On the Adaptive Computation of Integrals of Wavelets

The numerical solution of partial differential equations involves the computation of integrals of products of given functions and (derivatives of) trial and test functions. We study this problem using adaptively chosen wavelet bases. Firstly, we reduce this problem to the computation of 1--dimensional integrals. Then, we consider appropriate adaptive approximations and study the induced error. Finally, we present an algorithm for computing these integrals and give numerical results. ? This paper was written when the third author was in residence at the Istituto di Analisi Numerica del C.N.R. in Pavia, Italy. This work was supported by the European Commission within the TMR project (Training and Mobility for Researchers) Wavelets and Multiscale Methods in Numerical Analysis and Simulation, No. ERB FMRX CT98 0184, by the German Academic Exchange Service (DAAD) within the Vigoni--Project Multilevel-- Zerlegungsverfahren fur Partielle Differentialgleichungen and by the funds Murst 40% An..