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Adaptive and anisotropic piecewise polynomial approximation

In: Multiscale, Nonlinear and Adaptive Approximation

Author

Listed:
  • Albert Cohen

    (Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions)

  • Jean-Marie Mirebeau

    (Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions)

Abstract

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 L p -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.

Suggested Citation

  • Albert Cohen & Jean-Marie Mirebeau, 2009. "Adaptive and anisotropic piecewise polynomial approximation," Springer Books, in: Ronald DeVore & Angela Kunoth (ed.), Multiscale, Nonlinear and Adaptive Approximation, pages 75-135, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-03413-8_4
    DOI: 10.1007/978-3-642-03413-8_4
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