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), 21572190
MSC (2010):
Primary 15A69, 35B65, 41A25, 41A63, 42C40, 65N12, 65T60
Published electronically:
April 12, 2013
MathSciNet review:
3073195
Fulltext PDF
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Additional Information
Abstract: Following [Studia Math., 76(2) (1983), pp. 158 and 95136] by Z. Ciesielski and T. Figiel and [SIAM J. Math. Anal., 31 (1999), pp. 184230] 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 from corresponding bases on subdomains that form a nonoverlapping 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 . For two and threedimensional 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 waveletGalerkin scheme.
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 T. Apel.
Anisotropic finite elements: Local estimates and applications. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart, 1999. MR 1716824 (2000k:65002)
 [CDD01]
 A. Cohen, W. Dahmen, and R. DeVore.
Adaptive wavelet methods for elliptic operator equations  Convergence rates. Math. Comp, 70:2775, 2001. MR 1803124 (2002h:65201)
 [CDN10]
 M. Costabel, M. Dauge, and S. Nicaise.
Analytic regularity for linear elliptic systems in polygons and polyhedra. Technical report, 2010. arXiv:1002.1772v1 [math.AP].
 [CF83]
 Z. Ciesielski and T. Figiel.
Spline bases in classical function spaces on compact manifolds. I and II. Studia Math., 76(2):158 and 95136, 1983. MR 728195 (85f:46060a)
 [CS11]
 N.G. Chegini and R.P. Stevenson.
The adaptive tensor product wavelet scheme: Sparse matrices and the application to singularly perturbed problems. IMA J. Numer. Anal., 32(1):75104, 2011.
 [Dij09]
 T.J. Dijkema.
Adaptive tensor product wavelet methods for solving PDEs. PhD thesis, Utrecht University, 2009.
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 W. Dahmen, A. Kunoth, and K. Urban.
Biorthogonal splinewavelets on the interval  Stability and moment conditions. Appl. Comp. Harm. Anal., 6:132196, 1999. MR 1676771 (99m:42046)
 [DS99]
 W. Dahmen and R. Schneider.
Wavelets on manifolds I: Construction and domain decomposition. SIAM J. Math. Anal., 31:184230, 1999. MR 1742299 (2000k:65242)
 [DS10]
 M. Dauge and R.P. Stevenson.
Sparse tensor product wavelet approximation of singular functions. SIAM J. Math. Anal., 42(5):22032228, 2010. MR 2729437 (2011k:42073)
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 M.R. Hestenes.
Extension of the range of differentiable functions. Duke Math. J., 8(1):183192, 1941. MR 0003434 (2:219c)
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 A. Kunoth and J. Sahner.
Wavelets on manifolds: An optimized construction. Math. Comp., 75:13191349, 2006. MR 2219031 (2007d:42076)
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 V. G. Maz'ya and J. Roßmann.
Weighted estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains. ZAMM Z. Angew. Math. Mech., 83(7):435467, 2003. MR 1987897 (2004g:35061)
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 P.A. Nitsche.
Sparse approximation of singularity functions. Constr. Approx., 21(1):6381, 2005. MR 2105391 (2005h:41038)
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 P.A. Nitsche.
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 M. Primbs.
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 R.P. Stevenson.
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Tensor products of SobolevBesov spaces and applications to approximation from the hyperbolic cross. J. Approx. Theory, 161:748786, 2009. MR 2563079 (2010j:46056)
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Additional Information
Nabi Chegini
Affiliation:
Kortewegde 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, PhilippsUniversity Marburg, HansMeerwein Str., Lahnberge, 35032 Marburg, Germany
Email:
dahlke@mathematik.unimarburg.de
Ulrich Friedrich
Affiliation:
Department of Mathematics and Computer Sciences, PhilippsUniversity Marburg, HansMeerwein Str., Lahnberge, 35032 Marburg, Germany
Email:
friedrich@mathematik.unimarburg.de
Rob Stevenson
Affiliation:
Kortewegde 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/S002557182013026944
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 360/121. The second author also acknowledges support by the LOEWE Center for Synthetic Microbiology, Marburg.
Article copyright:
© Copyright 2013
American Mathematical Society
