I was wondering if it is necessary in FEM to reduce the derivatives from 2 to 1 through integration by parts or the Green-Gauss theorem. If quadratic shape functions are used that have defined 2nd derivatives, could they not be used directly rather than reducing the order of derivative and then substituting? Any references would also be appreciated.

The purpose of performing integration by parts is not so much to reduce the order of derivatives but to put the problem in a form where the Lax-Milgram lemma can be used. See

S. Larson, V. Thomee. Partial Differential Equations with Numerical Methods. Springer, Berlin, Heidelberg, New York, 2003

The use of 'integration by parts', especially in the momentum equations, allows the inclusion of boundary-conditions in a direct way.

diaw...

It also simplifies mapping elemental regions onto a reference element (allowing operations to be carried out at the element level). Having to transform derivetive terms becomes ugly and costly pretty fast as the order of the derivatives increase.

This to me seems more important than making it possible to prove something.

Sharif

Thanks for all of the input. The reference was checked out of my library, but I am researching the Lax-Milgram lemma. I am also looking into transforming the derivatives from global terms to element terms. Basically, I am going to be looking at systems in which the boundary is open. I am hoping to find a way to apply the FEM.