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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.

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This isn't exactly what you asked, but maybe this paper is helpful. – Christian Clason Nov 28 '12 at 18:12
Not bad, not bad at all :) Thanks! – Johntra Volta Nov 28 '12 at 18:18
@ChristianClason - That's an answer, as far as I'm concerned, feel free to move it out of comments, and perhaps paste the abstract in to your answer. – Aron Ahmadia Nov 28 '12 at 22:23

The paper [1] gives an explicit construction of the Bernstein form of a set of orthogonal polynomials on simplices based on Legendre polynomials.

[1] Farouki, R.T., Goodman, T.N.T and Sauer, T: Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains, Computer Aided Geometric Design 20 (2003), 209-230, DOI: 10.1016/S0167-8396(03)00025-6

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I think Rob Kirby told me once that he had written something on using Bernstein polynomials for FEM. Take a look at his web site at Texas Tech (or now at Baylor).

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Indeed, there are two recent papers in Numerische Mathematik: 121(2): 261 -- 279 (2012) and 117(4): 631 -- 652 (2011). – Christian Clason Nov 29 '12 at 7:55

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