Thirteen Ways to Estimate Global Error

In recent years there have been a number of surveys and comparison of techniques for estimating global discretization error in numerical solutions of differential equations [25, 29, 35, 36, 37, 39]. These have been an important stimulus to research in this area. In this paper we attempt to give an updated, organized, comprehensive survey of global error estimation techniques. For most of these techniques we outline an error analysis explaining why they work, and based on such theoretical considerations we discuss the relative merit of several of these techniques. To a large degree this paper is an extension of earlier work [30] by the author on deferred correction, which is one technique for global error estimation. This survey is limited to error estimates, most of which are asymptotically correct in some sense, and excludes approximate or rigorous error bounds. Approximate error bounds, in the same spirit as matrix condition number estimators, would probably be more desirable. Some work on this has been done by Dahlquist [5] and Eriksson [91. Rigorous error bounds are usually best achieved by interval analysis. Improvements in hardware and algorithms have made this approach feasible for problems of moderate size, and interval analy-