Intrepid
Public Member Functions | Private Member Functions
Intrepid::Basis_HDIV_TET_I1_FEM< Scalar, ArrayScalar > Class Template Reference

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

#include <Intrepid_HDIV_TET_I1_FEM.hpp>

Inheritance diagram for Intrepid::Basis_HDIV_TET_I1_FEM< Scalar, ArrayScalar >:
Intrepid::Basis< Scalar, ArrayScalar >

List of all members.

Public Member Functions

 Basis_HDIV_TET_I1_FEM ()
 Constructor.
void getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const EOperator operatorType) const
 Evaluation of a FEM basis on a reference Tetrahedron 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_TET_I1_FEM< Scalar, ArrayScalar >

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

Implements Raviart-Thomas basis of degree 1 on the reference Tetrahedron cell. The basis has cardinality 4 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    |       2      |       0      |       0      |      1      | L_0(u) = (u.n)(1/3,0,1/3)        |
  |---------|--------------|--------------|--------------|-------------|----------------------------------|
  |    1    |       2      |       1      |       0      |      1      | L_1(u) = (u.n)(1/3,1/3,1/3)      |
  |---------|--------------|--------------|--------------|-------------|----------------------------------|
  |    2    |       2      |       2      |       0      |      1      | L_2(u) = (u.n)(0,1/3,1/3)        |
  |---------|--------------|--------------|--------------|-------------|----------------------------------|
  |    3    |       2      |       3      |       0      |      1      | L_3(u) = (u.n)(1/3,1/3,0)        |
  |=========|==============|==============|==============|=============|==================================|
  |   MAX   |  maxScDim=2  |  maxScOrd=3  |  maxDfOrd=0  |      -      |                                  |
  |=========|==============|==============|==============|=============|==================================|
  
Remarks:
  • In the DoF functional ${\bf n}$ is a face normal. Direction of face normals is determined by the right-hand rule applied to faces oriented by their vertex order in the cell topology, from face vertex 0 to last face vertex, whereas their length is set equal to face area (see http://mathworld.wolfram.com/Right-HandRule.html for definition of right-hand rule). For example, face 1 of all Tetrahedron cells has vertex order {1,2,3} and its right-hand rule normal can be computed, e.g., by the vector product of edge tangents to edges {1,2} and {2,3}. On the reference Tetrahedron the coordinates of face 1 vertices are (1,0,0), (0,1,0), and (0,0,1), the edge tangents are (-1,1,0) and (0,-1,1) and the face normal direction is (-1,1,0) X (0,-1,1) = (1,1,1). Length of this raw face normal is twice the face area of face 1 and so the final face normal to face 1 is obtained by scaling the raw normal by 1/2: (1/2,1/2,1/2).
  • The length of the face normal equals the face area. As a result, the DoF functional is the value of the normal component of a vector field at the face center times the face area. The resulting basis is equivalent to a basis defined by using the face flux as a DoF functional. Note that faces 0, 2, and 3 of reference Tetrahedron<> cells have area 1/2 and face 1 has area Sqrt(3)/2.

Definition at line 89 of file Intrepid_HDIV_TET_I1_FEM.hpp.


Member Function Documentation

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HDIV_TET_I1_FEM< Scalar, ArrayScalar >::getValues ( ArrayScalar &  outputValues,
const ArrayScalar &  inputPoints,
const EOperator  operatorType 
) const [virtual]

Evaluation of a FEM basis on a reference Tetrahedron cell.

Returns values of operatorType acting on FEM basis functions for a set of points in the reference Tetrahedron 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 80 of file Intrepid_HDIV_TET_I1_FEMDef.hpp.


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