A nonsmooth extension of the Brezzi-Rappaz-Raviart approximation theorem via metric regularity techniques and applications to nonlinear PDEs

TL;DR

This paper extends the Brezzi-Rappaz-Raviart theorem to nonlinear PDEs with Lipschitz regularity using metric regularity, enabling error estimates for finite element solutions of complex equations.

Contribution

It introduces a nonsmooth generalization of the approximation theorem leveraging metric regularity, broadening applicability to less regular nonlinearities.

Findings

Derived quasi-optimal error estimates for finite element methods.

Extended the theorem to nonlinearities with only Lipschitz regularity.

Applied the results to viscous Hamilton-Jacobi equations and mean field games.

Abstract

We generalize the Brezzi-Rappaz-Raviart approximation theorem, which allows to obtain existence and a priori error estimates for approximations of solutions to some nonlinear partial differential equations. Our contribution lies in the fact that we typically allow for nonlinearities having merely Lipschitz regularity, while previous results required some form of differentiability. This is achieved by making use of the theory of metrically regular mappings, developed in the context of variational analysis. We apply this generalization to derive quasi-optimal error estimates for finite element approximations to solutions of viscous Hamilton-Jacobi equations and second order mean field game systems.