Sufficient conditions of the discrete maximum–minimum principle for parabolic problems on rectangular meshes

New numerical models for simulation of physical and chemical phenomena have to meet certain qualitative requirements, such as nonnegativity preservation, maximum–minimum principle, and maximum norm contractivity. For parabolic initial boundary value problems, these properties are generally guaranteed by certain geometrical conditions on the meshes used and by choosing the time-step according to some lower and upper bounds. The necessary and sufficient conditions of the qualitative properties and their relations have been already given. In this paper sufficient conditions are derived for the Galerkin finite element solution of a linear parabolic initial boundary value problem. We solve the problem on a 2D rectangular domain using bilinear basis function.