Preprint 637-1999

Discrete versions of Gronwall's lemma and their application to the numerical analysis of parabolic problems

Author(s) : Etienne Emmrich

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 637-1999

MSC 2000

26D10 Inequalities involving derivatives and differential and integral operators

26D15 Inequalities for sums, series and integrals

39A12 Discrete version of topics in analysis

65M12 Stability and convergence of numerical methods

Abstract :
The well-known Gronwall lemma often serves as a major tool for the analysis of time-dependent problems via energy methods. However, there is the need for a similar tool when considering temporal discretizations of evolutionary problems. With the paper in hand, the author wishes to give an overview of some discrete versions of the Gronwall lemma and presents a unified approach. In particular, new discrete versions of the lemma in its differential form and their application, showing decay behaviour for discretized parabolic problems, are studied. These versions give effective tools for the stability and error analysis of the temporal semi-discretization of parabolic problems covering non-homogeneous problems as well as approximations with variable step sizes.

Keywords : Gronwall-type inequality, differential inequality, difference inequality, parabolic differential equation, temporal discretization, Euler method