Discrete maximum principle for finite element parabolic models in higher dimensions

When we construct continuous and/or discrete mathematical models in order to describe a real-life problem, these models should have various qualitative properties, which typically arise from some basic principles of the modelled phenomena. In this paper we investigate this question for the numerical solution of initial-boundary problems for the parabolic problems in higher dimensions, with the first boundary condition, using the linear finite elements. We give the conditions for the geometry of the mesh and for the choice of the discretization parameters, i.e., for the step sizes under which the discrete qualitative properties hold. For the special regular uniform simplicial mesh we define the conditions for the discretization step-sizes.