What the best programming language to develop engineering software based on finite element method?

C will be the natural choice as wide open-source code-base is readily available.

Legacy solver code-base will be in Fortran and it can successfully and automatically converted to "C".

When starting to write new or improve the existing code-base try these;

1) Use C++ for better OO.

2) Use Python as it is much easier and equally good is performance (cpython).

3) Preferably look at server-side technology (node-js for eg) so that you target web-complaint software right from the inception.

-YD

finite element method=Mathematic analysis, So you have MATLAB