Optimized tenser-product approximation spaces

This paper is concerned with the construction of optimized grids and approx imation spaces for elliptic differential and integral equations. The main r esult is the analysis of the approximation of the embedding of the intersec tion of classes of functions with bounded mixed derivatives in standard Sob olev spaces. Based on the framework of censor-product biorthogonal wavelet bases and stable subspace splittings, the problem is reduced to diagonal ma ppings between Hilbert sequence spaces. We construct operator adapted finit e element subspaces with a lower dimension than the standard full-grid spac es. These new approximation spaces preserve the approximation order of the standard full-grid spaces, provided that certain additional regularity assu mptions are fulfilled. The form of the approximation spaces is governed by the ratios of the smoothness exponents of the considered classes of functio ns. We show in which cases the so-called curse of dimensionality can be bro ken. The theory covers elliptic boundary value problems as well as boundary integral equations.