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Iteratively solving a kind of signorinitransmission problem in a unbounded domain

Published online by Cambridge University Press:  15 August 2005

Qiya Hu  and
Dehao Yu
Qiya Hu
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics, Chinese Academy of Sciences, Beijing 100080, China. hqy@lsec.cc.ac.cn; ydh@lsec.cc.ac.cn 

Dehao Yu
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics, Chinese Academy of Sciences, Beijing 100080, China. hqy@lsec.cc.ac.cn; ydh@lsec.cc.ac.cn 

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Abstract

In this paper, we are concerned with a kind of Signorini transmission problem in a unbounded domain. A variational inequality is derived when discretizing this problem by coupled FEM-BEM. To solve such variational inequality, an iterative method, which can be viewed as a variant of the D-N alternative method, will be introduced. In the iterative method, the finite element part and the boundary element part can be solved independently. It will be shown that the convergence speed of this iteration is independent of the mesh size. Besides, a combination between this method and the steepest descent method is also discussed.

Keywords

Signorini contactFEM-BEM couplingvariational inequalityD-N alternationconvergence rate.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 4 , July 2005 , pp. 715 - 726
Copyright
© EDP Sciences, SMAI, 2005

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References

Carstensen, C., Interface problem in holonomic elastoplasticity. Math. Methods Appl. Sci. 16 (1993) 819–835. CrossRef
Carstensen, C. and Gwinner, J., BEM, FEM coupling for a nonlinear transmission problem with Signorini contact. SIAM J. Numer. Anal. 34 (1997) 1845–1864. CrossRef
Carstensen, C., Kuhn, M. and Langer, U., Fast parallel solvers for symmetric boundary element domain decomposition equations. Numer. Math. 79 (1998) 321–347. CrossRef
Costabel, M. and Stephan, E., Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal. 27 (1990) 1212–1226. CrossRef
G. Gatica and G. Hsiao, On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R 2. Numer. Math. 61(1992) 171–214.
R. Glowinski, Numerical methods for nonlinear variational problems. Springer-Verlag, New York (1984).
R. Glowinski, G. Golub, G. Meurant and J. Periaux, Eds., Proc. of the the First international symposium on domain decomposition methods for PDEs. SIAM Philadelphia (1988).
Hu, Q. and Yu, D., A solution method for a certain interface problem in unbounded domains. Computing 67 (2001) 119–140. CrossRef
N. Kikuchi and J. Oden, Contact problem in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia (1988).
J. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I. Springer-Verlag (1972).
P. Mund and E. Stephan, An adaptive two-level method for the coupling of nonlinear FEM-BEM equations, SIAM J. Numer. Anal. 36 (1999) 1001–1021.
J. Necas, Introduction to the theory of nonlinear elliptic equations. Teubner, Texte 52, Leipzig (1983).
E. Polak, Computational methods in optimization. Academic Press, New York (1971).
Schoberl, J., Solving the Signorini problem on the basis of domain decomposition techniques. Computing 60 (1998) 323–344. CrossRef
Stephan, E., Wendland, W. and Hsiao, G., On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Methods Appl. Sci. 1 (1979) 265–321. CrossRef
Tai, X. and Espedal, M., Rate of convergence of some space decomposition methods for linear and nonlinear problems. SIAM J. Numer. Anal. 35 (1998) 1558–1570. CrossRef
Tai, X. and Global, J. Xu convergence of space correction methods for convex optimization problems. Math. Comp. 71 (2002) 105–122. CrossRef
The, D. Yu relation between the Steklov-Poincare operator, the natural integral operator and Green functions. Chinese J. Numer. Math. Appl. 17 (1995) 95–106.
D. Yu, Discretization of non-overlapping domain decomposition method for unbounded domains and its convergence.Chinese J. Numer. Math. Appl. 18 (1996) 93–102.
D. Yu, Natural Boundary Integral Method and Its Applications. Science Press/Kluwer Academic Publishers, Beijing/New York (2002).