Fourier Analysis on Schwarz Domain Decomposition Methods for the Biharmonic Equation

SHANG Yue-qiang; HE Yin-nian

doi:10.3879/j.issn.1000-0887.2009.09.012

Volume 30 Issue 9

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SHANG Yue-qiang, HE Yin-nian. Fourier Analysis on Schwarz Domain Decomposition Methods for the Biharmonic Equation[J]. Applied Mathematics and Mechanics, 2009, 30(9): 1100-1106. doi: 10.3879/j.issn.1000-0887.2009.09.012

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Fourier Analysis on Schwarz Domain Decomposition Methods for the Biharmonic Equation

doi: 10.3879/j.issn.1000-0887.2009.09.012

SHANG Yue-qiang^1,2,
HE Yin-nian¹

1.
Faculty of Science, Xi'an Jiaotong University, Xi'an 710049, P. R. China;

Received Date: 2008-12-04
Rev Recd Date: 2009-07-10
Publish Date: 2009-09-15

Abstract

Abstract

Schwarz methods are an important type of domain decomposition methods.Using the Fourier transform tool,the error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation were deduced.It not only concisely proves the convergence of the Schwarz methods from a new point of view,but also provides detailed information about the convergence speeds and their dependence on the overlapping size of subdomains.The obtained results are independent of any unknown constant and discretization method,show that the Schwarz alternating method converges twice as quickly as the additive Schwarz method.
- domain decomposition algorithm,
- Schwarz method,
- Fourier transform,
- biharmonic equation

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References(11)

References

[1]	Schwarz H A.Gesammelte Mathematische Abhandlungen[M].Vol 2. Berlin: Springer,1890,133-143(First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich,Volume 15,1870).
[2]	Lions P L.On the Schwarz alternating method Ⅰ[A].In: Golub G H,Meurant G A,Periaux J,et al,Eds.Proceedings of the 1st International Symposium on Domain Decomposition Methods for Partial Differential Equations[C].Philadelphia: SIAM,1988,1-42.
[3]	吕涛,石济民,林振宝. 区域分解算法——偏微分方程数值解新技术[M].北京:科学出版社,1992.
[4]	Bjrstad P E. Multiplicative and additive Schwarz methods:convergence in the two subdomain case[A].In:Chan T F,Glowinski R,Periaux J,et al,Eds.Proceedings of the Second International Symposium on Domain Decomposition Methods[C].Philadelphia:SIAM,1989,147-159.
[5]	ZHANG Xue-jun. Two-level Schwarz method for biharmonic problems discretized by C1 conforming elements[J].SIAM J Numer Anal,1996,33(2):555-570. doi: 10.1137/0733029
[6]	XU Xue-jun,Lui S H,Rahman T.A two-level additive Schwarz method for the Morley nonconforming element approximation of a nonlinear biharmonic equation[J].IMA J Numer Anal,2004,24(1):97-122. doi: 10.1093/imanum/24.1.97
[7]	SHI Zhong-ci,XU Xue-jun.The mortar element method for a nonlinear biharmonic equation[J].J Comput Math,2005,23(5):537-560.
[8]	Gander M J. Optimized Schwarz methods[J].SIAM J Numer Anal,2006,44(2):699-731. doi: 10.1137/S0036142903425409
[9]	陈恕行. 现代偏微分方程导论[M].北京:科学出版社,2005.
[10]	Dolean V,Nataf F,Rapin G. Deriving a new domain decomposition method for the Stokes equations using the Smith factorization[J].Math Comp,2009,78(266):789-814.
[11]	李开泰,马逸尘,王立周. 广义函数和Sobolev空间[M].西安:西安交通大学出版社,2008.