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Mathematics of Computation

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Riesz bases of wavelets and applications to numerical solutions of elliptic equations


Authors: Rong-Qing Jia and Wei Zhao
Journal: Math. Comp. 80 (2011), 1525-1556
MSC (2010): Primary 42C40, 65N30, 41A15, 41A25, 46E35
DOI: https://doi.org/10.1090/S0025-5718-2011-02448-8
Published electronically: January 14, 2011
MathSciNet review: 2785467
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Abstract: We investigate Riesz bases of wavelets in Sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth-order.

First, we study bicubic splines on the unit square with homogeneous boundary conditions. The approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation. Second, we develop a fairly general theory for Riesz bases of Hilbert spaces equipped with induced norms. Under the guidance of the general theory, we are able to construct wavelet bases for Sobolev spaces on the unit square. The condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded. Third, we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient. Finally, we extend our study to general elliptic equations of fourth-order and demonstrate that our numerical schemes also have superb performance in the general case.


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Additional Information

Rong-Qing Jia
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
Email: rjia@ualberta.ca

Wei Zhao
Affiliation: Department of Mathematics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1
Email: wzhao@math.mcmaster.ca

DOI: https://doi.org/10.1090/S0025-5718-2011-02448-8
Keywords: Riesz bases, multilevel decompositions, splines, wavelets, wavelets on bounded domains, rate of convergence, biharmonic equation, elliptic equations
Received by editor(s): March 17, 2009
Received by editor(s) in revised form: May 13, 2010
Published electronically: January 14, 2011
Additional Notes: The authors were supported in part by NSERC Canada under Grant OGP 121336
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.