An Adaptive Collocation Method based on Interpolating Wavelets

S. Bertoluzza

We propose an adaptive collocation method based on the use of the Deslaurier-Dubuc interpolating bases. The use of a collocation approach and the particular structure of the chosen basis allows us to avoid computations in the uniform fine grid, which are usually needed in the framework of wavelet Galerkin schemes. A very simple refining strategy, based on the natural hierarchical decomposition relative to the basis chosen, is proposed in analogy with the Galerkin adaptive schemes based on hierarchical bases. Although it lacks a rigorous theoretical justification, mainly due to the fact that we are dealing with a collocation scheme, the proposed refining strategy turns out to be quite effective, as shown by the results of the numerical tests. In particular numerical results show that the proposed method gives good results when the continuous solution (which has to satisfy the minimal smoothness assumption $u\in C^2,$ needed in order to define the method) is very smooth in a large part of the domain, but has some localized singularity in some high order derivative.

To appear in "Multiresolution Analysis and Wavelets for the Numerical Solution of Partial Differential Equations", W. Dahmen, A. Kurdila, P. Oswald eds., Academic Press.

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