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On optimal global error bounds obtained by scaled local error estimates

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Summary

In the second section a general method of obtaining optimal global error bounds by scaling local error estimates is developed. It is reduced to the solution of a fixpoint problem. In Sect. 3 we show more concrete error estimates reflecting a singularity of order α. It is shown that under general circumstances an optimal global error bound is achieved by an (asymptotically) geometric mesh for the local error estimates. In the fourth section we specialize this to the best approximation ofg(x)x α by piecewise polynomials with variable knots and degrees having a total numberN of parameters. This generalizes the result of R. DeVore and the author forg(x)=1. In the last section this problem is studied for the functione −x on (0, ∞). The exact asymptotic behaviour of the approximation withN parameters is determined toe qoN, whereq o=0.895486 ....

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Germany (Fed. Rep.)

    Karl Scherer

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  1. Karl Scherer
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Scherer, K. On optimal global error bounds obtained by scaled local error estimates. Numer. Math. 36, 151–176 (1980). https://doi.org/10.1007/BF01396756

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