Error Estimation via Defect Computation and Reconstruction: Some Particular Techniques

The well-known technique of defect correction can be used in various ways for estimating local or global errors of numerical approximations to differential or integral equations. In this paper we describe the general principle in the context of linear and nonlinear problems and indicate the interplay between the auxiliary scheme involved and a correct definition of the defect. Applications discussed include collocation approximations to first and second order boundary value problems for nonlinear ODEs and, in particular, exponential splitting approximations for linear evolution equations. We describe the design of error estimators and their essential properties and give numerical examples. The theoretical tools for the analysis of the asymptotical correctness of such estimators are described, and references to original research papers are given where the complete analysis is provided. c ⃝ 2011 European Society of Computational Methods in Sciences and Engineering