Discrete comparison principles for quasilinear elliptic PDE
This work provides theoretical guarantees for finite element solutions of nonmonotone quasilinear problems, benefiting numerical analysts and engineers solving such PDEs.
The authors develop discrete comparison principles for quasilinear elliptic PDEs, identifying sufficient mesh conditions for piecewise linear finite element solutions to satisfy a comparison principle, which implies uniqueness. For problems without a lower-order term, the meshsize only needs local control based on solution variance.
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.