Adaptive methods for PDE's: wavelets or mesh refinement?

TL;DR

This paper discusses the use of wavelet bases as an alternative to adaptive mesh refinement techniques, as well as one of the approaches which has been followed in developing efficient adaptive wavelet solvers.

Abstract

Adaptive mesh refinement techniques are nowadays an established and powerful tool for the numerical discretization of PDE's. In recent years, wavelet bases have been proposed as an alternative to these techniques. The main motivation for the use of such bases in this context is their good performances in data compression and the approximation theoretic foundations which allow to analyze and optimize these performances. We shall discuss these theoretical foundations, as well as one of the approaches which has been followed in developing efficient adaptive wavelet solvers. We shall also discuss the similarities and differences between wavelet methods and adaptive mesh refinement.