A multiplicative Schwarz adaptive wavelet method for elliptic boundary value problems

A multiplicative Schwarz overlapping domain decomposition method is considered for solving elliptic boundary value problems. By equipping the relevant Sobolev spaces on the subdomains with wavelet bases, adaptive wavelet methods are used for approximately solving the subdomain problems. The union of the wavelet bases forms a frame for the Sobolev space on the domain as a whole. The resulting method is proven to be optimal in the sense that, in linear complexity, the iterands converge with the same rate as the sequence over N is an element of N of the best approximation from the span of the best N frame elements. Numerical results are given for the method applied to Poisson's equation.