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

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- Math. Comp.
**80**(2011), 1525-1556 Request permission

## 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.

## References

- Irfan Altas, Jonathan Dym, Murli M. Gupta, and Ram P. Manohar,
*Multigrid solution of automatically generated high-order discretizations for the biharmonic equation*, SIAM J. Sci. Comput.**19**(1998), no. 5, 1575–1585. MR**1618733**, DOI 10.1137/S1464827596296970 - H. Blum and R. Rannacher,
*On the boundary value problem of the biharmonic operator on domains with angular corners*, Math. Methods Appl. Sci.**2**(1980), no. 4, 556–581. MR**595625**, DOI 10.1002/mma.1670020416 - Carl de Boor,
*A practical guide to splines*, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR**507062** - C. de Boor and G. J. Fix,
*Spline approximation by quasiinterpolants*, J. Approximation Theory**8**(1973), 19–45. MR**340893**, DOI 10.1016/0021-9045(73)90029-4 - Susanne C. Brenner and L. Ridgway Scott,
*The mathematical theory of finite element methods*, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR**1894376**, DOI 10.1007/978-1-4757-3658-8 - Qianshun Chang and Zhaohui Huang,
*Efficient algebraic multigrid algorithms and their convergence*, SIAM J. Sci. Comput.**24**(2002), no. 2, 597–618. MR**1951057**, DOI 10.1137/S1064827501389850 - Qianshun Chang, Yau Shu Wong, and Hanqing Fu,
*On the algebraic multigrid method*, J. Comput. Phys.**125**(1996), no. 2, 279–292. MR**1388150**, DOI 10.1006/jcph.1996.0094 - Charles K. Chui and Jian-zhong Wang,
*On compactly supported spline wavelets and a duality principle*, Trans. Amer. Math. Soc.**330**(1992), no. 2, 903–915. MR**1076613**, DOI 10.1090/S0002-9947-1992-1076613-3 - Wolfgang Dahmen, Angela Kunoth, and Karsten Urban,
*Biorthogonal spline wavelets on the interval—stability and moment conditions*, Appl. Comput. Harmon. Anal.**6**(1999), no. 2, 132–196. MR**1676771**, DOI 10.1006/acha.1998.0247 - Oleg Davydov and Rob Stevenson,
*Hierarchical Riesz bases for $H^s(\Omega ),\ 1<s<{5\over 2}$*, Constr. Approx.**22**(2005), no. 3, 365–394. MR**2164141**, DOI 10.1007/s00365-004-0593-2 - Ronald A. DeVore, Björn Jawerth, and Vasil Popov,
*Compression of wavelet decompositions*, Amer. J. Math.**114**(1992), no. 4, 737–785. MR**1175690**, DOI 10.2307/2374796 - Lawrence C. Evans,
*Partial differential equations*, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR**1625845** - Bin Han and Qingtang Jiang,
*Multiwavelets on the interval*, Appl. Comput. Harmon. Anal.**12**(2002), no. 1, 100–127. MR**1874917**, DOI 10.1006/acha.2001.0370 - Bin Han and Zuowei Shen,
*Wavelets with short support*, SIAM J. Math. Anal.**38**(2006), no. 2, 530–556. MR**2237160**, DOI 10.1137/S0036141003438374 - Roger A. Horn and Charles R. Johnson,
*Topics in matrix analysis*, Cambridge University Press, Cambridge, 1991. MR**1091716**, DOI 10.1017/CBO9780511840371 - Rong-Qing Jia,
*Approximation with scaled shift-invariant spaces by means of quasi-projection operators*, J. Approx. Theory**131**(2004), no. 1, 30–46. MR**2103832**, DOI 10.1016/j.jat.2004.07.007 - Rong-Qing Jia,
*Bessel sequences in Sobolev spaces*, Appl. Comput. Harmon. Anal.**20**(2006), no. 2, 298–311. MR**2207841**, DOI 10.1016/j.acha.2005.11.001 - Rong-Qing Jia,
*Spline wavelets on the interval with homogeneous boundary conditions*, Adv. Comput. Math.**30**(2009), no. 2, 177–200. MR**2471447**, DOI 10.1007/s10444-008-9064-9 - Rong-Qing Jia,
*Approximation by quasi-projection operators in Besov spaces*, J. Approx. Theory**162**(2010), no. 1, 186–200. MR**2565832**, DOI 10.1016/j.jat.2009.04.003 - Rong-Qing Jia and Song-Tao Liu,
*Wavelet bases of Hermite cubic splines on the interval*, Adv. Comput. Math.**25**(2006), no. 1-3, 23–39. MR**2231693**, DOI 10.1007/s10444-003-7609-5 - Rong-Qing Jia and Song-Tao Liu,
*$C^1$ spline wavelets on triangulations*, Math. Comp.**77**(2008), no. 261, 287–312. MR**2353954**, DOI 10.1090/S0025-5718-07-02013-3 - Rong-Qing Jia, Jianzhong Wang, and Ding-Xuan Zhou,
*Compactly supported wavelet bases for Sobolev spaces*, Appl. Comput. Harmon. Anal.**15**(2003), no. 3, 224–241. MR**2010944**, DOI 10.1016/j.acha.2003.08.003 - Junjiang Lei, Rong-Qing Jia, and E. W. Cheney,
*Approximation from shift-invariant spaces by integral operators*, SIAM J. Math. Anal.**28**(1997), no. 2, 481–498. MR**1434046**, DOI 10.1137/S0036141095279869 - Randall J. LeVeque,
*Finite difference methods for ordinary and partial differential equations*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. Steady-state and time-dependent problems. MR**2378550**, DOI 10.1137/1.9780898717839 - Song Li and Jun Xian,
*Biorthogonal multiple wavelets generated by vector refinement equation*, Sci. China Ser. A**50**(2007), no. 7, 1015–1025. MR**2355873**, DOI 10.1007/s11425-007-0056-x - Yves Meyer,
*Wavelets and operators*, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. MR**1228209** - Peter Oswald,
*Multilevel preconditioners for discretizations of the biharmonic equation by rectangular finite elements*, Numer. Linear Algebra Appl.**2**(1995), no. 6, 487–505. MR**1367057**, DOI 10.1002/nla.1680020603 - D. J. Silvester and M. D. Mihajlović, Efficient preconditioning of the biharmonic equation, Numerical Analysis Report No.
**362**(2000), Manchester Center for Computational Mathematics, University of Manchester. - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095** - Rob Stevenson,
*Stable three-point wavelet bases on general meshes*, Numer. Math.**80**(1998), no. 1, 131–158. MR**1642527**, DOI 10.1007/s002110050363 - Gilbert Strang and George J. Fix,
*An analysis of the finite element method*, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR**0443377** - Jia Chang Sun,
*Domain decomposition and multilevel PCG method for solving $3$-D fourth order problems*, Domain decomposition methods in science and engineering (Como, 1992) Contemp. Math., vol. 157, Amer. Math. Soc., Providence, RI, 1994, pp. 71–78. MR**1262607**, DOI 10.1090/conm/157/01407 - Panayot S. Vassilevski and Junping Wang,
*Stabilizing the hierarchical basis by approximate wavelets. I. Theory*, Numer. Linear Algebra Appl.**4**(1997), no. 2, 103–126. MR**1443598**, DOI 10.1002/(SICI)1099-1506(199703/04)4:2<103::AID-NLA101>3.0.CO;2-J - Robert M. Young,
*An introduction to nonharmonic Fourier series*, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**591684**

## 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
- 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
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2785467