# Numerical Analysis Seminar

## Date:

Tuesday, 1st March 2005

Speaker: Dr. Andrej Nietsche
Eidgenössische Technische Hochschule (ETH) Züric

Title: Tensor Product Approximation Spaces & Elliptic PDEs

Date: Thursday March 10th
Time: 11am
Room: RC-4082

Abstract: We discuss approximation theory for tensor product bases. For linear
approximation (uniform and non-uniform sparse grid methods), the
approximation spaces are well known to be (weighted) Sobolev spaces of mixed
highest derivatives. In the nonlinear case (best $N$ term approximation,
adaptive approximation), the approximation spaces turn out to be $q$-tensor
products of suitable Besov spaces $B^s_q(L_q)$, which might be coined Besov
spaces of mixed highest derivatives.

In the second part of the talk, we apply these theoretical results to the
numerical solution of elliptic partial differential equations (PDEs). We
discuss Besov regularity of solutions to elliptic PDEs in the tensor product
setting as well as in the more standard setting using isotropically supported
bases. The results are quite different, showing an unlimited degree of
anisotropic Besov regularity (for the tensor product setting) but only a
limited degree of isotropic Besov regularity (for the setting using
isotropically supported basis functions) in dimension 3 and higher.