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Numerical approximation of a singularly perturbed contact problem. (English) Zbl 0955.74048
Summary: As a simplified model for contact problems, we study a mixed Neumann-Robin boundary value problem for the Laplace operator in a smooth domain in \(\mathbb{R}^2\). The Robin condition contains a small parameter \(\varepsilon\) inducing boundary layers of corner type at the transition points. We present an integral equation for the numerical solution of this problem together with estimates of the error. We also investigate an improvement obtained if we add to the discrete space some special functions.

74M15 Contact in solid mechanics
74S15 Boundary element methods applied to problems in solid mechanics
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
Full Text: DOI
[1] Bolley, P.; Camus, J.; Dauge, M., Régularité Gevrey pour le problème de Dirichlet dans des domaines á singularités coniques, Comm. partial diff. eqs., 10, 2, 391-432, (1985) · Zbl 0573.35024
[2] Franzone, P.Colli, Approssimazione mediante il metodo di penalizzazione di problemi misti di Dirichlet-Neumann per operatori lineari ellittici del seconde ordine, Bolletino U.M.I., 7, 4, 229-250, (1973) · Zbl 0266.35024
[3] Costabel, M., Boundary integral operators on Lipschitz domains: elementary results, SIAM J. math. anal., 19, 613-626, (1988) · Zbl 0644.35037
[4] Costabel, M.; Dauge, M., A singularly perturbed mixed boundary value problem, Comm. PDE, 21, 1919-1949, (1996) · Zbl 0879.35017
[5] M. Costabel, M. Dauge and M. Suri, Numerical methods for a singularly perturbed mixed boundary value problem, to appear. · Zbl 0955.74048
[6] Dauge, M., Elliptic boundary value problems in corner domains—smoothness and asymptotics of solutions, () · Zbl 0668.35001
[7] Grisvard, P., Boundary value problems in non-smooth domains, (1985), Pitman London · Zbl 0695.35060
[8] Hackbusch, W., Integral equations—theory and numerical treatment, () · Zbl 0997.65075
[9] Il’in, A.M., Matching of asymptotic expansions of solutions of boundary value problems, () · Zbl 0671.35002
[10] Kellogg, R.B., Boundary layers and corner singularities for a self-adjoint problem, (), 121-149 · Zbl 0824.35006
[11] Kondrat’ev, V.A., Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow math. soc., 16, 227-313, (1967) · Zbl 0194.13405
[12] Muskhelishvili, N.I., Some basic problems of the mathematical theory of elasticity, (1963), Noordhoff Groningen, The Netherlands · Zbl 0124.17404
[13] Nazarov, S.A., Vishik-lyusternik method for elliptic boundary value problems in regions with conical points, I. the problem in a cone, Siberian math. J., 22, 594-611, (1981) · Zbl 0502.35039
[14] Nazarov, S.A., Vishik-lyustemik method for elliptic boundary value problems in regions with conical points, II. the problem in a bounded region, Siberian math. J., 22, 753-769, (1981) · Zbl 0502.35040
[15] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
[16] Volpert, Y., Space enrichment methods for the numerical solution of contact problems, ()
[17] Volpert, Y.; Szabó, T.; Páczelt, I.; Szabó, B., Application of the space enrichment method to problems of mechanical contact, Finite elements anal. des., (1996), to appear
[18] Wendland, W.L., Boundary element methods for elliptic problems, (), 223-276 · Zbl 0712.65099
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