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    Alley-side Pickup: 10am - 7pm

    3038 Hennepin Ave Minneapolis, MN
    612-822-4611

Open Daily: 10am - 10pm | Alley-side Pickup: 10am - 7pm
3038 Hennepin Ave Minneapolis, MN
612-822-4611
Adaptive Wavelet Frame Methods for Nonlinear Elliptic Problems

Adaptive Wavelet Frame Methods for Nonlinear Elliptic Problems

Kappei, Jens

Paperback

General Mathematics

Currently unavailable to order

ISBN10: 3832530304
ISBN13: 9783832530303
Publisher: Logos Verlag Berlin
Published: Feb 6 2012
Pages: 175
Weight: 0.80
Height: 0.40 Width: 6.55 Depth: 9.46
Language: English
Over the last ten years, adaptive wavelet methods have turned out to be a powerful
tool in the numerical treatment of operator equations given on a bounded domain or
closed manifold. In this work, we consider semi-nonlinear operator equations,
including an elliptic linear operator as well as a nonlinear monotone one. Since the
classical approach to construct a wavelet Riesz basis for the solution space is
still afflicted with some notable problems, we use the weaker concept of wavelet
frames to design an adaptive algorithm for the numerical solution of problems of
this type.

Choosing an appropriate overlapping decomposition of the given domain, a suitable
frame system can be constructed easily. Applying it to the given continuous problem
yields a discrete, bi-infinite nonlinear system of equations, which is shown to be
solvable by a damped Richardson iteration method. We then successively introduce all
building blocks for the numerical implementation of the iteration method. Here, we
concentrate on the evaluation of the discrete nonlinearity, where we show that the
previously developed auxiliary of tree-structured index sets can be generalized to
the wavelet frame setting in a proper way. This allows an effective numerical
treatment of the nonlinearity by so-called aggregated trees. Choosing the error
tolerances appropriately, we show that our adaptive scheme is asymptotically optimal
with respect to aggregated tree-structured index sets, i.e., it realizes the same
convergence rate as the sequence of best N-term frame approximations of the solution
respecting aggregated trees. Moreover, under the assumption of a sufficiently
precise numerical quadrature method, the computational cost of our algorithm stays
the same order as the number of wavelets used by it.

The theoretical results are widely confirmed by one- and two-dimensional test
problems over non-trivial bounded domains.

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