An a posteriori estimate for the global error of collocation methods for two-point boundary value problems in ordinary differential equations is discussed. The estimate was first introduced and analyzed for regular problems in [1], and is based on the defect correction principle. Our analysis 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)
In [2], this error estimate was discussed for a restricted class of singular boundary value problems, where the coefficient matrix M(0) has only eigenvalues with nonpositive real parts. In that case, a shooting argument can be used to derive bounds for the error of the collocation solution (see also [3]) which enable an analysis of the error estimate. Here, we extend the results to the most general case, where eigenvalues of M(0) with positive real parts are permitted. To analyze the collocation solution and error estimate in this case, we derive a new representation of the global error. To accurately describe the structure of the terms in this representation, we have to derive new results for analytical, piecewise defined singular boundary value problems. Moreover, we have to extend the stability analysis for collocation methods (see [2]) to the case where eigenvalues of M(0) with positive real parts are present. To this end, we adapt some results from [4] for our purpose. Using the results for the collocation solution, we can analyze the error estimate similarly as in the regular case ([1]). We conclude that the error estimate is asymptotically correct when applied to our problem class. Consequently, collocation methods together with our a posteriori error estimation procedure can successfully be used in our Matlab code sbvp, which was designed especially for the efficient solution of singular boundary value problems, see [5].


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

[2] 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.

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

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

[5] W. Auzinger, G. Kneisl, O. Koch, and E. Weinmüller, A collocation code for boundary value problems in ordinary differential equations, to appear in Numer. Algorithms.