Discrete maximum-minimum principle for a linearly implicit scheme for nonlinear parabolic FEM problems under weakened time restrictions

Abstract

In this paper, we extend our earlier results in Faragó, I., Karátson, J. and Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems. IMA J. Numer. Anal., 32, 1541–1573) on the discrete maximum-minimum principle (DMP) for nonlinear parabolic systems of PDEs. We propose a linearly implicit scheme, where only linear problems have to be solved on the time layers. We obtain a DMP without the restrictive condition $\varDelta t\le O(h^{2})$. We show that we only need the lower bound $\varDelta t\ge O(h^{2})$, further, depending on the Lipschitz condition of the given nonlinearity, the upper bound is just $\varDelta t\le C$ (for globally Lipschitz) or $\varDelta t\le O(h^{\gamma })$ (for locally Lipschitz) for some constant $C>0$ arising from the PDE, or some $\gamma < 2$, respectively. In most situations in practical models, the latter condition becomes $\varDelta t \le O( h^{2/3} )$ in 2D and $\varDelta t \le O( h )$ in 3D. Various real-life examples are also presented where the results can be applied to obtain physically relevant numerical solutions.