We present an a posteriori error estimate for the global error of collocation methods for two-point boundary value problems in ordinary differential equations. The estimate was first introduced and analyzed for regular problems in [1], and is based on the defect correction principle, see for example [2]. Our analysis, cf. [3], is concerned with problems with a singularity of the first kind,
z'(t)=M(t)/t z(t)+f(t,z(t)), t∈ (0,1], (1)
Part I (by Ewa Weinmüller): The presentation deals with the linear case, where f(t,z(t))=f(t) in (1). Here, we recapitulate how the superposition principle is used to prove basic convergence results for collocation applied to linear singular problems, cf. [4] and [5]. With these prerequisites, we derive refined bounds for the collocation solution and its derivative and use these to analyze the error estimate. We show that the error estimate is asymptotically correct on the whole collocation grid if the underlying collocation method is not superconvergent.
Part II (by Othmar Koch): Here, we extend the above results to the nonlinear case. In particular, we choose a Banach space setting and use the stability results derived for collocation for linear problems in order to prove the convergence of the solution of the nonlinear collocation equations. This solution is shown to exist in a suitable neighborhood of an isolated solution of (1), (2). Moreover, Newton's method converges quadratically for the computation of this solution. Relying on these results, asymptotical correctness of our error estimate is shown in the case where the analytical problem is stable w.r.t. perturbations in the right-hand side of (1).


[1] W. Auzinger, O. Koch, and E. Weinmüller, Efficient collocation schemes for singular boundary value problems. Numer. Algorithms 31 (2003), pp. 5-25.

[2] H.J. Stetter, The defect correction principle and discretization methods, Numer. Math., 29 (1978), pp. 425-443.

[3] W. Auzinger, O. Koch, and E. Weinmüller}, Analysis of a new error estimate for collocation methods applied to singular boundary value problems. Submitted to SIAM J. Numer. Anal.

[4] F. de Hoog, and R. Weiss, Collocation methods for singular boundary value problems, SIAM J. Numer. Anal., 15 (1978), pp. 198-217.

[5] E. Weinmüller, Collocation for singular boundary value problems of second order, SIAM J. Numer. Anal., 23 (1986), pp. 1062-1095.