Discrete comparison principles for quasilinear elliptic PDE

TL;DR

This paper develops discrete comparison principles for quasilinear elliptic PDEs, ensuring solution uniqueness under certain mesh conditions, with implications for finite element methods in nonlinear PDE analysis.

Contribution

It introduces new local and global discretization conditions for comparison principles in nonlinear elliptic PDEs, including nonmonotone Leray-Lions problems.

Findings

Comparison principles ensure uniqueness of finite element solutions.

Mesh size control depends on solution variance, not global parameters.

Simpler semilinear case allows sharper mesh conditions.

Abstract

Comparison principles are developed for discrete quasilinear elliptic partial differential equations. We consider the analysis of a class of nonmonotone Leray-Lions problems featuring both nonlinear solution and gradient dependence in the principal coefficient, and a solution dependent lower-order term. Sufficient local and global conditions on the discretization are found for piecewise linear finite element solutions to satisfy a comparison principle, which implies uniqueness of the solution. For problems without a lower-order term, our analysis shows the meshsize is only required to be locally controlled, based on the variance of the computed solution over each element. We include a discussion of the simpler semilinear case where a linear algebra argument allows a sharper mesh condition for the lower order term.