This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := . The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincaré-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart-Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory.
@article{M2AN_2011__45_4_779_0,
author = {Gatica, Gabriel N. and Maischak, Matthias and Stephan, Ernst P.},
title = {Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {45},
year = {2011},
pages = {779-802},
doi = {10.1051/m2an/2010102},
mrnumber = {2804659},
zbl = {1267.74110},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2011__45_4_779_0}
}
Gatica, Gabriel N.; Maischak, Matthias; Stephan, Ernst P. Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) pp. 779-802. doi : 10.1051/m2an/2010102. http://gdmltest.u-ga.fr/item/M2AN_2011__45_4_779_0/
[1] and , Survey Lectures on the Mathematical Foundations of the Finite Element Method. Academic Press, New York (1972) 3-359. | MR 421106 | Zbl 0268.65052
[2] and , On the mixed finite element method with Lagrange multipliers. Numer. Methods Partial Differ. Equ. 19 (2003) 192-210. | MR 1958060 | Zbl 1021.65056
[3] and , Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002
[4] , and , Error estimates for the finite element solution of variational inequalities. Numer. Math. 28 (1977) 431-443. | MR 448949 | Zbl 0369.65030
[5] , Interface problem in holonomic elastoplasticity. Math. Methods Appl. Sci. 16 (1993) 819-835. | MR 1245631 | Zbl 0792.73017
[6] and , FEM and BEM coupling for a nonlinear transmission problem with Signorini contact. SIAM J. Numer. Anal. 34 (1997) 1845-1864. | MR 1472200 | Zbl 0896.65079
[7] and , Mathematical Analysis and Numerical Methods for Science and Technology 4. Springer (1990). | MR 1081946 | Zbl 0755.35001
[8] and , Inequalities in Mechanics and Physics. Springer, Berlin (1976). | MR 521262 | Zbl 0331.35002
[9] and , Analyse Convexe et Problèmes Variationnels. Études mathématiques, Dunod, Gauthier-Villars, Paris-Bruxelles-Montreal (1974). | MR 463993 | Zbl 0281.49001
[10] , Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28 (1974) 963-971. | MR 391502 | Zbl 0297.65061
[11] and , Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems. Appl. Anal. 63 (1996) 39-75. | MR 1622612 | Zbl 0865.65077
[12] , and , Numerical Analysis of Variational Inequalities, Studies in Mathematics and its Applications 8. North-Holland Publishing Co., Amsterdam-New York (1981). | MR 635927 | Zbl 0463.65046
[13] , , and , Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences 66. Springer-Verlag (1988). | MR 952855 | Zbl 0654.73019
[14] , Linear Partial Differential Operators. Springer-Verlag, Berlin (1969). | Zbl 0321.35001
[15] and , Contact Problems in Elasticity: a Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988). | MR 961258 | Zbl 0685.73002
[16] and , An Introduction to Variational Inequalities and their Applications. Academic Press (1980). | MR 567696 | Zbl 0457.35001
[17] and , Non-Homogeneous Boundary Value Problems and Applications I. Springer-Verlag, Berlin (1972). | MR 350177 | Zbl 0223.35039
[18] and , Mixed and Hybrid Methods, in Handbook of Numerical Analysis II, P.G. Ciarlet and J.-L. Lions Eds., North-Holland, Amsterdam (1991) 523-639. | MR 1115239 | Zbl 0875.65090
[19] , Finite Element Procedures for Contact-Impact Problems. Oxford University Press (1993).