Error Estimation via Defect Computation and Reconstruction: Some Particular Techniques

W. Auzinger
Institute for Analysis and Scientific Computing,
Vienna University of Technology,
1040 Vienna, Austria
Received 20 February, 2011; accepted in revised form 10 March, 2011
Abstract: 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 de-
sign 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.


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