Intrepid
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Intrepid::Basis_HDIV_QUAD_I1_FEM< Scalar, ArrayScalar > Class Template Reference

Implementation of the default H(div)-compatible FEM basis of degree 1 on Quadrilateral cell. More...

#include <Intrepid_HDIV_QUAD_I1_FEM.hpp>

Inheritance diagram for Intrepid::Basis_HDIV_QUAD_I1_FEM< Scalar, ArrayScalar >:
Intrepid::Basis< Scalar, ArrayScalar >

Public Member Functions

 Basis_HDIV_QUAD_I1_FEM ()
 Constructor.
 
void getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const EOperator operatorType) const
 Evaluation of a FEM basis on a reference Quadrilateral cell. More...
 
void getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const ArrayScalar &cellVertices, const EOperator operatorType=OPERATOR_VALUE) const
 FVD basis evaluation: invocation of this method throws an exception.
 
- Public Member Functions inherited from Intrepid::Basis< Scalar, ArrayScalar >
virtual ~Basis ()
 Destructor.
 
virtual int getCardinality () const
 Returns cardinality of the basis. More...
 
virtual int getDegree () const
 Returns the degree of the basis. More...
 
virtual const shards::CellTopology getBaseCellTopology () const
 Returns the base cell topology for which the basis is defined. See Shards documentation http://trilinos.sandia.gov/packages/shards for definition of base cell topology. More...
 
virtual EBasis getBasisType () const
 Returns the basis type. More...
 
virtual ECoordinates getCoordinateSystem () const
 Returns the type of coordinate system for which the basis is defined. More...
 
virtual int getDofOrdinal (const int subcDim, const int subcOrd, const int subcDofOrd)
 DoF tag to ordinal lookup. More...
 
virtual const std::vector
< std::vector< std::vector
< int > > > & 
getDofOrdinalData ()
 DoF tag to ordinal data structure.
 
virtual const std::vector< int > & getDofTag (const int dofOrd)
 DoF ordinal to DoF tag lookup. More...
 
virtual const std::vector
< std::vector< int > > & 
getAllDofTags ()
 Retrieves all DoF tags. More...
 

Private Member Functions

void initializeTags ()
 Initializes tagToOrdinal_ and ordinalToTag_ lookup arrays.
 

Additional Inherited Members

- Protected Attributes inherited from Intrepid::Basis< Scalar, ArrayScalar >
int basisCardinality_
 Cardinality of the basis, i.e., the number of basis functions/degrees-of-freedom.
 
int basisDegree_
 Degree of the largest complete polynomial space that can be represented by the basis.
 
shards::CellTopology basisCellTopology_
 Base topology of the cells for which the basis is defined. See the Shards package http://trilinos.sandia.gov/packages/shards for definition of base cell topology.
 
EBasis basisType_
 Type of the basis.
 
ECoordinates basisCoordinates_
 The coordinate system for which the basis is defined.
 
bool basisTagsAreSet_
 "true" if tagToOrdinal_ and ordinalToTag_ have been initialized
 
std::vector< std::vector< int > > ordinalToTag_
 DoF ordinal to tag lookup table. More...
 
std::vector< std::vector
< std::vector< int > > > 
tagToOrdinal_
 DoF tag to ordinal lookup table. More...
 

Detailed Description

template<class Scalar, class ArrayScalar>
class Intrepid::Basis_HDIV_QUAD_I1_FEM< Scalar, ArrayScalar >

Implementation of the default H(div)-compatible FEM basis of degree 1 on Quadrilateral cell.

      Implements Raviart-Thomas basis of degree 1 on the reference Quadrilateral cell. The basis has
      cardinality 4 and spans a INCOMPLETE bi-linear polynomial space. Basis functions are dual 
      to a unisolvent set of degrees-of-freedom (DoF) defined and enumerated as follows:
===================================================================================================
|         |           degree-of-freedom-tag table                    |                            |
|   DoF   |----------------------------------------------------------|       DoF definition       |
| ordinal |  subc dim    | subc ordinal | subc DoF ord |subc num DoF |                            |
|=========|==============|==============|==============|=============|============================|
|    0    |       1      |       0      |       0      |      1      |   L_0(u) = (u.n)(0,-1)     |
|---------|--------------|--------------|--------------|-------------|----------------------------|
|    1    |       1      |       1      |       0      |      1      |   L_1(u) = (u.n)(1,0)      |
|---------|--------------|--------------|--------------|-------------|----------------------------|
|    2    |       1      |       2      |       0      |      1      |   L_2(u) = (u.n)(0,1)      |
|---------|--------------|--------------|--------------|-------------|----------------------------|
|    3    |       1      |       3      |       0      |      1      |   L_3(u) = (u.n)(-1,0)     |
|=========|==============|==============|==============|=============|============================|
|   MAX   |  maxScDim=2  |  maxScOrd=5  |  maxDfOrd=0  |      -      |                            |
|=========|==============|==============|==============|=============|============================|
Remarks
  • In the DOF functional ${\bf n}=(t_2,-t_1)$ where ${\bf t}=(t_1,t_2)$ is the side (edge) tangent, i.e., the choice of normal direction is such that the pair $({\bf n},{\bf t})$ is positively oriented.
  • Direction of side tangents is determined by the vertex order of the sides in the cell topology and runs from side vertex 0 to side vertex 1, whereas their length is set equal to the side length. For example, side 1 of all Quadrilateral reference cells has vertex order {1,2}, i.e., its tangent runs from vertex 1 of the reference Quadrilateral to vertex 2 of that cell. On the reference Quadrilateral the coordinates of these vertices are (1,-1) and (1,1), respectively. Therefore, the tangent to side 1 is (1,1)-(1,-1) = (0,2) and the normal to that side is (2,0). Because its length already equals side length, no further rescaling of the side tangent is needed.
  • The length of the side normal equals the length of the side. As a result, the DoF functional is the value of the normal component of a vector field at the side center times the side length. The resulting basis is equivalent to a basis defined by using the side flux as a DoF functional. Note that all sides of the reference Quadrilateral<> cells have length 2.

Definition at line 103 of file Intrepid_HDIV_QUAD_I1_FEM.hpp.

Member Function Documentation

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HDIV_QUAD_I1_FEM< Scalar, ArrayScalar >::getValues ( ArrayScalar &  outputValues,
const ArrayScalar &  inputPoints,
const EOperator  operatorType 
) const
virtual

Evaluation of a FEM basis on a reference Quadrilateral cell.

    Returns values of <var>operatorType</var> acting on FEM basis functions for a set of
    points in the <strong>reference Quadrilateral</strong> cell. For rank and dimensions of
    I/O array arguments see Section \ref basis_md_array_sec.
Parameters
outputValues[out] - rank-3 or 4 array with the computed basis values
inputPoints[in] - rank-2 array with dimensions (P,D) containing reference points
operatorType[in] - operator applied to basis functions

Implements Intrepid::Basis< Scalar, ArrayScalar >.

Definition at line 93 of file Intrepid_HDIV_QUAD_I1_FEMDef.hpp.


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