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

Discrete comparison principles for quasilinear elliptic PDE