A priori error estimates for finite element discretization of semilinear elliptic equations with non-Lipschitz nonlinearities

TL;DR

This paper develops error estimates for finite element methods applied to semilinear elliptic equations with non-Lipschitz nonlinearities, including cases with non-Hölder continuity, supported by numerical examples.

Contribution

It provides the first a priori error estimates for direct finite element discretization of such equations without regularization.

Findings

Error estimates derived for various norms

Applicable to non-Lipschitz and non-Hölder continuous nonlinearities

Numerical examples confirm theoretical results

Abstract

In this paper we develop numerical analysis for finite element discretization of semilinear elliptic equations with potentially non-Lipschitz nonlinearites. The nonlinearity is essecially assumed to be continuous and monotonically decreasing including the case of non-Lipschitz or even non-H\"older continuous nonlinearities. For a direct discretization (without any regularization) with linear finite elements we derive error estimates with respect to various norms and illustrate these results with a numerical example.