Strong uniform stability and exact discretizations of a model singular perturbation problem and its finite-difference approximations

摘要

This paper is concerned with stability with respect to norms like ⦀u⦀ϵ ≔ ∥u∥∞ + ϵ∥u'∥∞, uniformly in ϵ, for the operator Lϵu ≔ -ϵu + a(x)u' + b(x)u (with a(x)⩾ ā 0) and compact finite-difference discretizations of it. Strong uniform stability of the continuous problem is established for all ϵ 0 sufficiently small. Strong stability of the discrete operators is analyzed using the general stability results of Niederdrenk and Yserentant [Numer. Math., 41 (1983), pp. 223–253] as well as from the stand-point of exact discretizations. Examples illustrating some of the anomalous behavior associated with these problems—schemes can be compact and uniformly consistent but not uniformly stable, for instance—are presented.