Iteratively solving a kind of Signorini transmission problem in a unbounded domain
ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 715-726.

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.

DOI : 10.1051/m2an:2005031
Classification : 65N30, 65R20, 73C50
Mots clés : Signorini contact, FEM-BEM coupling, variational inequality, D-N alternation, convergence rate
@article{M2AN_2005__39_4_715_0,
     author = {Hu, Qiya and Yu, Dehao},
     title = {Iteratively solving a kind of {Signorini} transmission problem in a unbounded domain},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {715--726},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {4},
     year = {2005},
     doi = {10.1051/m2an:2005031},
     mrnumber = {2165676},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2005031/}
}
TY  - JOUR
AU  - Hu, Qiya
AU  - Yu, Dehao
TI  - Iteratively solving a kind of Signorini transmission problem in a unbounded domain
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2005
SP  - 715
EP  - 726
VL  - 39
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2005031/
DO  - 10.1051/m2an:2005031
LA  - en
ID  - M2AN_2005__39_4_715_0
ER  - 
%0 Journal Article
%A Hu, Qiya
%A Yu, Dehao
%T Iteratively solving a kind of Signorini transmission problem in a unbounded domain
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2005
%P 715-726
%V 39
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2005031/
%R 10.1051/m2an:2005031
%G en
%F M2AN_2005__39_4_715_0
Hu, Qiya; Yu, Dehao. Iteratively solving a kind of Signorini transmission problem in a unbounded domain. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 715-726. doi : 10.1051/m2an:2005031. http://www.numdam.org/articles/10.1051/m2an:2005031/

[1] C. Carstensen, Interface problem in holonomic elastoplasticity. Math. Methods Appl. Sci. 16 (1993) 819-835. | Zbl

[2] C. Carstensen and J. Gwinner, FEM and BEM coupling for a nonlinear transmission problem with Signorini contact. SIAM J. Numer. Anal. 34 (1997) 1845-1864. | Zbl

[3] C. Carstensen, M. Kuhn and U. Langer, Fast parallel solvers for symmetric boundary element domain decomposition equations. Numer. Math. 79 (1998) 321-347. | Zbl

[4] M. Costabel and E. Stephan, Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal. 27 (1990) 1212-1226. | Zbl

[5] 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. | Zbl

[6] R. Glowinski, Numerical methods for nonlinear variational problems. Springer-Verlag, New York (1984). | MR | Zbl

[7] 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). | MR

[8] Q. Hu and D. Yu, A solution method for a certain interface problem in unbounded domains. Computing 67 (2001) 119-140. | Zbl

[9] N. Kikuchi and J. Oden, Contact problem in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia (1988). | MR | Zbl

[10] J. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I. Springer-Verlag (1972). | MR | Zbl

[11] 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. | Zbl

[12] J. Necas, Introduction to the theory of nonlinear elliptic equations. Teubner, Texte 52, Leipzig (1983). | MR | Zbl

[13] E. Polak, Computational methods in optimization. Academic Press, New York (1971). | MR

[14] J. Schoberl, Solving the Signorini problem on the basis of domain decomposition techniques. Computing 60 (1998) 323-344. | Zbl

[15] E. Stephan, W. Wendland and G. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Methods Appl. Sci. 1 (1979) 265-321. | Zbl

[16] X. Tai and M. Espedal, Rate of convergence of some space decomposition methods for linear and nonlinear problems. SIAM J. Numer. Anal. 35 (1998) 1558-1570. | Zbl

[17] X. Tai and J. Xu, Global convergence of space correction methods for convex optimization problems. Math. Comp. 71 (2002) 105-122. | Zbl

[18] D. Yu, The relation between the Steklov-Poincare operator, the natural integral operator and Green functions. Chinese J. Numer. Math. Appl. 17 (1995) 95-106. | Zbl

[19] D. Yu, Discretization of non-overlapping domain decomposition method for unbounded domains and its convergence.Chinese J. Numer. Math. Appl. 18 (1996) 93-102. | Zbl

[20] D. Yu, Natural Boundary Integral Method and Its Applications. Science Press/Kluwer Academic Publishers, Beijing/New York (2002). | MR | Zbl

Cité par Sources :