
* function using a Q1-mapping for cells.
*
* @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, typename VectorType, int spacedim>
Tensor<1, spacedim, typename VectorType::value_type>
point_gradient (const DoFHandler<dim,spacedim> &dof,
const VectorType &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, typename VectorType, int spacedim>
Tensor<1, spacedim, typename VectorType::value_type>
point_gradient (const hp::DoFHandler<dim,spacedim> &dof,
const VectorType &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
* gradients of this function through the last argument.
*
* Compared with the other function of the same name, this function uses an
* arbitrary mapping for evaluation.
*
* @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, typename VectorType, int spacedim>
void
point_gradient (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const VectorType &fe_function,
const Point<spacedim> &point,
std::vector<Tensor<1, spacedim, typename VectorType::value_type> > &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, typename VectorType, int spacedim>
void
point_gradient (const hp::MappingCollection<dim, spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const VectorType &fe_function,
const Point<spacedim> &point,
std::vector<Tensor<1, spacedim, typename VectorType::value_type> > &value);
/**
* Evaluate a scalar finite element function defined by the given DoFHandler
* and nodal vector at the given point, and return the gradient of this
* function.
*
* Compared with the other function of the same name, this function uses an
* arbitrary mapping for evaluation.
*
* @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, typename VectorType, int spacedim>
Tensor<1, spacedim, typename VectorType::value_type>
point_gradient (const Mapping<dim,spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const VectorType &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, typename VectorType, int spacedim>
Tensor<1, spacedim, typename VectorType::value_type>
point_gradient (const hp::MappingCollection<dim,spacedim> &mapping,
const hp::DoFHandler<dim,spacedim> &dof,
const VectorType &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 <typename VectorType>
void subtract_mean_value(VectorType &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 $\frac{1}{|\Omega|}\int_\Omega
* [u_h(x)]_c \; dx$ 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, typename VectorType, int spacedim>
double compute_mean_value (const Mapping<dim, spacedim> &mapping,
const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &quadrature,
const VectorType &v,
const unsigned int component);
/**
* Calls the other compute_mean_value() function, see above, with
* <tt>mapping=MappingQGeneric@<dim@>(1)</tt>.
*/
template <int dim, typename VectorType, int spacedim>
double compute_mean_value (const DoFHandler<dim,spacedim> &dof,
const Quadrature<dim> &quadrature,
const VectorType &v,
const unsigned int component);
//@}
/**
* Geometrical interpolation
*/
//@{
/**
* Given a DoFHandler containing at least a spacedim vector field, this
* function interpolates the Triangulation at the support points of a FE_Q()
* finite element of the same degree as the degree of the required
* components.
*
* Curved manifold are respected, and the resulting VectorType will be
* geometrically consistent. The resulting map is guaranteed to be
* interpolatory at the support points of a FE_Q() finite element of the
* same degree as the degree of the required components.
*
* If the underlying finite element is an FE_Q(1)^spacedim, then the
* resulting @p VectorType is a finite element field representation of the
* vertices of the Triangulation.
*
* The optional ComponentMask argument can be used to specify what
* components of the FiniteElement to use to describe the geometry. If no
* mask is specified at construction time, then a default one is used, i.e.,
* the first spacedim components of the FiniteElement are assumed to
* represent the geometry of the problem.
*
* This function is only implemented for FiniteElements where the specified
* components are primitive.
*
* @author Luca Heltai, 2015
*/
template<typename DoFHandlerType, typename VectorType>
void get_position_vector(const DoFHandlerType &dh,
VectorType &vector,
const ComponentMask &mask = ComponentMask());
//@}
/**
* Exception
*/
DeclExceptionMsg (ExcNonInterpolatingFE,
"You are attempting an operation that requires the "
"finite element involved to be 'interpolating', i.e., "
"it needs to have support points. The finite element "
"you are using here does not appear to have those.");
/**
* Exception
*/
DeclExceptionMsg (ExcPointNotAvailableHere,
"The given point is inside a cell of a "
"parallel::distributed::Triangulation that is not "
"locally owned by this processor.");
}
DEAL_II_NAMESPACE_CLOSE
#endif
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