Discrete maximum principles for nonlinear elliptic finite element problems on Riemannian manifolds with boundary

TL;DR

This paper establishes discrete maximum principles for nonlinear elliptic finite element problems on Riemannian manifolds, extending classical maximum principles to complex geometric settings with practical applications.

Contribution

It introduces novel discrete maximum principles for nonlinear surface finite element problems on Riemannian manifolds, broadening the theoretical framework for elliptic PDEs on curved surfaces.

Findings

DMPs are proven for nonlinear surface finite element problems on Riemannian manifolds.

The results are supported by various real-life examples demonstrating applicability.

The work extends classical maximum principles to nonlinear geometric PDEs on manifolds.

Abstract

The maximum principle forms an important qualitative property of second order elliptic equations, therefore its discrete analogues, the so-called discrete maximum principles (DMPs) have drawn much attention. In this paper DMPs are established for nonlinear surface finite element problems on Riemannian manifolds, corresponding to the classical pointwise maximum principles on surfaces in the spirit of Pucci et al. Various real-life examples illustrate the scope of the results.