# Uniqueness of discrete solutions of nonmonotone PDEs without a globally fine mesh condition

## TL;DR

This paper proves the uniqueness of finite element solutions for certain nonmonotone elliptic PDEs in 1D and 2D without needing a globally fine mesh, using local comparison theorems.

## Contribution

It introduces a local comparison theorem approach to establish solution uniqueness without a global mesh size restriction.

## Key findings

- Uniqueness proven for nonmonotone quasilinear elliptic PDEs in 1D and 2D.
- Local bounds on solution variation suffice for uniqueness.
- No globally small mesh condition required for finite element solutions.

## Abstract

Uniqueness of the finite element solution for nonmonotone quasilinear problems of elliptic type is established in one and two dimensions. In each case, we prove a comparison theorem based on locally bounding the variation of the discrete so- lution over each element. The uniqueness follows from this result, and does not require a globally small meshsize.