Adaptive Methods for PDE's Wavelets or Mesh Refinement?

Adaptive mesh re nement 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 developping ecient adaptive wavelet solvers. We shall also discuss the similarities and dierences between wavelet methods and adaptive mesh re nement