/usr/include/deal.II/fe/fe

// ---------------------------------------------------------------------
// $Id: fe_nothing.h 30036 2013-07-18 16:55:32Z maier $
//
// Copyright (C) 2009 - 2013 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------

#ifndef __deal2__fe_nothing_h
#define __deal2__fe_nothing_h

#include <deal.II/base/config.h>
#include <deal.II/fe/fe.h>

DEAL_II_NAMESPACE_OPEN


/*!@addtogroup fe */
/*@{*/

/**
 * Definition of a finite element with zero degrees of freedom.  This class is
 * useful (in the context of an hp method) to represent empty cells in the
 * triangulation on which no degrees of freedom should be allocated, or to
 * describe a field that is extended by zero to a part of the domain where we
 * don't need it.  Thus a triangulation may be divided into two regions: an
 * active region where normal elements are used, and an inactive region where
 * FE_Nothing elements are used.  The hp::DoFHandler will therefore assign no
 * degrees of freedom to the FE_Nothing cells, and this subregion is therefore
 * implicitly deleted from the computation. step-46 shows a use case for this
 * element. An interesting application for this element is also presented in the
 * paper A. Cangiani, J. Chapman, E. Georgoulis, M. Jensen:
 * <b>Implementation of the Continuous-Discontinuous Galerkin Finite Element Method</b>,
 * arXiv:1201.2878v1 [math.NA], 2012 (see http://arxiv.org/abs/1201.2878).
 *
 * Note that some care must be taken that the resulting mesh topology
 * continues to make sense when FE_Nothing elements are introduced.
 * This is particularly true when dealing with hanging node constraints,
 * because the library makes some basic assumptions about the nature
 * of those constraints.  The following geometries are acceptable:
 * @code
 * +---------+----+----+
 * |         | 0  |    |
 * |    1    +----+----+
 * |         | 0  |    |
 * +---------+----+----+
 * @endcode
 * @code
 * +---------+----+----+
 * |         | 1  |    |
 * |    0    +----+----+
 * |         | 1  |    |
 * +---------+----+----+
 * @endcode
 * Here, 0 denotes an FE_Nothing cell, and 1 denotes some other
 * element type.  The library has no difficulty computing the necessary
 * hanging node constraints in these cases (i.e. no constraint).
 * However, the following geometry is NOT acceptable (at least
 * in the current implementation):
 * @code
 * +---------+----+----+
 * |         | 0  |    |
 * |    1    +----+----+
 * |         | 1  |    |
 * +---------+----+----+
 * @endcode
 * The distinction lies in the mixed nature of the child faces,
 * a case we have not implemented as of yet.
 *
 * @author Joshua White, Wolfgang Bangerth
 */
template <int dim>
class FE_Nothing : public FiniteElement<dim>
{
public:

  /**
    * Constructor. Argument denotes the
    * number of components to give this
    * finite element (default = 1).
    */
  FE_Nothing (unsigned int n_components = 1);

  /**
   * A sort of virtual copy
   * constructor. Some places in
   * the library, for example the
   * constructors of FESystem as
   * well as the hp::FECollection
   * class, need to make copied of
   * finite elements without
   * knowing their exact type. They
   * do so through this function.
   */
  virtual
  FiniteElement<dim> *
  clone() const;

  /**
   * Return a string that uniquely
   * identifies a finite
   * element. In this case it is
   * <code>FE_Nothing@<dim@></code>.
   */
  virtual
  std::string
  get_name() const;

  /**
   * Determine the values a finite
   * element should compute on
   * initialization of data for
   * FEValues.
   *
   * Given a set of flags
   * indicating what quantities are
   * requested from a FEValues
   * object, update_once() and
   * update_each() compute which
   * values must really be
   * computed. Then, the
   * <tt>fill_*_values</tt> functions
   * are called with the result of
   * these.
   *
   * In this case, since the element
   * has zero degrees of freedom and
   * no information can be computed on
   * it, this function simply returns
   * the default (empty) set of update
   * flags.
   */

  virtual
  UpdateFlags
  update_once (const UpdateFlags flags) const;

  /**
   * Complementary function for
   * update_once().
   *
   * While update_once() returns
   * the values to be computed on
   * the unit cell for yielding the
   * required data, this function
   * determines the values that
   * must be recomputed on each
   * cell.
   *
   * Refer to update_once() for
   * more details.
   */
  virtual
  UpdateFlags
  update_each (const UpdateFlags flags) const;

  /**
   * Return the value of the
   * @p ith shape function at the
   * point @p p. @p p is a point
   * on the reference element. Because the
   * current element has no degrees of freedom,
   * this function should obviously not be
   * called in practice.  All this function
   * really does, therefore, is trigger an
   * exception.
   */
  virtual
  double
  shape_value (const unsigned int i, const Point<dim> &p) const;

  /**
   * Fill the fields of
   * FEValues. This function
   * performs all the operations
   * needed to compute the data of an
   * FEValues object.
   *
   * In the current case, this function
   * returns no meaningful information,
   * since the element has no degrees of
   * freedom.
   */
  virtual
  void
  fill_fe_values (const Mapping<dim> &mapping,
                  const typename Triangulation<dim>::cell_iterator &cell,
                  const Quadrature<dim> &quadrature,
                  typename Mapping<dim>::InternalDataBase &mapping_data,
                  typename Mapping<dim>::InternalDataBase &fedata,
                  FEValuesData<dim,dim> &data,
                  CellSimilarity::Similarity &cell_similarity) const;

  /**
   * Fill the fields of
   * FEFaceValues. This function
   * performs all the operations
   * needed to compute the data of an
   * FEFaceValues object.
   *
   * In the current case, this function
   * returns no meaningful information,
   * since the element has no degrees of
   * freedom.
   */
  virtual
  void
  fill_fe_face_values (const Mapping<dim> &mapping,
                       const typename Triangulation<dim> :: cell_iterator &cell,
                       const unsigned int face,
                       const Quadrature<dim-1> & quadrature,
                       typename Mapping<dim> :: InternalDataBase &mapping_data,
                       typename Mapping<dim> :: InternalDataBase &fedata,
                       FEValuesData<dim,dim> &data) const;

  /**
   * Fill the fields of
   * FESubFaceValues. This function
   * performs all the operations
   * needed to compute the data of an
   * FESubFaceValues object.
   *
   * In the current case, this function
   * returns no meaningful information,
   * since the element has no degrees of
   * freedom.
   */
  virtual
  void
  fill_fe_subface_values (const Mapping<dim> &mapping,
                          const typename Triangulation<dim>::cell_iterator &cell,
                          const unsigned int face,
                          const unsigned int subface,
                          const Quadrature<dim-1> & quadrature,
                          typename Mapping<dim>::InternalDataBase &mapping_data,
                          typename Mapping<dim>::InternalDataBase &fedata,
                          FEValuesData<dim,dim> &data) const;

  /**
   * Prepare internal data
   * structures and fill in values
   * independent of the
   * cell. Returns a pointer to an
   * object of which the caller of
   * this function then has to
   * assume ownership (which
   * includes destruction when it
   * is no more needed).
   *
   * In the current case, this function
   * just returns a default pointer, since
   * no meaningful data exists for this
   * element.
   */
  virtual
  typename Mapping<dim>::InternalDataBase *
  get_data (const UpdateFlags     update_flags,
            const Mapping<dim>     &mapping,
            const Quadrature<dim> &quadrature) const;

  /**
   * Return whether this element dominates
   * the one given as argument when they
   * meet at a common face,
   * whether it is the other way around,
   * whether neither dominates, or if
   * either could dominate.
   *
   * For a definition of domination, see
   * FiniteElementBase::Domination and in
   * particular the @ref hp_paper "hp paper".
   *
   * In the current case, this element
   * is always assumed to dominate, unless
   * it is also of type FE_Nothing().  In
   * that situation, either element can
   * dominate.
   */
  virtual
  FiniteElementDomination::Domination
  compare_for_face_domination (const FiniteElement<dim> &fe_other) const;



  virtual
  std::vector<std::pair<unsigned int, unsigned int> >
  hp_vertex_dof_identities (const FiniteElement<dim> &fe_other) const;

  virtual
  std::vector<std::pair<unsigned int, unsigned int> >
  hp_line_dof_identities (const FiniteElement<dim> &fe_other) const;

  virtual
  std::vector<std::pair<unsigned int, unsigned int> >
  hp_quad_dof_identities (const FiniteElement<dim> &fe_other) const;

  virtual
  bool
  hp_constraints_are_implemented () const;

  /**
  * Return the matrix
  * interpolating from a face of
  * of one element to the face of
  * the neighboring element.
  * The size of the matrix is
  * then <tt>source.#dofs_per_face</tt> times
  * <tt>this->#dofs_per_face</tt>.
  *
  * Since the current finite element has no
  * degrees of freedom, the interpolation
  * matrix is necessarily empty.
  */

  virtual
  void
  get_face_interpolation_matrix (const FiniteElement<dim> &source_fe,
                                 FullMatrix<double>       &interpolation_matrix) const;


  /**
   * Return the matrix
   * interpolating from a face of
   * of one element to the subface of
   * the neighboring element.
   * The size of the matrix is
   * then <tt>source.#dofs_per_face</tt> times
   * <tt>this->#dofs_per_face</tt>.
   *
   * Since the current finite element has no
   * degrees of freedom, the interpolation
   * matrix is necessarily empty.
   */

  virtual
  void
  get_subface_interpolation_matrix (const FiniteElement<dim> &source_fe,
                                    const unsigned int index,
                                    FullMatrix<double>  &interpolation_matrix) const;


};


/*@}*/

DEAL_II_NAMESPACE_CLOSE

#endif
libdeal.ii-dev 8.1.0-4 / usr / include / deal.II / fe / fe_nothing.h
