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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.