# Analysis of mixed discontinuous Galerkin formulations for quasilinear elliptic problems

## TL;DR

This paper develops a unified framework for analyzing discontinuous Galerkin methods applied to nonlinear quasilinear elliptic problems, establishing stability, existence, uniqueness, and optimal error estimates.

## Contribution

It extends well-known DG schemes to nonlinear problems in a canonical way and proves their stability and optimal error bounds.

## Key findings

- Stability of DG solutions for nonlinear problems established
- Existence and uniqueness proven for monotone, Lipschitz cases
- Optimal error estimates in energy and L2 norms obtained

## Abstract

In this manuscript we present an approach to analyze the discontinuous Galerkin solution for general quasilinear elliptic problems. This approach is sufficiently general to extend most of the well-known discretization schemes, including BR1, BR2, SIPG and LDG, to nonlinear cases in a canonical way, and to establish the stability of their solution. Furthermore, in case of monotone and globally Lipschitz problems, we prove the existence and uniqueness of the approximated solution and the $h$-optimality of the error estimate in the energy norm as well as in the $L_2$ norm.