Adaptive wavelet schemes for elliptic problems implementation and numerical experiments

Recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution behaves asymptotically like the smallest possible error that can be realized by any linear combination of the corresponding number of wavelets. On one hand, the results are purely asymptotic. On the other hand, the analysis suggests new algorithmic ingredients for which no prototypes seem to exist yet. It is therefore the objective of this paper to develop suitable data structures for the new algorithmic components and to obtain a quantitative validation of the theoretical results. We briefly review rst the main theoretical facts, describe the main ingredients of the algorithm, highlight the essential data structures, and illustrate the results by one- and two-dimensional numerical examples including comparisons with an adaptive finite element scheme.