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.

Classification: 65N30, 65R20, 73C50

Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {39}, number = {4}, year = {2005}, pages = {715-726}, doi = {10.1051/m2an:2005031}, zbl = {pre02213936}, mrnumber = {2165676}, language = {en}, url = {http://www.numdam.org/item/M2AN_2005__39_4_715_0} }

Hu, Qiya; Yu, Dehao. Iteratively solving a kind of Signorini transmission problem in a unbounded domain. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 4, pp. 715-726. doi : 10.1051/m2an:2005031. http://www.numdam.org/item/M2AN_2005__39_4_715_0/

[1] Interface problem in holonomic elastoplasticity. Math. Methods Appl. Sci. 16 (1993) 819-835. | Zbl 0792.73017

,[2] FEM and BEM coupling for a nonlinear transmission problem with Signorini contact. SIAM J. Numer. Anal. 34 (1997) 1845-1864. | Zbl 0896.65079

and ,[3] Fast parallel solvers for symmetric boundary element domain decomposition equations. Numer. Math. 79 (1998) 321-347. | Zbl 0907.65119

, and ,[4] Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal. 27 (1990) 1212-1226. | Zbl 0725.73090

and ,[5] On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in ${R}^{2}$. Numer. Math. 61(1992) 171-214. | Zbl 0741.65084

and ,[6] Numerical methods for nonlinear variational problems. Springer-Verlag, New York (1984). | MR 737005 | Zbl 0536.65054

,[7] | MR 972509

, , and , Eds., Proc. of the the First international symposium on domain decomposition methods for PDEs. SIAM Philadelphia (1988).[8] A solution method for a certain interface problem in unbounded domains. Computing 67 (2001) 119-140. | Zbl 1109.65309

and ,[9] Contact problem in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia (1988). | MR 961258 | Zbl 0685.73002

and ,[10] Non-homogeneous boundary value problems and applications, Vol. I. Springer-Verlag (1972). | MR 350177 | Zbl 0223.35039

and ,[11] An adaptive two-level method for the coupling of nonlinear FEM-BEM equations, SIAM J. Numer. Anal. 36 (1999) 1001-1021. | Zbl 0938.65138

and ,[12] Introduction to the theory of nonlinear elliptic equations. Teubner, Texte 52, Leipzig (1983). | MR 731261 | Zbl 0526.35003

,[13] Computational methods in optimization. Academic Press, New York (1971). | MR 282511

,[14] Solving the Signorini problem on the basis of domain decomposition techniques. Computing 60 (1998) 323-344. | Zbl 0915.73077

,[15] On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Methods Appl. Sci. 1 (1979) 265-321. | Zbl 0461.65082

, and ,[16] Rate of convergence of some space decomposition methods for linear and nonlinear problems. SIAM J. Numer. Anal. 35 (1998) 1558-1570. | Zbl 0915.65063

and ,[17] Global convergence of space correction methods for convex optimization problems. Math. Comp. 71 (2002) 105-122. | Zbl 0985.65065

and ,[18] The relation between the Steklov-Poincare operator, the natural integral operator and Green functions. Chinese J. Numer. Math. Appl. 17 (1995) 95-106. | Zbl 0885.35027

,[19] Discretization of non-overlapping domain decomposition method for unbounded domains and its convergence.Chinese J. Numer. Math. Appl. 18 (1996) 93-102. | Zbl 0928.65146

,[20] Natural Boundary Integral Method and Its Applications. Science Press/Kluwer Academic Publishers, Beijing/New York (2002). | MR 1961132 | Zbl 1028.65129

,