Shannon wavelet approximations of linear differential operators

TL;DR

This paper explores the use of Shannon wavelet approximations for linear differential operators, focusing on convergence properties of iterative algorithms and extending wavelet methods for PDE solutions.

Contribution

It extends previous wavelet-based PDE approximation results by analyzing convergence of iterative algorithms using Shannon wavelets and related wavelet structures.

Findings

Proved convergence of the Leray projector algorithm with divergence-free Shannon wavelets.

Analyzed tensorial wavelets and wavelet packets for PDE approximation.

Extended wavelet theory applications to linear differential operators.

Abstract

Recent works emphasized the interest of numerical solution of PDE's with wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non linear approximation aspect of the wavelet approximation of PDE's to prove the relevance of such methods. In order to extend these results, we focuss on the convergence of the iterative algorithm, and we consider different possibilities offered by the wavelet theory: the tensorial wavelets and the derivation/integration of wavelet bases. We also investigate the use of wavelet packets. We apply these extended results to prove in the case of the Shannon wavelets, the convergence of the Leray projector algorithm with divergence-free wavelets.