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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Piecewise tensor product wavelet bases by extensions and approximation rates


Authors: Nabi Chegini, Stephan Dahlke, Ulrich Friedrich and Rob Stevenson
Journal: Math. Comp. 82 (2013), 2157-2190
MSC (2010): Primary 15A69, 35B65, 41A25, 41A63, 42C40, 65N12, 65T60
Published electronically: April 12, 2013
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Abstract: Following [Studia Math., 76(2) (1983), pp. 1-58 and 95-136] by Z. Ciesielski and T. Figiel and [SIAM J. Math. Anal., 31 (1999), pp. 184-230] by W. Dahmen and R. Schneider, by the application of extension operators we construct a basis for a range of Sobolev spaces on a domain $ \Omega $ from corresponding bases on subdomains that form a non-overlapping decomposition. As subdomains, we take hypercubes, or smooth parametric images of those, and equip them with tensor product wavelet bases. We prove approximation rates from the resulting piecewise tensor product basis that are independent of the spatial dimension of $ \Omega $. For two- and three-dimensional polytopes we show that the solution of Poisson type problems satisfies the required regularity condition. The dimension independent rates will be realized numerically in linear complexity by the application of the adaptive wavelet-Galerkin scheme.


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Additional Information

Nabi Chegini
Affiliation: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
Email: n.godarzvandchegini@uva.nl

Stephan Dahlke
Affiliation: Department of Mathematics and Computer Sciences, Philipps-University Marburg, Hans-Meerwein Str., Lahnberge, 35032 Marburg, Germany
Email: dahlke@mathematik.uni-marburg.de

Ulrich Friedrich
Affiliation: Department of Mathematics and Computer Sciences, Philipps-University Marburg, Hans-Meerwein Str., Lahnberge, 35032 Marburg, Germany
Email: friedrich@mathematik.uni-marburg.de

Rob Stevenson
Affiliation: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
Email: r.p.stevenson@uva.nl

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02694-4
PII: S 0025-5718(2013)02694-4
Keywords: Wavelets, tensor product approximation, domain decomposition, extension operators, weighted anisotropic Sobolev space, regularity, adaptive wavelet scheme, best approximation rates, Fichera corner
Received by editor(s): September 2, 2011
Received by editor(s) in revised form: February 5, 2012, and February 14, 2012
Published electronically: April 12, 2013
Additional Notes: The first author was supported by the Netherlands Organization for Scientific Research (NWO) under contract no. 613.000.902
The second and third authors were supported by Deutsche Forschungsgemeinschaft, grant number DA 360/12-1. The second author also acknowledges support by the LOEWE Center for Synthetic Microbiology, Marburg.
Article copyright: © Copyright 2013 American Mathematical Society