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// $Id: fe_nedelec.h 18933 2009-06-12 08:53:35Z bangerth $
// Version: $Name$
//
// Copyright (C) 2002, 2003, 2004, 2005, 2006 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_nedelec_h
#define __deal2__fe_nedelec_h
#include <base/config.h>
#include <base/geometry_info.h>
#include <fe/fe.h>
DEAL_II_NAMESPACE_OPEN
template <int dim, int spacedim> class MappingQ;
/*!@addtogroup fe */
/*@{*/
/**
* Implementation of continuous Nedelec elements for the space
* H_curl. Note, however, that continuity only concerns the tangential
* component of the vector field.
*
* The constructor of this class takes the degree @p p of this finite
* element. However, presently, only lowest order elements
* (i.e. <tt>p==1</tt>) are implemented. For a general overview of this
* element and its properties, see the report by Anna Schneebeli that
* is linked from the general documentation page of the library.
*
* This class has not yet been implemented for the codimension one case
* (<tt>spacedim != dim</tt>).
*
*
* <h3>Restriction on transformations</h3>
*
* In some sense, the implementation of this element is not complete,
* but you will rarely notice. Here is the fact: since the element is
* vector-valued already on the unit cell, the Jacobian matrix (or its
* inverse) is needed already to generate the values of the shape
* functions on the cells in real space. This is in contrast to most
* other elements, where you only need the Jacobian for the
* gradients. Thus, to generate the gradients of Nedelec shape
* functions, one would need to have the derivatives of the inverse of
* the Jacobian matrix.
*
* Basically, the Nedelec shape functions can be understood as the
* gradients of scalar shape functions on the real cell. They are thus
* the inverse Jacobian matrix times the gradients of scalar shape
* functions on the unit cell. The gradient of Nedelec shape functions
* is then, by the product rule, the sum of first the derivative (with
* respect to true coordinates) of the inverse Jacobian times the
* gradient (in unit coordinates) of the scalar shape function, plus
* second the inverse Jacobian times the derivative (in true
* coordinates) of the gradient (in unit coordinates) of the scalar
* shape functions. Note that each of the derivatives in true
* coordinates can be expressed as inverse Jacobian times gradient in
* unit coordinates.
*
* The problem is the derivative of the inverse Jacobian. This rank-3
* tensor can actually be computed (and we did so in very early
* versions of the library), but is a large task and very time
* consuming, so we dropped it. Since it is not available, we simply
* drop this first term.
*
* What this means for the present case: first the computation of
* gradients of Nedelec shape functions is wrong. Second, you will not
* notice this usually, for two reasons:
*
* The first reason is that the gradient of the Jacobian vanishes if
* the cells are mapped by an affine mapping, to which the usual
* bilinear mapping reduces if the cell is a parallelogram. Then the
* gradient of the shape functions is computed exact, since the first
* term is zero.
*
* Second, with the Nedelec elements, you will usually want to compute
* the curl, and extract and sum up the respective elements of the
* full gradient tensor. However, the curl of the Jacobian vanishes,
* so for the curl of shape functions the first term is irrelevant,
* and the curl will be computed correctly as well.
*
*
* <h3>Interpolation to finer and coarser meshes</h3>
*
* Each finite element class in deal.II provides matrices that are
* used to interpolate from coarser to finer meshes and the other way
* round. Interpolation from a mother cell to its children is usually
* trivial, since finite element spaces are normally nested and this
* kind of interpolation is therefore exact. On the other hand, when
* we interpolate from child cells to the mother cell, we usually have
* to throw away some information.
*
* For continuous elements, this transfer usually happens by
* interpolating the values on the child cells at the support points
* of the shape functions of the mother cell. However, for
* discontinuous elements, we often use a projection from the child
* cells to the mother cell. The projection approach is only possible
* for discontinuous elements, since it cannot be guaranteed that the
* values of the projected functions on one cell and its neighbor
* match. In this case, only an interpolation can be
* used. (Internally, whether the values of a shape function are
* interpolated or projected, or better: whether the matrices the
* finite element provides are to be treated with the properties of a
* projection or of an interpolation, is controlled by the
* @p restriction_is_additive flag. See there for more information.)
*
* Here, things are not so simple: since the element has some
* continuity requirements across faces, we can only resort to some
* kind of interpolation. On the other hand, for the lowest order
* elements, the values of generating functionals are the (constant)
* tangential values of the shape functions. We would therefore really
* like to take the mean value of the tangential values of the child
* faces, and make this the value of the mother face. Then, however,
* taking a mean value of two piecewise constant function is not an
* interpolation, but a restriction. Since this is not possible, we
* cannot use this.
*
* To make a long story somewhat shorter, when interpolating from
* refined edges to a coarse one, we do not take the mean value, but
* pick only one (the one from the first child edge). While this is
* not optimal, it is certainly a valid choice (using an interpolation
* point that is not in the middle of the cell, but shifted to one
* side), and it also preserves the order of the interpolation.
*
*
* <h3>Numbering of the degrees of freedom (DoFs)</h3>
*
* Nedelec elements have their degrees of freedom on edges, with shape
* functions being vector valued and pointing in tangential
* direction. We use the standard enumeration and direction of edges
* in deal.II, yielding the following shape functions in 2d:
*
* @verbatim
* 3
* 2-->--3
* | |
* 0^ ^1
* | |
* 0-->--1
* 2
* @endverbatim
*
* For the 3d case, the ordering follows the same scheme: the lines
* are numbered as described in the documentation of the
* Triangulation class, i.e.
* @verbatim
* *---7---* *---7---*
* /| | / /|
* 4 | 11 4 5 11
* / 10 | / / |
* * | | *---6---* |
* | *---3---* | | *
* | / / | 9 /
* 8 0 1 8 | 1
* |/ / | |/
* *---2---* *---2---*
* @endverbatim
* and their directions are as follows:
* @verbatim
* *--->---* *--->---*
* /| | / /|
* ^ | ^ ^ ^ ^
* / ^ | / / |
* * | | *--->---* |
* | *--->---* | | *
* | / / | ^ /
* ^ ^ ^ ^ | ^
* |/ / | |/
* *--->---* *--->---*
* @endverbatim
*
* The element does not make much sense in 1d, so it is not
* implemented there.
*
*
* @author Wolfgang Bangerth, Anna Schneebeli, 2002, 2003
*/
template <int dim, int spacedim=dim>
class FE_Nedelec : public FiniteElement<dim,spacedim>
{
public:
/**
* Constructor for the Nedelec
* element of degree @p p.
*/
FE_Nedelec (const unsigned int p);
/**
* Return a string that uniquely
* identifies a finite
* element. This class returns
* <tt>FE_Nedelec<dim>(degree)</tt>, with
* @p dim and @p degree
* replaced by appropriate
* values.
*/
virtual std::string get_name () const;
/**
* Return the value of the
* @p componentth vector
* component of the @p ith shape
* function at the point
* @p p. See the
* FiniteElement base
* class for more information
* about the semantics of this
* function.
*/
virtual double shape_value_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the gradient of the
* @p componentth vector
* component of the @p ith shape
* function at the point
* @p p. See the
* FiniteElement base
* class for more information
* about the semantics of this
* function.
*/
virtual Tensor<1,dim> shape_grad_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the second derivative
* of the @p componentth vector
* component of the @p ith shape
* function at the point
* @p p. See the
* FiniteElement base
* class for more information
* about the semantics of this
* function.
*/
virtual Tensor<2,dim> shape_grad_grad_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the polynomial degree
* of this finite element,
* i.e. the value passed to the
* constructor.
*/
unsigned int get_degree () 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,spacedim> &
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;
/**
* This function returns
* @p true, if the shape
* function @p shape_index has
* non-zero values on the face
* @p face_index. For the lowest
* order Nedelec elements, this
* is actually the case for the
* one on which the shape
* function is defined and all
* neighboring ones.
*
* 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 unsigned int memory_consumption () const;
/**
* Declare a nested class which
* will hold static definitions of
* various matrices such as
* constraint and embedding
* matrices. The definition of
* the various static fields are
* in the files <tt>fe_nedelec_[23]d.cc</tt>
* in the source directory.
*/
struct Matrices
{
/**
* Embedding matrices. For
* each element type (the
* first index) there are as
* many embedding matrices as
* there are children per
* cell. The first index
* starts with linear
* elements and goes up in
* polynomial degree. The
* array may grow in the
* future with the number of
* elements for which these
* matrices have been
* computed. If for some
* element, the matrices have
* not been computed then you
* may use the element
* nevertheless but can not
* access the respective
* fields.
*/
static const double * const
embedding[][GeometryInfo<dim>::max_children_per_cell];
/**
* Number of elements (first
* index) the above field
* has. Equals the highest
* polynomial degree for
* which the embedding
* matrices have been
* computed.
*/
static const unsigned int n_embedding_matrices;
/**
* As the
* @p embedding_matrices
* field, but for the
* interface constraints. One
* for each element for which
* it has been computed.
*/
static const double * const constraint_matrices[];
/**
* Like
* @p n_embedding_matrices,
* but for the number of
* interface constraint
* matrices.
*/
static const unsigned int n_constraint_matrices;
};
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;
/**
* Prepare internal data
* structures and fill in values
* independent of the cell.
*/
virtual
typename Mapping<dim,spacedim>::InternalDataBase *
get_data (const UpdateFlags,
const Mapping<dim,spacedim>& mapping,
const Quadrature<dim>& quadrature) const ;
/**
* Implementation of the same
* function in
* FiniteElement.
*/
virtual void
fill_fe_values (const Mapping<dim,spacedim> &mapping,
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const Quadrature<dim> &quadrature,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
typename Mapping<dim,spacedim>::InternalDataBase &fe_internal,
FEValuesData<dim,spacedim> &data,
CellSimilarity::Similarity &cell_similarity) const;
/**
* Implementation of the same
* function in
* FiniteElement.
*/
virtual void
fill_fe_face_values (const Mapping<dim,spacedim> &mapping,
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const Quadrature<dim-1> &quadrature,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
typename Mapping<dim,spacedim>::InternalDataBase &fe_internal,
FEValuesData<dim,spacedim>& data) const ;
/**
* Implementation of the same
* function in
* FiniteElement.
*/
virtual void
fill_fe_subface_values (const Mapping<dim,spacedim> &mapping,
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const unsigned int sub_no,
const Quadrature<dim-1> &quadrature,
typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
typename Mapping<dim,spacedim>::InternalDataBase &fe_internal,
FEValuesData<dim,spacedim>& data) 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);
/**
* Initialize the hanging node
* constraints matrices. Called
* from the constructor.
*/
void initialize_constraints ();
/**
* Initialize the embedding
* matrices. Called from the
* constructor.
*/
void initialize_embedding ();
/**
* Initialize the restriction
* matrices. Called from the
* constructor.
*/
void initialize_restriction ();
/**
* Initialize the
* @p unit_support_points field
* of the FiniteElement
* class. Called from the
* constructor.
*/
void initialize_unit_support_points ();
/**
* Initialize the
* @p unit_face_support_points field
* of the FiniteElement
* class. Called from the
* constructor.
*/
void initialize_unit_face_support_points ();
/**
* 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;
/**
* Degree of the polynomials.
*/
const unsigned int degree;
/**
* 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,spacedim>::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;
};
/**
* Allow access from other
* dimensions.
*/
template <int , int> friend class FE_Nedelec;
};
/*@}*/
#ifndef DOXYGEN
/* -------------- declaration of explicit specializations ------------- */
template <>
void FE_Nedelec<1,1>::initialize_unit_face_support_points ();
template <>
double
FE_Nedelec<1,1>::shape_value_component (const unsigned int ,
const Point<1> &,
const unsigned int ) const;
template <>
double
FE_Nedelec<2,2>::shape_value_component (const unsigned int ,
const Point<2> &,
const unsigned int ) const;
template <>
double
FE_Nedelec<3,3>::shape_value_component (const unsigned int ,
const Point<3> &,
const unsigned int ) const;
template <>
Tensor<1,1>
FE_Nedelec<1,1>::shape_grad_component (const unsigned int ,
const Point<1> &,
const unsigned int ) const;
template <>
Tensor<1,2>
FE_Nedelec<2,2>::shape_grad_component (const unsigned int ,
const Point<2> &,
const unsigned int ) const;
template <>
Tensor<1,3>
FE_Nedelec<3,3>::shape_grad_component (const unsigned int ,
const Point<3> &,
const unsigned int ) const;
template <>
Tensor<2,1>
FE_Nedelec<1,1>::shape_grad_grad_component (const unsigned int ,
const Point<1> &,
const unsigned int ) const;
template <>
Tensor<2,2>
FE_Nedelec<2,2>::shape_grad_grad_component (const unsigned int ,
const Point<2> &,
const unsigned int ) const;
template <>
Tensor<2,3>
FE_Nedelec<3,3>::shape_grad_grad_component (const unsigned int ,
const Point<3> &,
const unsigned int ) const;
// declaration of explicit specializations of member variables, if the
// compiler allows us to do that (the standard says we must)
#ifndef DEAL_II_MEMBER_VAR_SPECIALIZATION_BUG
template <>
const double * const
FE_Nedelec<1,1>::Matrices::embedding[][GeometryInfo<1>::max_children_per_cell];
template <>
const unsigned int FE_Nedelec<1,1>::Matrices::n_embedding_matrices;
template <>
const double * const FE_Nedelec<1,1>::Matrices::constraint_matrices[];
template <>
const unsigned int FE_Nedelec<1,1>::Matrices::n_constraint_matrices;
template <>
const double * const
FE_Nedelec<2,2>::Matrices::embedding[][GeometryInfo<2>::max_children_per_cell];
template <>
const unsigned int FE_Nedelec<2,2>::Matrices::n_embedding_matrices;
template <>
const double * const FE_Nedelec<2,2>::Matrices::constraint_matrices[];
template <>
const unsigned int FE_Nedelec<2,2>::Matrices::n_constraint_matrices;
template <>
const double * const
FE_Nedelec<3,3>::Matrices::embedding[][GeometryInfo<3>::max_children_per_cell];
template <>
const unsigned int FE_Nedelec<3,3>::Matrices::n_embedding_matrices;
template <>
const double * const FE_Nedelec<3,3>::Matrices::constraint_matrices[];
template <>
const unsigned int FE_Nedelec<3,3>::Matrices::n_constraint_matrices;
#endif
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#endif
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