FE_Q_Hierarchical< dim > Class Template Reference
[Finite element space descriptions]

Inheritance diagram for FE_Q_Hierarchical< dim >:
Inheritance graph
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List of all members.

Public Member Functions

 FE_Q_Hierarchical (const unsigned int p)
virtual std::string get_name () const
virtual bool has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const
virtual unsigned int memory_consumption () const
std::vector< unsigned intget_embedding_dofs (const unsigned int sub_degree) const
template<>
void initialize_unit_face_support_points ()
template<>
bool has_support_on_face (const unsigned int, const unsigned int) const
template<>
std::vector< unsigned intface_fe_q_hierarchical_to_hierarchic_numbering (const unsigned int)

Protected Member Functions

virtual FiniteElement< dim > * clone () const

Private Member Functions

void build_dofs_cell (std::vector< FullMatrix< double > > &dofs_cell, std::vector< FullMatrix< double > > &dofs_subcell) const
void initialize_constraints (const std::vector< FullMatrix< double > > &dofs_subcell)
void initialize_embedding_and_restriction (const std::vector< FullMatrix< double > > &dofs_cell, const std::vector< FullMatrix< double > > &dofs_subcell)
void initialize_unit_support_points ()
void initialize_unit_face_support_points ()

Static Private Member Functions

static std::vector< unsigned intget_dpo_vector (const unsigned int degree)
static std::vector< unsigned inthierarchic_to_fe_q_hierarchical_numbering (const FiniteElementData< dim > &fe)
static std::vector< unsigned intface_fe_q_hierarchical_to_hierarchic_numbering (const unsigned int degree)

Private Attributes

const std::vector< unsigned intface_renumber

Friends

class FE_Q_Hierarchical

Detailed Description

template<int dim>
class FE_Q_Hierarchical< dim >

Implementation of Hierarchical finite elements Qp that yield the finite element space of continuous, piecewise polynomials of degree p. This class is realized using tensor product polynomials based on a hierarchical basis Hierarchical of the interval [0,1] which is suitable for building an hp tensor product finite element, if we assume that each element has a single degree.

There are not many differences between FE_Q_Hierarchical and FE_Q, except that we add a function embedding_dofs that takes a given integer q, between 1 and p, and returns the numbering of basis functions of the element of order q in basis of order p. This function is useful if one wants to make calculations using the hierarchical nature of these shape functions.

The unit support points now are reduced to 0, 1, and 0.5 in one dimension, and tensor products in higher dimensions. Thus, various interpolation functions will only work correctly for the linear case. Future work will involve writing projection--interpolation operators that can interpolate onto the higher order bubble functions.

The various constraint, prolongation, and restriction matrices are now available in all dimensions for all degrees p, currently up to order 19.

The constructor of this class takes the degree p of this finite element.

This class is not implemented for the codimension one case (spacedim != dim).

Implementation

The constructor creates a TensorProductPolynomials object that includes the tensor product of Hierarchical polynomials of degree p. This TensorProductPolynomials object provides all values and derivatives of the shape functions.

Numbering of the degrees of freedom (DoFs)

The original ordering of the shape functions represented by the TensorProductPolynomials is a tensor product numbering. However, the shape functions on a cell are renumbered beginning with the shape functions whose support points are at the vertices, then on the line, on the quads, and finally (for 3d) on the hexes. To be explicit, these numberings are listed in the following:

Q1 elements

Q2 elements

Q3 elements

Q4 elements

Author:
Brian Carnes, 2002, Ralf Hartmann 2004, 2005

Constructor & Destructor Documentation

template<int dim>
FE_Q_Hierarchical< dim >::FE_Q_Hierarchical ( const unsigned int  p  ) 

Constructor for tensor product polynomials of degree p.


Member Function Documentation

template<int dim>
virtual std::string FE_Q_Hierarchical< dim >::get_name (  )  const [virtual]

Return a string that uniquely identifies a finite element. This class returns FE_Q_Hierarchical<dim>(degree), with dim and degree replaced by appropriate values.

Implements FiniteElement< dim, dim >.

template<int dim>
virtual bool FE_Q_Hierarchical< dim >::has_support_on_face ( const unsigned int  shape_index,
const unsigned int  face_index 
) const [virtual]

Check for non-zero values on a face.

This function returns true, if the shape function shape_index has non-zero values on the face face_index.

Implementation of the interface in FiniteElement

Reimplemented from FiniteElement< dim, dim >.

template<int dim>
virtual unsigned int FE_Q_Hierarchical< dim >::memory_consumption (  )  const [virtual]

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.

Reimplemented from FiniteElement< dim, dim >.

template<int dim>
std::vector<unsigned int> FE_Q_Hierarchical< dim >::get_embedding_dofs ( const unsigned int  sub_degree  )  const

For a finite element of degree sub_degree < degree, we return a vector which maps the numbering on an FE of degree sub_degree into the numbering on this element.

template<int dim>
virtual FiniteElement<dim>* FE_Q_Hierarchical< dim >::clone (  )  const [protected, virtual]

clone function instead of a copy constructor.

This function is needed by the constructors of FESystem.

Implements FiniteElement< dim, dim >.

template<int dim>
static std::vector<unsigned int> FE_Q_Hierarchical< dim >::get_dpo_vector ( const unsigned int  degree  )  [static, private]

Only for internal use. Its full name is get_dofs_per_object_vector function and it creates the dofs_per_object vector that is needed within the constructor to be passed to the constructor of FiniteElementData.

template<int dim>
static std::vector<unsigned int> FE_Q_Hierarchical< dim >::hierarchic_to_fe_q_hierarchical_numbering ( const FiniteElementData< dim > &  fe  )  [static, private]

The numbering of the degrees of freedom in continous finite elements is hierarchic, i.e. in such a way that we first number the vertex dofs, in the order of the vertices as defined by the triangulation, then the line dofs in the order and respecting the direction of the lines, then the dofs on quads, etc.

The dofs associated with 1d hierarchical polynomials are ordered with the vertices first ($phi_0(x)=1-x$ and $phi_1(x)=x$) and then the line dofs (the higher degree polynomials). The 2d and 3d hierarchical polynomials originate from the 1d hierarchical polynomials by tensor product. In the following, the resulting numbering of dofs will be denoted by `fe_q_hierarchical numbering`.

This function constructs a table which fe_q_hierarchical index each degree of freedom in the hierarchic numbering would have.

This function is anologous to the FETools::hierarchical_to_lexicographic_numbering() function. However, in contrast to the fe_q_hierarchical numbering defined above, the lexicographic numbering originates from the tensor products of consecutive numbered dofs (like for LagrangeEquidistant).

It is assumed that the size of the output argument already matches the correct size, which is equal to the number of degrees of freedom in the finite element.

template<int dim>
static std::vector<unsigned int> FE_Q_Hierarchical< dim >::face_fe_q_hierarchical_to_hierarchic_numbering ( const unsigned int  degree  )  [static, private]

This is an analogon to the previous function, but working on faces.

template<int dim>
void FE_Q_Hierarchical< dim >::build_dofs_cell ( std::vector< FullMatrix< double > > &  dofs_cell,
std::vector< FullMatrix< double > > &  dofs_subcell 
) const [private]

Initialize two auxiliary fields that will be used in setting up the various matrices in the constructor.

template<int dim>
void FE_Q_Hierarchical< dim >::initialize_constraints ( const std::vector< FullMatrix< double > > &  dofs_subcell  )  [private]

Initialize the hanging node constraints matrices. Called from the constructor.

template<int dim>
void FE_Q_Hierarchical< dim >::initialize_embedding_and_restriction ( const std::vector< FullMatrix< double > > &  dofs_cell,
const std::vector< FullMatrix< double > > &  dofs_subcell 
) [private]

Initialize the embedding matrices. Called from the constructor.

template<int dim>
void FE_Q_Hierarchical< dim >::initialize_unit_support_points (  )  [private]

Initialize the unit_support_points field of the FiniteElement class. Called from the constructor.

template<int dim>
void FE_Q_Hierarchical< dim >::initialize_unit_face_support_points (  )  [private]

Initialize the unit_face_support_points field of the FiniteElement class. Called from the constructor.

template<>
void FE_Q_Hierarchical< 1 >::initialize_unit_face_support_points (  )  [inline]
template<>
bool FE_Q_Hierarchical< 1 >::has_support_on_face ( const unsigned int  shape_index,
const unsigned int  face_index 
) const [inline, virtual]

Check for non-zero values on a face in order to optimize out matrix elements.

This function returns true, if the shape function shape_index has non-zero values on the face face_index.

A default implementation is provided in this basis class which always returns true. This is the safe way to go.

Reimplemented from FiniteElement< dim, dim >.

template<>
std::vector< unsigned int > FE_Q_Hierarchical< 1 >::face_fe_q_hierarchical_to_hierarchic_numbering ( const unsigned  int  )  [inline]

Friends And Related Function Documentation

template<int dim>
friend class FE_Q_Hierarchical [friend]

Allow access from other dimensions. We need this since we want to call the functions get_dpo_vector and lexicographic_to_hierarchic_numbering for the faces of the finite element of dimension dim+1.


Member Data Documentation

template<int dim>
const std::vector<unsigned int> FE_Q_Hierarchical< dim >::face_renumber [private]

Mapping from lexicographic to shape function numbering on first face.


The documentation for this class was generated from the following file:

deal.II documentation generated on Mon Nov 23 22:57:42 2009 by doxygen 1.6.1