  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.
