Discrete maximum principles for FE solutions of nonstationary diffusion-reaction problems with mixed boundary conditions

Abstract

In this paper we derive and discuss sufficient conditions that provide the validity of the discrete maximum principle for nonstationary diffusion-reaction problems with mixed boundary conditions solved by means of simplicial finite elements and the θ time discretization method. Theoretical analysis is supported by numerical experiments. AMS subject classifications: 65M60, 65M50, 35B50Keywords: nonstationary diffusion-reaction problem, maximum principle, mixed boundary conditions, linear finite elements, discrete maximum principle, simplicial partition, angle condition

Qualitative properties of discrete nonlinear parabolic operators

“…For linear parabolic problems the most extensively studied properties are the different maximum and minimum principles, the nonnegativity and nonpositivity preservation and the maximum norm contractivity. Such results can be found in the works [1,5,7,15,16,[20][21][22][24][25][26][27] and in the references therein. Not only the preservation of the qualitative properties are important themselves but also their relations.…”

Preserving nonnegativity of an affine finite element approximation for a convection–diffusion–reaction problem

“…In [10] they established sufficient conditions for the θ-method time discretization of the linear finite element approximation to the time dependent diffusion-reaction problem to satisfy a discrete maximum principle. Discrete maximum principles for a nonlinear parabolic equation and a nonlinear system of equations were analyzed in [9,11], respectively.…”

Space dependent models for studying the spread of some diseases

“…The most frequently investigated properties are the ∧ maximum-minimum principles, the non-negativity and non-positivity preservation properties and the maximum norm contractivity properties, e.g. [1][2][3][4][5][6]. Monotonicity and sign-stability properties were also discussed [7,8].…”

“…Let L be a fixed real value. If the implicit Euler finite difference numerical solution of problem (1) produced by(2) exists, then the above constructed B G sets…”