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

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

Public Member Functions

 FE_DGPMonomial (const unsigned int p)
virtual std::string get_name () const
virtual void get_interpolation_matrix (const FiniteElement< dim > &source, FullMatrix< double > &matrix) const
virtual void get_face_interpolation_matrix (const FiniteElement< dim > &source, FullMatrix< double > &matrix) const
virtual void get_subface_interpolation_matrix (const FiniteElement< dim > &source, const unsigned int subface, FullMatrix< double > &matrix) const
virtual bool has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const
virtual unsigned int memory_consumption () const
Functions to support hp



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
virtual
FiniteElementDomination::Domination 
compare_for_face_domination (const FiniteElement< dim > &fe_other) const

Protected Member Functions

virtual FiniteElement< dim > * clone () const

Private Member Functions

void initialize_embedding ()
void initialize_restriction ()

Static Private Member Functions

static std::vector< unsigned intget_dpo_vector (const unsigned int degree)

Detailed Description

template<int dim>
class FE_DGPMonomial< dim >

Discontinuous finite elements based on monomials.

This finite element implements complete polynomial spaces, that is, dim-dimensional polynomials of degree p. For example, in 2d the element FE_DGP(1) would represent the span of the functions $\{1,\hat x,\hat y\}$, which is in contrast to the element FE_DGQ(1) that is formed by the span of $\{1,\hat x,\hat y,\hat x\hat y\}$. Since the DGP space has only three unknowns for each quadrilateral, it is immediately clear that this element can not be continuous.

The basis functions for this element are chosen to be the monomials listed above. Note that this is the main difference to the FE_DGP class that uses a set of polynomials of complete degree p that form a Legendre basis on the unit square. Thus, there, the mass matrix is diagonal, if the grid cells are parallelograms. The basis here does not have this property; however, it is simpler to compute. On the other hand, this element has the additional disadvantage that the local cell matrices usually have a worse condition number than the ones originating from the FE_DGP element.

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

Transformation properties

It is worth noting that under a (bi-, tri-)linear mapping, the space described by this element does not contain $P(k)$, even if we use a basis of polynomials of degree $k$. Consequently, for example, on meshes with non-affine cells, a linear function can not be exactly represented by elements of type FE_DGP(1) or FE_DGPMonomial(1).

This can be understood by the following 2-d example: consider the cell with vertices at $(0,0),(1,0),(0,1),(s,s)$:

dgp_doesnt_contain_p.png

For this cell, a bilinear transformation $F$ produces the relations $x=\hat x+\hat x\hat y$ and $y=\hat y+\hat x\hat y$ that correlate reference coordinates $\hat x,\hat y$ and coordinates in real space $x,y$. Under this mapping, the constant function is clearly mapped onto itself, but the two other shape functions of the $P_1$ space, namely $\phi_1(\hat x,\hat y)=\hat x$ and $\phi_2(\hat x,\hat y)=\hat y$ are mapped onto $\phi_1(x,y)=\frac{x-t}{t(s-1)},\phi_2(x,y)=t$ where $t=\frac{y}{s-x+sx+y-sy}$.

For the simple case that $s=1$, i.e. if the real cell is the unit square, the expressions can be simplified to $t=y$ and $\phi_1(x,y)=x,\phi_2(x,y)=y$. However, for all other cases, the functions $\phi_1(x,y),\phi_2(x,y)$ are not linear any more, and neither is any linear combincation of them. Consequently, the linear functions are not within the range of the mapped $P_1$ polynomials.

Author:
Ralf Hartmann, 2004

Constructor & Destructor Documentation

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

Constructor for the polynomial space of degree p.


Member Function Documentation

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

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

Implements FiniteElement< dim, dim >.

template<int dim>
virtual std::vector<std::pair<unsigned int, unsigned int> > FE_DGPMonomial< dim >::hp_vertex_dof_identities ( const FiniteElement< dim > &  fe_other  )  const [virtual]

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 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, and returns a list of pairs of global dof indices in identities. 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.

template<int dim>
virtual std::vector<std::pair<unsigned int, unsigned int> > FE_DGPMonomial< dim >::hp_line_dof_identities ( const FiniteElement< dim > &  fe_other  )  const [virtual]

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.

template<int dim>
virtual std::vector<std::pair<unsigned int, unsigned int> > FE_DGPMonomial< dim >::hp_quad_dof_identities ( const FiniteElement< dim > &  fe_other  )  const [virtual]

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.

template<int dim>
virtual bool FE_DGPMonomial< dim >::hp_constraints_are_implemented (  )  const [virtual]

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_DGPMonomial class the result is always true (independent of the degree of the element), as it has no hanging nodes (being a discontinuous element).

Reimplemented from FiniteElement< dim, dim >.

template<int dim>
virtual FiniteElementDomination::Domination FE_DGPMonomial< dim >::compare_for_face_domination ( const FiniteElement< dim > &  fe_other  )  const [virtual]

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 hp paper.

template<int dim>
virtual void FE_DGPMonomial< dim >::get_interpolation_matrix ( const FiniteElement< dim > &  source,
FullMatrix< double > &  matrix 
) const [virtual]

Return the matrix interpolating from the given finite element to the present one. The size of the matrix is then dofs_per_cell times source.dofs_per_cell.

These matrices are only available if the source element is also a FE_Q element. Otherwise, an exception of type FiniteElement<dim>::ExcInterpolationNotImplemented is thrown.

template<int dim>
virtual void FE_DGPMonomial< dim >::get_face_interpolation_matrix ( const FiniteElement< dim > &  source,
FullMatrix< double > &  matrix 
) const [virtual]

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 dofs_per_face times source.dofs_per_face.

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.

template<int dim>
virtual void FE_DGPMonomial< dim >::get_subface_interpolation_matrix ( const FiniteElement< dim > &  source,
const unsigned int  subface,
FullMatrix< double > &  matrix 
) const [virtual]

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 dofs_per_face times source.dofs_per_face.

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.

template<int dim>
virtual bool FE_DGPMonomial< 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_DGPMonomial< 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>
virtual FiniteElement<dim>* FE_DGPMonomial< 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_DGPMonomial< 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>
void FE_DGPMonomial< dim >::initialize_embedding (  )  [private]

Initialize the embedding matrices. Called from the constructor.

template<int dim>
void FE_DGPMonomial< dim >::initialize_restriction (  )  [private]

Initialize the restriction matrices. Called from the constructor.


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

deal.II documentation generated on Mon Nov 23 22:57:40 2009 by doxygen 1.6.1