On the Dirichlet problem in elasticity for a domain exterior to an arc. (English) Zbl 0742.73026

The Dirichlet problem for the two-dimensional linear elasticity in the domain exterior to an open arc in the plane is considered. The problem is reduced to a system of boundary integral equations with the unknown density function being the jump of stresses across the arc. Existence, uniqueness and regularity conditions are established in appropriate Sobolev spaces. The asymptotic expansions concerning the singular behaviour for the solution near the tips of the arc are obtained. An augmented Galerkin procedure is used for the corresponding boundary integral equations to obtain a quasi-optimal rate of convergence for the approximate solutions.


74S15 Boundary element methods applied to problems in solid mechanics
74B99 Elastic materials
74H99 Dynamical problems in solid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
35B65 Smoothness and regularity of solutions to PDEs
74R99 Fracture and damage
35C20 Asymptotic expansions of solutions to PDEs
Full Text: DOI


[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] Ahner, J. F.; Hsiao, G. C., On the two-dimensional exterior boundary-value problems of elasticity, SIAM J. Appl. Math., 31, 677-685 (1976) · Zbl 0355.73019
[3] Baiocchi, C.; Capelo, A., Variational and Quasivariational Inequalities, Applications to Free-Boundary Problems (1984), Wiley: Wiley Chichester · Zbl 0551.49007
[4] Costabel, M., boundary integral operators on curved polygons, Ann. Mat. Pura Appl., 33, 305-326 (1983) · Zbl 0533.45009
[5] Costabel, M.; Stephan, E. P., Curvature terms in the asymptotic expansion for solutions of boundary integral equations on curved polygons, J. Integral Equations, 5, 353-371 (1983) · Zbl 0538.35022
[6] Costabel, M.; Stephan, E. P., The method of Mellin transformation for boundary integral equations on curves with corners, (Gerasoulis, A.; Vichnevetsky, R., Numerical Solution of Singular Integral Equations (1984), IMACS: IMACS New Brunswick, NJ), 95-102
[7] Costabel, M.; Stephan, E. P., A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl., 106, 367-413 (1985) · Zbl 0597.35021
[8] Costabel, M.; Stephan, E. P., Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, (Fiszdon, W.; Wilmański, K., Mathematical Models and Methods in Mechanics. Mathematical Models and Methods in Mechanics, Banach Center Publ., 15 (1985), PWN: PWN Warsaw), 175-251
[9] Costabel, M.; Stephan, E. P., Duality estimates for the numerical approximation of boundary integral equations, Numer. Math., 54, 339-353 (1988) · Zbl 0663.65141
[10] Costabel, M.; Stephan, E. P.; Wendland, W. L., On the boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10, 197-242 (1983) · Zbl 0563.45007
[11] Eskin, G. I., Boundary Problems for Elliptic Pseudo-differential Operators, (Transl. Math. Monographs, 52 (1981), Amer. Mathematical Soc.,: Amer. Mathematical Soc., Providence, RI) · Zbl 0458.35002
[12] Fichera, G., Linear elliptic equations of higher order in two independent variables and singular integral equations, (Langer, R. E., Proc. Conf. Partial Differential Equations and Conf. Mechanics (1961), Univ. Wisconsin Press), 55-80
[13] Fichera, G., Existence theorems in elasticity. Unilateral constraints in elasticity, (Flügge, S., Handbuch der Physik, VIa/2 (1972), Springer: Springer Berlin), 347-424
[14] Hörmander, L., Linear Partial Differential Operators (1969), Springer: Springer Berlin · Zbl 0177.36401
[15] Hsiao, G. C., Boundary element methods for exterior problems in elasticity and fluid mechanics, (Whiteman, J. R., Mathematics of Finite Elements and Applications VI (1988), Academic Press: Academic Press London), 323-341
[16] Hsiao, G. C.; MacCamy, R. C., Solution of boundary value problems by integral equations of the first kind, SIAM Rev., 15, 687-705 (1973) · Zbl 0235.45006
[17] Hsiao, G. C.; Kopp, P.; Wendland, W. L., Some applications of a Galerkin-collocation method for boundary integral equations of the first kind, Math. Methods Appl. Sci., 6, 280-325 (1984) · Zbl 0546.65091
[18] Hsiao, G. C.; Stephan, E. P.; Wendland, W. L., An integral equation formulation for a boundary value problem of elasticity in the domain exterior to an arc, (Grisvard, P.; Wendland, W.; Whiteman, J. R., Singularities and Constructive Methods for their Treatment. Singularities and Constructive Methods for their Treatment, Lecture Notes in Math., 1121 (1985), Springer: Springer Berlin), 153-165
[19] Hsiao, G. C.; Wendland, W. L., A finite element method for some integral equations of the first kind, J. Math. Appl., 58, 449-481 (1977) · Zbl 0352.45016
[20] Hsiao, G. C.; Wendland, W. L., The Aubin-Nitsche lemma for integral equations, J. Integral Equations, 3, 299-315 (1981) · Zbl 0478.45004
[21] Hsiao, G. C.; Wendland, W. L., On a boundary integral method for some exterior problems in elasticity, Proc. Tbilisi Univ., 257, 31-60 (1985) · Zbl 0624.73096
[22] Kondratiev, V. A., Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 16, 227-313 (1967) · Zbl 0194.13405
[23] Kupradze, V. D., Potential Methods in the Theory of Elasticity (1965), Israel Program Scientific Transl. · Zbl 0188.56901
[24] Lamp, U.; Schleicher, K.-T.; Stephan, E. P.; Wendland, W. L., Galerkin collocation for an improved boundary element method for a plane mixed boundary value problem, Computing, 33, 269-296 (1984) · Zbl 0546.65080
[25] Lions, I. L.; Magenes, E., Non-homogeneous Boundary Value Problems and Applications I (1972), Springer: Springer Berlin · Zbl 0223.35039
[26] Panagiotopoulos, P. D., Inequality Problems in Mechanics, and Applications, Convex and Nonconvex Energy Functions (1985), Birkhäuser: Birkhäuser Boston · Zbl 0579.73014
[27] Ruotsalainen, K., On the convergence of some boundary element methods in the plane, (Doctoral Thesis (1987), Univ. Yyväskylä: Univ. Yyväskylä Finland) · Zbl 0807.65121
[28] Seelev, R., Topics in pseudo-differential operators, (Nirenberg, L., Pseudo-Differential Operators (1969), CIME: CIME Cremonese), 169-305
[29] Stephan, E. P., Boundary integral equations for mixed boundary value problems, screen and transmission problems in \(R^3\), (Habilitationsschrift (1984), Techn. Univ. Darmstadt)
[30] Stephan, E. P.; Wendland, W. L., Remarks to Galerkin and least squares methods with finite elements for general elliptic problems, (Everitt, W. N.; Sleeman, B. D., Ordinary and Partial Differential Equations. Ordinary and Partial Differential Equations, Lecture Notes in Math., 564 (1976), Springer: Springer Berlin). (Everitt, W. N.; Sleeman, B. D., Manuscripta Geodaetica, 1 (1976)), 93-123 · Zbl 0353.65067
[31] Stephan, E. P.; Wendland, W. L., Boundary element method for membrane and torsion crack problems, Comput. Methods Appl. Mech. Engrg., 36, 3, 331-358 (1983) · Zbl 0488.73096
[32] Stephan, E. P.; Wendland, W. L., An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems, Appl. Anal., 18, 183-219 (1984) · Zbl 0582.73093
[34] Wendland, W. L., I. Asymptotic convergence of boundary element methods; II. Integral equation methods for mixed boundary value problems, (Babuška, I.; Liu, I.-P.; Osborn, J., Lectures on the Numerical Solution of Partial Differential Equations (1981), Univ. of Maryland: Univ. of Maryland College Park, MD), 453-528
[35] Wendland, W. L., On applications and the convergence of boundary integral methods, (Baker, C. T.; Miller, G. F., Treatment of Integral Equations by Numerical Methods (1982), Academic Press: Academic Press London), 465-476
[36] Wendland, W. L.; Stephan, E.; Hsiao, G. C., On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Methods Appl. Sci., 1, 265-321 (1979) · Zbl 0461.65082
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