Adaptive wavelet methods for solving operator equations: An overview

Abstract

In [Math. Comp, 70 (2001), 27–75] and [Found. Comput. Math., 2(3) (2002), 203–245], Cohen, Dahmen and DeVore introduced adaptive wavelet methods for solving operator equations. These papers meant a break-through in the field, because their adaptive methods were not only proven to converge, but also with a rate better than that of their non-adaptive counterparts in cases where the latter methods converge with a reduced rate due a lacking regularity of the solution. Until then, adaptive methods were usually assumed to converge via a saturation assumption. An exception was given by the work of Dorfler in [SIAM J. Numer. Anal., 33 (1996), 1106–1124], where an adaptive finite element method was proven to converge, with no rate though.