Discrete nonnegativity for nonlinear cooperative parabolic PDE systems with non-monotone coupling

Abstract

Discrete nonnegativity principles are established for finite element approximations of nonlinear parabolic PDE systems with mixed boundary conditions. Previous results of the authors are extended such that diagonal dominance (or essentially monotonicity) of the nonlinear coupling can be relaxed, allowing to include much more general situations in suitable models.

A Numerical Method for SolvingTwo-Dimensional Nonlinear Parabolic ProblemsBased on a Preconditioning Operator

“…In order to improve the condition number, we can use preconditioning operator in iterative methods such as Sobolev space gradient method, conjugate gradient method, Newton-like methods and so on. In most cases, finding a suitable preconditioner is the fundamental part of these iterative methods (see [1,2,3,5,10,12]). Farago and Karatson in [4] provide an overview of existing preconditioned iterative methods, especially on nonlinear elliptic problems.…”