We discuss new variants of the acceleration technique known as Iterated Defect Correction (IDeC). In the classical version of the IDeC procedure, the defect is chosen as the pointwise residual with respect to the given ODE. Here, we show that the classical method can be improved if the defect is suitably modified. As an example we consider the backward Euler method for boundary value problems in ODEs. For the defect, we choose a locally integrated form which amounts to the residual with respect to certain Runge-Kutta schemes associated with a particular set of quadrature rules. It is shown that this results in a possible improvement in the maximal attainable convergence order. Moreover, the grids for this variant do not have to be chosen as piecewise equidistant, as is the case for the classical version. It is further demonstrated that the attainable orders are determined by the fixed points of the procedure. Finally, an alternative implementation using the box scheme as the basic method is shown to approach a high-order solution very efficiently. In particular, the application of these procedures to singular boundary value problems of the form
z'(t)=M(t)z(t)/t+f(t,z(t)), t∈ (0,1],
is discussed. Further applications of the defect correction principle to this particular problem class are discussed for example in [1], [2] and [3].


[1] W. Auzinger, O. Koch, and E. Weinmüller. Efficient collocation schemes for singular boundary value problems. To appear in Numer. Algorithms.

[2] W. Auzinger, O. Koch, and E. Weinmüller. Theory and Solution Techniques for Singular Boundary Value Problems for Ordinary Differential Equations. To appear in Proceedings of PPAM 2001.

[3] O. Koch and E. Weinmüller. Iterated Defect Correction for the Solution of Singular Initial Value Problems. SIAM J. Numer. Anal., 38(6):1784-1799, 2001.