Discrete maximum principles for nonlinear parabolic PDE systems

Abstract

Discrete maximum principles are established for finite element approximations of nonlinear parabolic PDE systems with mixed boundary and interface conditions. The results are based on an algebraic discrete maximum principle for suitable ODE systems.

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

“…This also shows that if especially µ 0 = 0 (i.e. the coupling is diagonally dominant as in [8]) then no restriction remains on ∆t, i.e. the new result improves the previous one in this respect, too.…”

“…Second, we only use implicit time discretization instead of a more general θ-method. We note, however, that the results of [8] using θ-methods were more restrictive for θ < 1 than for the implicit case θ = 1: for the latter it sufficed to assume the lower estimate ∆t ≥ ch 2 , whereas for the former one needed the two-sided estimate ∆t = O(h 2 ). Moreover, in the present paper we do not even require ∆t ≥ ch 2 , instead, we assume ∆t ≤ 1/µ 0 , where −µ 0 is the lower bound of the sums of Jacobians.…”

“…In our recent paper [8], we proved the discrete analogue (discrete maximum principle, or DMP, in short) of the maximum principle for the case of finite element space discretizations for some nonlinear parabolic PDE systems. Besides standard general smoothness and growth conditions, we assumed cooperativity and diagonal dominance for the nonlinear coupling of the equations.…”

“…In fact, we consider a more special situation in some other respects as compared to the problem in [8]. First, we only include mixed boundary conditions but no interface conditions inside the domain.…”

“…We first summarize the problem and then its discretization, the latter built on [8]. Then we give some background on elliptic DMPs, and based on it, we derive the desired result for our parabolic system.…”