Defect Correction is an efficient method to obtain an a-posteriori error estimate for the discretization error. It was originally proposed by Zadunaisky to estimate the global error of Runge-Kutta schemes, cf. . The idea later became a basis for the acceleration technique called Iterated Defect Correction (IDeC) used to iteratively improve the numerical solution, see  and . Numerical experiments show that classical variants of the Defect Correction work well on meshes which are at least locally equidistant. Otherwise, the convergence of the procedure shows order reductions. In order to overcome this difficulty, certain algorithmic modification are necessary. We discuss a new, carefully designed modification of the error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind,
 z'(t)=M(t)z(t)/t+f(t,z(t)), t∈ (0,1], B0z(0)+B1z(1)=β, z∈ C[0,1], (1)
where B0, B1 and β are suitably chosen to yield a well-posed problem. This global error estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. Most importantly, we observe that the grid is refined in a way reflecting only the smoothness of the solution. The above strategies have been implemented in a MATLAB solver, SBVP 1.0, freely available from http://www.math.tuwien.ac.at/~ewa, cf. . Finally, we present the application of another variant of the IDeC procedure to stiff ODEs.

# References

 P.E. Zadunaisky, On the Estimation of Errors Propagated in the Numerical Integration of ODEs, Numer. Math. 27 (1976), pp. 21-39.

 R. Frank, and C. Überhuber, Iterated Defect Correction for Differential Equations, Part I: Theoretical Results, Computing 20 (1978), pp. 207-228.

 H.J. Stetter, The Defect Correction Principle and Discretization Methods, Numer. Math., 29 (1978), pp. 425-443.

 W. Auzinger, G. Kneisl, O. Koch, and E. Weinmüller, SBVP 1.0 - A MATLAB Solver for Singular Boundary Value Problems, Manual, ANUM Preprint No. 2/02, Vienna University of Technology, Austria (2002).