Optimized Tensor-Product Approximation Spaces |
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Authors: | M Griebel S Knapek |
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Institution: | Institut für Angewandte Mathematik griebel@iam.uni-bonn.de, knapek@iam.uni-bonn.de, Abteilung für Wissenschaftliches Rechnen und Numerische Simulation Universit?t Bonn D-53115 Bonn Germany, DE
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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.
September 17, 1998. Date revised: March 5, 1999. Date accepted: September 20, 1999. |
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Keywords: | , Biorthogonal wavelets, Optimized approximation spaces, Hyperbolic cross, Sparse grids, Norm equivalences, Subspace,,,,,splittings, Partial differential equations, Integral equations, Elliptic problems, AMS Classification, 41A25, 41A30, 41A65,,,,,,45L10, 65N12, 65N30, 65Y20, 68Q25, |
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