Basis function/shape function of P2 triangular & tetrahedral element

General Discussion
1

I’m doing a postprocessing work which needs to calculate the Gauss integration points’ Cartesian coordinates. One way to implement this is substituting the Gauss coordinates into the basis functions or shape functions, and then multiplying the basis function vector with the element nodal coordinates vector.

Here, my question is about the basis functions used in FreeFem++. I found it is a bit different from that in the monograph of “The Finite Element Method Its Basis and Fundamentals” by Zienkiewicz, Taylor & Govindjee. According to the source code

FreeFem-sources/src/femlib/P012_2d.cpp at master · FreeFem/FreeFem-sources · GitHub

FreeFem-sources/src/femlib/P012_3d.cpp at master · FreeFem/FreeFem-sources · GitHub

where the nvedge array can be found in

FreeFem-sources/src/femlib/Mesh2dn.cpp at master · FreeFem/FreeFem-sources · GitHub

FreeFem-sources/src/femlib/Mesh3dn.cpp at master · FreeFem/FreeFem-sources · GitHub

The basis functions for P2 triangular element is

// P2 Lagrange shape functions
[l0 * (2 * l0 - 1), 
 l1 * (2 * l1 - 1), 
 l2 * (2 * l2 - 1), 
 4 * l1 * l2, 
 4 * l2 * l0, 
 4 * l0 * l1];

and for P2 tetrahedral element is

// P2 Lagrange shape functions
[l0 * (2 * l0 - 1),
 l1 * (2 * l1 - 1),
 l2 * (2 * l2 - 1),
 l3 * (2 * l3 - 1),
 4 * l0 * l1,
 4 * l0 * l2,
 4 * l0 * l3,
 4 * l1 * l2,
 4 * l1 * l3,
 4 * l2 * l3];

Are the above results correct?