Analysis of nonlinear finite element methods

Question

I have been doing a lot of reading on the development of finite element methods and their analysis using, e.g., functional analysis. I am clear on the formulation of the weak form of a PDE and properties defined on the bilinear form.

However, for my research I am pursuing the solution of nonlinear PDEs using Discontinuous Galerkin (DG) methods. I would like to prove interesting properties about the PDEs subject to the specific DG schemes I will employ such as stability, consistency, well-posedness, etc. It seems to me that all these properties are proven using functional analysis techniques which rely heavily on the linearity of the bilinear form.

With this concern, my questions are: Am I understanding these methods correctly? Is there a straightforward path to prove, e.g., that my problem is well-posed? If not, is it meaningful to prove these properties about linearizations of my problem using a Newton's method, e.g.?

Any references that can address these concerns are much appreciated! Thanks!

For what it's worth, I am dealing with highly convective flows, so the convective term in Navier Stokes equations is the dominating nonlinear term.

Answer 1

When one learn about functional analysis methods for PDEs, usually starts from common theorems like the Riesz representation and the Lax-Milgram lemmas, which work quite good with linear PDEs.

When dealing with non-linear PDEs the story is far more complicated. There are some results from functional analysis that may come in hand, and usually these take the form of a fixed-point theorem. You can rely on the Banach theorem, the Schauder theorem or the Leray-Schauder theorem. These last two are generalizations of the Brower theorem.

Regarding your final questions, in the Navier-Stokes case the proof of well-posedness is all but straightforward. Rather, for sake of completeness, you can find an example here where the authors prove experimentally the evidence of multiple solutions obtained with a finite element solver converging up to machine zero, hence, without bothering functional analysis.