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W. Auzinger:
"An overview on defect-based a posteriori error estimation for ODEs and DAEs";
Keynote Lecture: International Conference dedicated to the 100th anniversary of M.M.Bogolyubov and to the 70th anniversary of M.I.Nahnybida, Chernivtsi, Ukraine (invited); 2006-06-08 - 2006-06-13.

For a given numerical approximation to the solution of an ODE (initial or boundary value problem) we consider an efficient and reliable procedure to estimate its global error. In particular, we consider the case of estimating the error of a piecewise polynomial collocation solution. Our a posteriori error estimate is based on the defect correction principle. It is realized by considering an 'exact difference scheme', computing the defect of the given approximation with respect to this scheme, and using this as the right hand side in a simple auxiliary difference scheme. Solution of this auxiliary problem results in the error estimate. The integral means involved in this process, in particular for computation of the defect, are approximated by local interpolatory quadrature of an appropriate order.

We show how to realize this approach for ODEs of first and second order, and sketch the proof of the asymptotic correctness for these cases. Numerical examples are given for a number of different problems, including singular ODEs and differential-algebraic equations, which can be successfully treated in this way. Some open problems are also discussed.