Iteratively solving a kind of Signorini transmission problem in a unbounded domain

Qiya Hu; Dehao Yu

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 4, page 715-726
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Hu, Qiya, and 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 39.4 (2005): 715-726. <http://eudml.org/doc/245582>.

@article{Hu2005,
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.},
author = {Hu, Qiya, Yu, Dehao},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Signorini contact; FEM-BEM coupling; variational inequality; D-N alternation; convergence rate},
language = {eng},
number = {4},
pages = {715-726},
publisher = {EDP-Sciences},
title = {Iteratively solving a kind of Signorini transmission problem in a unbounded domain},
url = {http://eudml.org/doc/245582},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Hu, Qiya
AU - Yu, Dehao
TI - Iteratively solving a kind of Signorini transmission problem in a unbounded domain
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 4
SP - 715
EP - 726
AB - 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.
LA - eng
KW - Signorini contact; FEM-BEM coupling; variational inequality; D-N alternation; convergence rate
UR - http://eudml.org/doc/245582
ER -

References

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  1. [1] C. Carstensen, Interface problem in holonomic elastoplasticity. Math. Methods Appl. Sci. 16 (1993) 819–835. Zbl0792.73017
  2. [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. Zbl0896.65079
  3. [3] C. Carstensen, M. Kuhn and U. Langer, Fast parallel solvers for symmetric boundary element domain decomposition equations. Numer. Math. 79 (1998) 321–347. Zbl0907.65119
  4. [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. Zbl0725.73090
  5. [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. Zbl0741.65084
  6. [6] R. Glowinski, Numerical methods for nonlinear variational problems. Springer-Verlag, New York (1984). Zbl0536.65054MR737005
  7. [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). MR972509
  8. [8] Q. Hu and D. Yu, A solution method for a certain interface problem in unbounded domains. Computing 67 (2001) 119–140. Zbl1109.65309
  9. [9] N. Kikuchi and J. Oden, Contact problem in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia (1988). Zbl0685.73002MR961258
  10. [10] J. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I. Springer-Verlag (1972). Zbl0223.35039MR350177
  11. [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. Zbl0938.65138
  12. [12] J. Necas, Introduction to the theory of nonlinear elliptic equations. Teubner, Texte 52, Leipzig (1983). Zbl0526.35003MR731261
  13. [13] E. Polak, Computational methods in optimization. Academic Press, New York (1971). MR282511
  14. [14] J. Schoberl, Solving the Signorini problem on the basis of domain decomposition techniques. Computing 60 (1998) 323–344. Zbl0915.73077
  15. [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. Zbl0461.65082
  16. [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. Zbl0915.65063
  17. [17] X. Tai and J. Xu, Global convergence of space correction methods for convex optimization problems. Math. Comp. 71 (2002) 105–122. Zbl0985.65065
  18. [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. Zbl0885.35027
  19. [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. Zbl0928.65146
  20. [20] D. Yu, Natural Boundary Integral Method and Its Applications. Science Press/Kluwer Academic Publishers, Beijing/New York (2002). Zbl1028.65129MR1961132

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