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// $Id: fe_abf.h 17866 2008-12-05 22:27:44Z bangerth $
// Version: $Name$
//
// Copyright (C) 2003, 2004, 2005, 2006, 2007 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
// to the file deal.II/doc/license.html for the text and
// further information on this license.
//
//---------------------------------------------------------------------------
#ifndef __deal2__fe_abf_h
#define __deal2__fe_abf_h
#include <base/config.h>
#include <base/table.h>
#include <base/polynomials_abf.h>
#include <base/polynomial.h>
#include <base/tensor_product_polynomials.h>
#include <base/geometry_info.h>
#include <fe/fe.h>
#include <fe/fe_poly_tensor.h>
#include <vector>
DEAL_II_NAMESPACE_OPEN
template <int dim, int spacedim> class MappingQ;
/*!@addtogroup fe */
/*@{*/
/**
* Implementation of Arnold-Boffi-Falk (ABF) elements, conforming with the
* space H<sup>div</sup>. These elements generate vector fields with
* normal components continuous between mesh cells.
*
* These elements are based on an article from Arnold, Boffi and Falk:
* Quadrilateral H(div) finite elements, SIAM J. Numer. Anal.
* Vol.42, No.6, pp.2429-2451
*
* In this article, the authors demonstrate that the usual RT elements
* and also BDM and other proposed finite dimensional subspaces of
* H(div) do not work properly on arbitrary FE grids. I.e. the
* convergence rates deteriorate on these meshes. As a solution the
* authors propose the ABF elements, which are implemented in this
* module.
*
* This class is not implemented for the codimension one case
* (<tt>spacedim != dim</tt>).
*
* @todo Even if this element is implemented for two and three space
* dimensions, the definition of the node values relies on
* consistently oriented faces in 3D. Therefore, care should be taken
* on complicated meshes.
*
* <h3>Interpolation</h3>
*
* The @ref GlossInterpolation "interpolation" operators associated
* with the RT element are constructed such that interpolation and
* computing the divergence are commuting operations. We require this
* from interpolating arbitrary functions as well as the #restriction
* matrices. It can be achieved by two interpolation schemes, the
* simplified one in FE_RaviartThomasNodal and the original one here:
*
* <h4>Node values on edges/faces</h4>
*
* On edges or faces, the @ref GlossNodes "node values" are the moments of
* the normal component of the interpolated function with respect to
* the traces of the RT polynomials. Since the normal trace of the RT
* space of degree <i>k</i> on an edge/face is the space
* <i>Q<sub>k</sub></i>, the moments are taken with respect to this
* space.
*
* <h4>Interior node values</h4>
*
* Higher order RT spaces have interior nodes. These are moments taken
* with respect to the gradient of functions in <i>Q<sub>k</sub></i>
* on the cell (this space is the matching space for RT<sub>k</sub> in
* a mixed formulation).
*
* <h4>Generalized support points</h4>
*
* The node values above rely on integrals, which will be computed by
* quadrature rules themselves. The generalized support points are a
* set of points such that this quadrature can be performed with
* sufficient accuracy. The points needed are those of
* QGauss<sub>k+1</sub> on each face as well as QGauss<sub>k</sub> in
* the interior of the cell (or none for RT<sub>0</sub>).
*
*
* @author Oliver Kayser-Herold, 2006, based on previous work
* by Guido Kanschat and Wolfgang Bangerth
*/
template <int dim>
class FE_ABF : public FE_PolyTensor<PolynomialsABF<dim>, dim>
{
public:
/**
* Constructor for the ABF
* element of degree @p p.
*/
FE_ABF (const unsigned int p);
/**
* Return a string that uniquely
* identifies a finite
* element. This class returns
* <tt>FE_ABF<dim>(degree)</tt>, with
* @p dim and @p degree
* replaced by appropriate
* values.
*/
virtual std::string get_name () const;
/**
* Number of base elements in a
* mixed discretization. Here,
* this is of course equal to
* one.
*/
virtual unsigned int n_base_elements () const;
/**
* Access to base element
* objects. Since this element is
* atomic, <tt>base_element(0)</tt> is
* @p this, and all other
* indices throw an error.
*/
virtual const FiniteElement<dim> &
base_element (const unsigned int index) const;
/**
* Multiplicity of base element
* @p index. Since this is an
* atomic element,
* <tt>element_multiplicity(0)</tt>
* returns one, and all other
* indices will throw an error.
*/
virtual unsigned int element_multiplicity (const unsigned int index) const;
/**
* Check whether a shape function
* may be non-zero on a face.
*
* Right now, this is only
* implemented for RT0 in
* 1D. Otherwise, returns always
* @p true.
*/
virtual bool has_support_on_face (const unsigned int shape_index,
const unsigned int face_index) const;
virtual void interpolate(std::vector<double>& local_dofs,
const std::vector<double>& values) const;
virtual void interpolate(std::vector<double>& local_dofs,
const std::vector<Vector<double> >& values,
unsigned int offset = 0) const;
virtual void interpolate(
std::vector<double>& local_dofs,
const VectorSlice<const std::vector<std::vector<double> > >& values) const;
virtual unsigned int memory_consumption () const;
virtual FiniteElement<dim> * clone() const;
private:
/**
* The order of the
* ABF element. The
* lowest order elements are
* usually referred to as RT0,
* even though their shape
* functions are piecewise
* quadratics.
*/
const unsigned int rt_order;
/**
* 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);
/**
* Initialize the @p
* generalized_support_points
* field of the FiniteElement
* class and fill the tables with
* interpolation weights
* (#boundary_weights and
* #interior_weights). Called
* from the constructor.
*/
void initialize_support_points (const unsigned int rt_degree);
/**
* Initialize the interpolation
* from functions on refined mesh
* cells onto the father
* cell. According to the
* philosophy of the
* Raviart-Thomas element, this
* restriction operator preserves
* the divergence of a function
* weakly.
*/
void initialize_restriction ();
/**
* Given a set of flags indicating
* what quantities are requested
* from a @p FEValues object,
* return which of these can be
* precomputed once and for
* all. Often, the values of
* shape function at quadrature
* points can be precomputed, for
* example, in which case the
* return value of this function
* would be the logical and of
* the input @p flags and
* @p update_values.
*
* For the present kind of finite
* element, this is exactly the
* case.
*/
virtual UpdateFlags update_once (const UpdateFlags flags) const;
/**
* This is the opposite to the
* above function: given a set of
* flags indicating what we want
* to know, return which of these
* need to be computed each time
* we visit a new cell.
*
* If for the computation of one
* quantity something else is
* also required (for example, we
* often need the covariant
* transformation when gradients
* need to be computed), include
* this in the result as well.
*/
virtual UpdateFlags update_each (const UpdateFlags flags) const;
/**
* Fields of cell-independent data.
*
* For information about the
* general purpose of this class,
* see the documentation of the
* base class.
*/
class InternalData : public FiniteElement<dim>::InternalDataBase
{
public:
/**
* Array with shape function
* values in quadrature
* points. There is one row
* for each shape function,
* containing values for each
* quadrature point. Since
* the shape functions are
* vector-valued (with as
* many components as there
* are space dimensions), the
* value is a tensor.
*
* In this array, we store
* the values of the shape
* function in the quadrature
* points on the unit
* cell. The transformation
* to the real space cell is
* then simply done by
* multiplication with the
* Jacobian of the mapping.
*/
std::vector<std::vector<Tensor<1,dim> > > shape_values;
/**
* Array with shape function
* gradients in quadrature
* points. There is one
* row for each shape
* function, containing
* values for each quadrature
* point.
*
* We store the gradients in
* the quadrature points on
* the unit cell. We then
* only have to apply the
* transformation (which is a
* matrix-vector
* multiplication) when
* visiting an actual cell.
*/
std::vector<std::vector<Tensor<2,dim> > > shape_gradients;
};
/**
* These are the factors
* multiplied to a function in
* the
* #generalized_face_support_points
* when computing the
* integration. They are
* organized such that there is
* one row for each generalized
* face support point and one
* column for each degree of
* freedom on the face.
*/
Table<2, double> boundary_weights;
/**
* Precomputed factors for
* interpolation of interior
* degrees of freedom. The
* rationale for this Table is
* the same as for
* #boundary_weights. Only, this
* table has a third coordinate
* for the space direction of the
* component evaluated.
*/
Table<3, double> interior_weights;
/**
* These are the factors
* multiplied to a function in
* the
* #generalized_face_support_points
* when computing the
* integration. They are
* organized such that there is
* one row for each generalized
* face support point and one
* column for each degree of
* freedom on the face.
*/
Table<2, double> boundary_weights_abf;
/**
* Precomputed factors for
* interpolation of interior
* degrees of freedom. The
* rationale for this Table is
* the same as for
* #boundary_weights. Only, this
* table has a third coordinate
* for the space direction of the
* component evaluated.
*/
Table<3, double> interior_weights_abf;
/**
* Allow access from other
* dimensions.
*/
template <int dim1> friend class FE_ABF;
};
/*@}*/
/* -------------- declaration of explicit specializations ------------- */
#ifndef DOXYGEN
template <>
std::vector<unsigned int> FE_ABF<1>::get_dpo_vector (const unsigned int);
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#endif
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