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W. Auzinger, O. Koch, E. Weinmüller:
"Global Error Estimates for Collocation Methods Applied to BVPs with an Essential Singularity";
Talk: ANODE 2003, Auckland Numerical ODEs, Auckland, New Zealand; 2003-07-14 - 2003-07-18.

We consider boundary value problems with an essential singularity (singularity of the second kind). Boundary value problems with an essential singularity are frequently encountered in applications. In particular, problems posed on infinite intervals belong to the above problem class when they are suitably transformed to a finite interval. Flow problems (Blasius equation, von Karman swirling flow) and the classical electromagnetic self-interaction problem, are sources for the models we are interested in.

Our aim is to investigate a numerical approach which may be successfully applied to obtain high-order solutions for boundary value problems. First, we examine the empirical convergence order of collocation methods at either equidistant or Gaussian points. The motivation to apply these methods was their satisfactory performance when solving boundary value problems with a singularity of the first kind. Therefore collocation has been implemented in our MATLAB code SBVP 1.0 designed for the latter class of problems, together with an a posteriori global error estimate based on the defect correction principle. This error estimate with backward Euler scheme as an auxiliary method is asymptotically correct when a singularity of the first kind is present, but unfortunately, it does not work for problems with an essential singularity. Consequently, we examine three alternative approaches to estimate the global error: the defect correction principle with the box scheme or collocation replacing the
backward Euler method, and a strategy based on mesh halving.