A matrix analysis approach to discrete comparison principles for nonmonotone PDE

Abstract

We present 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, local conditions on the mesh are required 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.