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.