I was wondering, before trying to do that myself, has anyone attempted to do orthonormalization of Bernstein polynomials using Gram-Schmidt?
I discussed this with several people and have been told that Bernstein polynomials don't make a good basis for FEM because they are not orthogonal.
I didn't use FEM, instead I made a (pseudospectral-like) collocation method formulation, and documented my attempts to solve elliptic problems in 2D domains in an arXiv article. I had exponential convergence with polynomial orders $n<20$. After that approximation become worse as $n$ was increased. One of the reasons may be non-orthogonality of Bernstein polynomial basis functions.
The code discussed is here.
My idea is to make a new orthogonal basis using Gram-Schmidt and try again.