Adaptive and anisotropic piecewise polynomial approximation

Details Adaptive and anisotropic piecewise polynomial approximation Adaptive and anisotropic piecewise polynomial approximation

2011-01-08

arxiv.org/abs/1101.1587v1

Chapter of the book : Multiscale, Nonlinear and Adaptive Approximation, editors R. Devore, A. Kunoth, Springer, 2009

Mathematics

Numerical Analysis

40 pages, 9 figures

Scientific paper

We survey the main results of approximation theory for adaptive piecewise polynomial functions. In such methods, the partition on which the piecewise polynomial approximation is defined is not fixed in advance, but adapted to the given function f which is approximated. We focus our discussion on (i) the properties that describe an optimal partition for f, (ii) the smoothness properties of f that govern the rate of convergence of the approximation in the Lp-norms, and (iii) fast refinement algorithms that generate near optimal partitions. While these results constitute a fairly established theory in the univariate case and in the multivariate case when dealing with elements of isotropic shape, the approximation theory for adaptive and anisotropic elements is still building up. We put a particular emphasis on some recent results obtained in this direction.

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