libdeal.ii-dev 8.1.0-4 / usr / include / deal.II / fe / fe_dgq.h

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// $Id: fe_dgq.h 30036 2013-07-18 16:55:32Z maier $
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#ifndef __deal2__fe_dgq_h
#define __deal2__fe_dgq_h

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

DEAL_II_NAMESPACE_OPEN

template <int dim, int spacedim> class MappingQ;
template <int dim> class Quadrature;

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

/**
 * Implementation of scalar, discontinuous tensor product elements
 * based on equidistant support points.
 *
 * This is a discontinuous finite element based on tensor products of
 * Lagrangian polynomials. The shape functions are Lagrangian
 * interpolants of an equidistant grid of points on the unit cell. The
 * points are numbered in lexicographical order, with <i>x</i> running
 * fastest, then <i>y</i>, then <i>z</i> (if these coordinates are present for a
 * given space dimension at all). For example, these are the node
 * orderings for <tt>FE_DGQ(1)</tt> in 3d:
 *  @verbatim
 *         6-------7        6-------7
 *        /|       |       /       /|
 *       / |       |      /       / |
 *      /  |       |     /       /  |
 *     4   |       |    4-------5   |
 *     |   2-------3    |       |   3
 *     |  /       /     |       |  /
 *     | /       /      |       | /
 *     |/       /       |       |/
 *     0-------1        0-------1
 *  @endverbatim
 * and <tt>FE_DGQ(2)</tt>:
 *  @verbatim
 *         24--25--26       24--25--26
 *        /|       |       /       /|
 *      21 |       |     21  22  23 |
 *      /  15  16  17    /       /  17
 *    18   |       |   18--19--20   |
 *     |12 6---7---8    |       |14 8
 *     9  /       /     9  10  11  /
 *     | 3   4   5      |       | 5
 *     |/       /       |       |/
 *     0---1---2        0---1---2
 *  @endverbatim
 * with node 13 being placed in the interior of the hex.
 *
 * Note, however, that these are just the Lagrange interpolation
 * points of the shape functions. Even though they may physically be
 * on the surface of the cell, they are logically in the interior
 * since there are no continuity requirements for these shape
 * functions across cell boundaries.
 * This class if partially implemented for the codimension one case
 * (<tt>spacedim != dim </tt>), since no passage of information
 * between meshes of different refinement level is possible because
 * the embedding and projection matrices are not computed in the class
 * constructor.
 *
 * @author Ralf Hartmann, Guido Kanschat 2001, 2004
 */
template <int dim, int spacedim=dim>
class FE_DGQ : public FE_Poly<TensorProductPolynomials<dim>, dim, spacedim>
{
public:
  /**
   * Constructor for tensor product
   * polynomials of degree
   * <tt>p</tt>. The shape
   * functions created using this
   * constructor correspond to
   * Lagrange interpolation
   * polynomials for equidistantly
   * spaced support points in each
   * coordinate direction.
   */
  FE_DGQ (const unsigned int p);

  /**
   * Return a string that uniquely
   * identifies a finite
   * element. This class returns
   * <tt>FE_DGQ<dim>(degree)</tt>, with
   * <tt>dim</tt> and <tt>degree</tt>
   * replaced by appropriate
   * values.
   */
  virtual std::string get_name () const;

  /**
   * Return the matrix
   * interpolating from the given
   * finite element to the present
   * one. The size of the matrix is
   * then @p dofs_per_cell times
   * <tt>source.dofs_per_cell</tt>.
   *
   * These matrices are only
   * available if the source
   * element is also a @p FE_DGQ
   * element. Otherwise, an
   * exception of type
   * FiniteElement<dim>::ExcInterpolationNotImplemented
   * is thrown.
   */
  virtual void
  get_interpolation_matrix (const FiniteElement<dim, spacedim> &source,
                            FullMatrix<double>           &matrix) 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>.
   *
   * Derived elements will have to
   * implement this function. They
   * may only provide interpolation
   * matrices for certain source
   * finite elements, for example
   * those from the same family. If
   * they don't implement
   * interpolation from a given
   * element, then they must throw
   * an exception of type
   * FiniteElement<dim>::ExcInterpolationNotImplemented.
   */
  virtual void
  get_face_interpolation_matrix (const FiniteElement<dim, spacedim> &source,
                                 FullMatrix<double>       &matrix) 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>.
   *
   * Derived elements will have to
   * implement this function. They
   * may only provide interpolation
   * matrices for certain source
   * finite elements, for example
   * those from the same family. If
   * they don't implement
   * interpolation from a given
   * element, then they must throw
   * an exception of type
   * FiniteElement<dim>::ExcInterpolationNotImplemented.
   */
  virtual void
  get_subface_interpolation_matrix (const FiniteElement<dim, spacedim> &source,
                                    const unsigned int        subface,
                                    FullMatrix<double>       &matrix) const;

  /**
   * @name Functions to support hp
   * @{
   */

  /**
   * If, on a vertex, several finite elements are active, the hp code
   * first assigns the degrees of freedom of each of these FEs
   * different global indices. It then calls this function to find out
   * which of them should get identical values, and consequently can
   * receive the same global DoF index. This function therefore
   * returns a list of identities between DoFs of the present finite
   * element object with the DoFs of @p fe_other, which is a reference
   * to a finite element object representing one of the other finite
   * elements active on this particular vertex. The function computes
   * which of the degrees of freedom of the two finite element objects
   * are equivalent, both numbered between zero and the corresponding
   * value of dofs_per_vertex of the two finite elements. The first
   * index of each pair denotes one of the vertex dofs of the present
   * element, whereas the second is the corresponding index of the
   * other finite element.
   *
   * This being a discontinuous element, the set of such constraints
   * is of course empty.
   */
  virtual
  std::vector<std::pair<unsigned int, unsigned int> >
  hp_vertex_dof_identities (const FiniteElement<dim, spacedim> &fe_other) const;

  /**
   * Same as hp_vertex_dof_indices(), except that the function treats
   * degrees of freedom on lines.
   *
   * This being a discontinuous element, the set of such constraints
   * is of course empty.
   */
  virtual
  std::vector<std::pair<unsigned int, unsigned int> >
  hp_line_dof_identities (const FiniteElement<dim, spacedim> &fe_other) const;

  /**
   * Same as hp_vertex_dof_indices(), except that the function treats
   * degrees of freedom on quads.
   *
   * This being a discontinuous element, the set of such constraints
   * is of course empty.
   */
  virtual
  std::vector<std::pair<unsigned int, unsigned int> >
  hp_quad_dof_identities (const FiniteElement<dim, spacedim> &fe_other) const;

  /**
   * Return whether this element
   * implements its hanging node
   * constraints in the new way,
   * which has to be used to make
   * elements "hp compatible".
   *
   * For the FE_DGQ class the result is
   * always true (independent of the degree
   * of the element), as it has no hanging
   * nodes (being a discontinuous element).
   */
  virtual bool hp_constraints_are_implemented () 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".
   */
  virtual
  FiniteElementDomination::Domination
  compare_for_face_domination (const FiniteElement<dim, spacedim> &fe_other) const;

  /**
   * @}
   */

  /**
   * Check for non-zero values on a face.
   *
   * This function returns
   * @p true, if the shape
   * function @p shape_index has
   * non-zero values on the face
   * @p face_index.
   *
   * Implementation of the
   * interface in
   * FiniteElement
   */
  virtual bool has_support_on_face (const unsigned int shape_index,
                                    const unsigned int face_index) const;

  /**
   * Determine an estimate for the
   * memory consumption (in bytes)
   * of this object.
   *
   * This function is made virtual,
   * since finite element objects
   * are usually accessed through
   * pointers to their base class,
   * rather than the class itself.
   */
  virtual std::size_t memory_consumption () const;


protected:
  /**
   * Constructor for tensor product
   * polynomials based on
   * Polynomials::Lagrange
   * interpolation of the support
   * points in the quadrature rule
   * <tt>points</tt>. The degree of
   * these polynomials is
   * <tt>points.size()-1</tt>.
   *
   * Note: The FE_DGQ::clone function
   * does not work properly for FE with
   * arbitrary nodes!
   */
  FE_DGQ (const Quadrature<1> &points);

  /**
   * @p clone function instead of
   * a copy constructor.
   *
   * This function is needed by the
   * constructors of @p FESystem.
   */
  virtual FiniteElement<dim, spacedim> *clone() const;

private:
  /**
   * Only for internal use. Its
   * full name is
   * @p get_dofs_per_object_vector
   * function and it creates the
   * @p dofs_per_object vector that is
   * needed within the constructor to
   * be passed to the constructor of
   * @p FiniteElementData.
   */
  static std::vector<unsigned int> get_dpo_vector (const unsigned int degree);

  /**
   * Compute renumbering for rotation
   * of degrees of freedom.
   *
   * Rotates a tensor product
   * numbering of degrees of
   * freedom by 90 degrees. It is
   * used to compute the transfer
   * matrices of the children by
   * using only the matrix for the
   * first child.
   *
   * The direction parameter
   * determines the type of
   * rotation. It is one character
   * of @p xXyYzZ. The character
   * determines the axis of
   * rotation, case determines the
   * direction. Lower case is
   * counter-clockwise seen in
   * direction of the axis.
   *
   * Since rotation around the
   * y-axis is not used, it is not
   * implemented either.
   */
  void rotate_indices (std::vector<unsigned int> &indices,
                       const char                 direction) const;

  /**
   * Allow access from other dimensions.
   */
  template <int dim1, int spacedim1> friend class FE_DGQ;

  /**
   * Allows @p MappingQ class to
   * access to build_renumbering
   * function.
   */
  template <int dim1, int spacedim1> friend class MappingQ;
};



/**
 * Implementation of scalar, discontinuous tensor product elements based on
 * Lagrange polynomials with arbitrary nodes. The primary purpose of this
 * class is to provide an element for which the mass matrix can be made
 * diagonal by choosing basis functions that are not either zero or one at the
 * vertices of the cell, but instead are zero or one at a given set of
 * quadrature points. If this set of quadrature points is then also used in
 * integrating the mass matrix, then it will be diagonal. The number of
 * quadrature points automatically determines the polynomial degree chosen for
 * this element.
 *
 * See the base class documentation in FE_DGQ for details.
 *
 * @author F. Prill 2006
 */
template <int dim,int spacedim=dim>
class FE_DGQArbitraryNodes : public FE_DGQ<dim,spacedim>
{
public:
  /**
   * Constructor for tensor product
   * polynomials based on
   * Polynomials::Lagrange
   * interpolation of the support
   * points in the quadrature rule
   * <tt>points</tt>. The degree of
   * these polynomials is
   * <tt>points.size()-1</tt>.
   */
  FE_DGQArbitraryNodes (const Quadrature<1> &points);

  /**
   * Return a string that uniquely
   * identifies a finite
   * element. This class returns
   * <tt>FE_DGQArbitraryNodes<dim>(degree)</tt>,
   * with <tt>dim</tt> and <tt>degree</tt>
   * replaced by appropriate
   * values.
   */
  virtual std::string get_name () const;

protected:
  /**
   * @p clone function instead of
   * a copy constructor.
   *
   * This function is needed by the
   * constructors of @p FESystem.
   */
  virtual FiniteElement<dim,spacedim> *clone() const;
};


/*@}*/

DEAL_II_NAMESPACE_CLOSE

#endif