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O. Koch, E. Weinmüller:
"A Posteriori Error Estimation for Collocation Methods Applied to Singular Boundary Value Problems, Part II";
Vortrag: SciCADE 2003, Trondheim, Norway; 28.06.2003 - 04.07.2003.

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 by the authors for regular problems, and is based on the defect correction principle. Our analysis is concerned with boundary value problems with a singularity of the first kind.

Part I: The presentation deals with the linear case. Here, we recapitulate how the superposition principle is used to prove basic convergence results for collocation applied to linear singular problems. With these prerequisites, we derive refined bounds for the
collocation solution and its derivative and use them 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: 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 the linear problems in order to show the existence of a solution of the nonlinear collocation equations. Such solution exists in a suitable neighborhood of an isolated solution of the boundary value problem. Moreover, quadratic convergence of Newton's method for the computation of this solution is discussed. Relying on these results, asymptotical correctness of our error estimate is shown, in case that the analytical problem is stable w.r.t. perturbations in the right-hand side of the underlying system of ODEs.