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.


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.


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