@TechReport{ Hochmuth.Knapek.Zumbusch:2000,
author = {R.~Hochmuth and S.~Knapek and G.~Zumbusch},
title = {Tensor products of {S}obolev spaces and applications},
year = {2000},
number = {685},
institution = {SFB 256, Univ. Bonn},
abstract = {In many cases the approximation of solutions to
variational problems involving isotropic Sobolev spaces has
a complexity which depends exponentially on the dimension.
However, if the solutions possess dominating mixed
derivatives one can find discretizations to the
corresponding variational problems with a lower complexity
-- sometimes even independent of the dimension. In order to
analyse these effects, we relate tensor products of Sobolev
spaces with spaces with dominating mixed derivatives. Based
on these considerations we construct families of finite
dimensional anisotropic approximation spaces which
generalize in particular sparse grids. The obtained
estimates demonstrate, in which cases a complexity
independent or nearly independent of the dimension can be
expected. Finally numerical experiments demonstrate the
usefulness of the suggested approximation spaces.}
}