Quantitative error analysis of numerical methods for partial differential equations

The classical theory of numerical methods for partial differential equations is concerned to a large extent with problems of consistency, stability and convergence of algorithms. While these aspects of the theory are useful in establishing the validity of difference approximations, they fail to provide quantitative measures of errors which may be used in practice to characterize the effect of truncation. Nor do they provide relevant criteria whereby a specified accuracy requirement in a numerical solution may be converted into the choice of algorithms and/or the choice of grid sizes. When one attempts to develop such criteria, one readily finds out that they are very much dependent upon the initial and boundary conditions of the problem: by contrast, the classical properties of consistency, stability and convergence depend upon those initial/boundary conditions in a much weaker sense. It also becomes readily apparent that something other than the conventional tools must be used to perform such quantative error analyses. One approach to this problem, which has proved to be useful in several case studies, is outlined in the remainder of this paper.Technical report DCS-TR-2