caa a22 4500 475743725 CHVBK 20180406123507.0 cr unu---uuuuu 170329e20001001xx s 000 0 eng 10.1007/s003650010010 doi (NATIONALLICENCE)springer-10.1007/s003650010010 Optimized Tensor-Product Approximation Spaces [Elektronische Daten] [M. Griebel, S. Knapek] Abstract. : This paper is concerned with the construction of optimized grids and approximation spaces for elliptic differential and integral equations. The main result is the analysis of the approximation of the embedding of the intersection of classes of functions with bounded mixed derivatives in standard Sobolev spaces. Based on the framework of tensor-product biorthogonal wavelet bases and stable subspace splittings, the problem is reduced to diagonal mappings between Hilbert sequence spaces. We construct operator adapted finite element subspaces with a lower dimension than the standard full-grid spaces. These new approximation spaces preserve the approximation order of the standard full-grid spaces, provided that certain additional regularity assumptions are fulfilled. The form of the approximation spaces is governed by the ratios of the smoothness exponents of the considered classes of functions. We show in which cases the so-called curse of dimensionality can be broken. The theory covers elliptic boundary value problems as well as boundary integral equations. Springer-Verlag New York, 2000 Key words. Biorthogonal wavelets, Optimized approximation spaces, Hyperbolic cross, Sparse grids, Norm equivalences, Subspace splittings nationallicence Partial differential equations, Integral equations nationallicence Elliptic problems nationallicence AMS Classification nationallicence 41A25 nationallicence 41A30 nationallicence 41A65 nationallicence 45L10 nationallicence 65N12 nationallicence 65N30 nationallicence 65Y20 nationallicence 68Q25 nationallicence Griebel M. Institut für Angewandte Mathematik griebel@iam.uni-bonn.de, knapek@iam.uni-bonn.de, Abteilung für Wissenschaftliches Rechnen und Numerische Simulation Universität Bonn D-53115 Bonn Germany, DE aut Knapek S. Institut für Angewandte Mathematik griebel@iam.uni-bonn.de, knapek@iam.uni-bonn.de, Abteilung für Wissenschaftliches Rechnen und Numerische Simulation Universität Bonn D-53115 Bonn Germany, DE aut https://doi.org/10.1007/s003650010010 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s003650010010 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Griebel M. Institut für Angewandte Mathematik griebel@iam.uni-bonn.de, knapek@iam.uni-bonn.de, Abteilung für Wissenschaftliches Rechnen und Numerische Simulation Universität Bonn D-53115 Bonn Germany, DE aut NATIONALLICENCE 700 1- Knapek S. Institut für Angewandte Mathematik griebel@iam.uni-bonn.de, knapek@iam.uni-bonn.de, Abteilung für Wissenschaftliches Rechnen und Numerische Simulation Universität Bonn D-53115 Bonn Germany, DE aut Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer