A matrix analysis approach to discrete comparison principles for nonmonotone PDE

TL;DR

This paper introduces a matrix analysis method to establish discrete comparison principles for nonmonotone quasilinear elliptic PDEs, providing sharper constants and relaxed mesh conditions compared to previous approaches.

Contribution

It presents a novel linear algebra approach to prove discrete comparison principles for nonmonotone PDEs, improving upon existing methods with sharper bounds and less restrictive mesh conditions.

Findings

Matrix analysis yields sharper constants.

Relaxed mesh conditions for comparison principle.

Establishes maximum principle for linearized problem.

Abstract

We consider a linear algebra approach to establishing a discrete comparison principle for a nonmonotone class of quasilinear elliptic partial differential equations. In the absence of a lower order term, we require local conditions on the mesh to establish the comparison principle and uniqueness of the piecewise linear finite element solution. We consider the assembled matrix corresponding to the linearized problem satisfied by the difference of two solutions to the nonlinear problem. Monotonicity of the assembled matrix establishes a maximum principle for the linear problem and a comparison principle for the nonlinear problem. The matrix analysis approach to the discrete comparison principle yields sharper constants and more relaxed mesh conditions than does the argument by contradiction used in previous work.