Intrepid::Basis_HDIV_TRI_I1_FEM< Scalar, ArrayScalar > Class Template Reference

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

#include <Intrepid_HDIV_TRI_I1_FEM.hpp>

Inheritance diagram for Intrepid::Basis_HDIV_TRI_I1_FEM< Scalar, ArrayScalar >:
Intrepid::Basis< Scalar, ArrayScalar >

List of all members.

Public Member Functions

 Basis_HDIV_TRI_I1_FEM ()
 Constructor.
void getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const EOperator operatorType) const
 Evaluation of a FEM basis on a reference Triangle cell.
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.

Private Member Functions

void initializeTags ()
 Initializes tagToOrdinal_ and ordinalToTag_ lookup arrays.

Detailed Description

template<class Scalar, class ArrayScalar>
class Intrepid::Basis_HDIV_TRI_I1_FEM< Scalar, ArrayScalar >

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

Implements Raviart-Thomas basis of degree 1 on the reference Triangle cell. The basis has cardinality 3 and spans an INCOMPLETE 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)(1/2,0)            |
  |---------|--------------|--------------|--------------|-------------|----------------------------------|
  |    1    |       1      |       1      |       0      |      1      | L_1(u) = (u.n)(1/2,1/2)          |
  |---------|--------------|--------------|--------------|-------------|----------------------------------|
  |    2    |       1      |       2      |       0      |      1      | L_2(u) = (u.n)(0,1/2)            |
  |=========|==============|==============|==============|=============|==================================|
  |   MAX   |  maxScDim=1  |  maxScOrd=2  |  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 Triangle reference cells has vertex order {1,2}, i.e., its tangent runs from vertex 1 of the reference Triangle to vertex 2 of that cell. On the reference Triangle the coordinates of these vertices are (1,0) and (0,1), respectively. Therefore, the tangent to side 1 is (0,1)-(1,0) = (-1,1) and the normal to that side is (1,1). 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 sides 0 and 2 of reference Triangle<> cells have length 1 and side 1 has length Sqrt(2).

Definition at line 88 of file Intrepid_HDIV_TRI_I1_FEM.hpp.


Member Function Documentation

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HDIV_TRI_I1_FEM< Scalar, ArrayScalar >::getValues ( ArrayScalar &  outputValues,
const ArrayScalar &  inputPoints,
const EOperator  operatorType 
) const [inline, virtual]

Evaluation of a FEM basis on a reference Triangle cell.

Returns values of operatorType acting on FEM basis functions for a set of points in the reference Triangle cell. For rank and dimensions of I/O array arguments see Section MD array template arguments for basis methods.

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 82 of file Intrepid_HDIV_TRI_I1_FEMDef.hpp.

References Intrepid::Basis< Scalar, ArrayScalar >::getBaseCellTopology(), and Intrepid::Basis< Scalar, ArrayScalar >::getCardinality().


The documentation for this class was generated from the following files:
Generated on Mon Jan 31 09:55:52 2011 for Intrepid by  doxygen 1.6.3