Adaptive wavelet frame methods for nonlinear elliptic problems

Jens Kappei

ISBN 978-3-8325-3030-3
175 pages, year of publication: 2012
price: 37.00 €
Adaptive wavelet frame methods for nonlinear elliptic problems

Over the last ten years, adaptive wavelet methods have turned out to be a powerf ul tool in the numerical treatment of operator equations given on a bounded doma in or closed manifold. In this work, we consider semi-nonlinear operator equatio ns, including an elliptic linear operator as well as a nonlinear monotone one. S ince 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 o f wavelet frames to design an adaptive algorithm for the numerical solution of p roblems of this type.

Choosing an appropriate overlapping decomposition of the given domain, a suitabl e frame system can be constructed easily. Applying it to the given continuous pr oblem yields a discrete, bi-infinite nonlinear system of equations, which is sho wn 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, whe re 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 fram e approximations of the solution respecting aggregated trees. Moreover, under th e assumption of a sufficiently precise numerical quadrature method, the computat ional 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 pr oblems over non-trivial bounded domains.