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Contemporary Mathematics New Variants of Defect Correction for Boundary Value Problems in Ordinary Differential Equations

In this paper we discuss new variants of the acceleration technique known as Iterated Defect Correction (IDeC) for the numerical solution of boundary value problems in ordinary differential equations. A first approximation, computed by the backward Euler scheme, is iteratively improved to obtain a high order solution. Typically, the maximal attainable accuracy is limited by the smoothness of the exact solution and by technical details of the procedure. We propose a new version of the IDeC algorithm with maximal achievable order higher than in the classical setting. Moreover, our procedure can be shown to be convergent on arbitrary grids, while the classical IDeC iteration requires piecewise equidistant grids. Finally, the performance of this new algorithm for singular boundary value problems is discussed.