Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
[1] R. Hochmuth, S. Knapek, and G. Zumbusch. Tensor products of Sobolev spaces and applications. submitted, 2000. also as Technical Report 685, SFB 256, Univ. Bonn.
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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.