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W. Auzinger, G Kitzler, O. Koch, A. Saboor Bagherzadeh:
"Robust and highly accurate error estimators for second order ODEs";
Talk: 6th Austrian Numerical Analysis Day, Salzburg; 2010-05-06 - 2010-05-07.

Based on the notion of exact finite difference schemes (EDS), the error of a given approximation to a boundary value problem can be estimated by computing its defect with respect to the EDS, possibly after local reconstruction via interpolation, and solving back for the estimate using a simple finite-difference scheme. Implementation using appropriate quadrature for defect evaluation yields a robust and efficient, asymptotically correct a posteriori error estimate. In particular, it is designed to work on arbitrary, nonequidistant, meshes, useful for its deployment in adaptive schemes.

In this talk, this way of estimating the error is presented for the particular case of collocation or finite-difference approximations to second order two-point boundary value problems. The particular construction of the estimator is discussed and the proof of its high asymptotic correctness for h to 0 is indicated, with a O(h^(p+2)) deviation for a scheme of order p on arbitrary meshes. We also show how to compute the defect for boundary conditions involving derivatives, and we give several numerical examples, including singular ODEs.