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A domain decomposition method for solving a Helmholtz-like problem in elasticity based on the Wilson nonconforming element. (English) Zbl 0877.73061
A parallelizable iterative procedure based on a domain decomposition technique is proposed and analyzed for a sequence of elliptic systems with first-order absorbing boundary conditions. This sequence of systems, which are not coercive and have characteristics similar to the Helmholtz equation, describes the motion of a nearly elastic solid in the frequency domain. As an application, the procedure is used to solve an approximation to elliptic systems by Wilson non-conforming finite elements. The convergence of the procedure is demonstrated, and the rate of convergence is obtained when the domain can be decomposed into subdomains consisting of an individual element associated with the Wilson finite element method. The hybridization of the Wilson elements is essentially used in the construction of the discrete procedure.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity
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[1] R. A. ADAMS, 1975, Sobolev Spaces, Academic Press, New York. Zbl0314.46030 MR450957 · Zbl 0314.46030
[2] L. BERS, F. JOHN and M. SCHECHTER, 1964, Partial Differential Equations, John Wiley & Sons, New York. Zbl0126.00207 MR162045 · Zbl 0126.00207
[3] P. G. CIARLET, 1978, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam. Zbl0383.65058 MR520174 · Zbl 0383.65058
[4] B. E. J. DAHLBERG, C. E. KENIG and G. C. VERCHOTA, 1988, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J., 57, pp. 795-818. Zbl0699.35073 MR975122 · Zbl 0699.35073
[5] R. DAUTRAY and J. L. LIONS, 1990, Mathematical Analysis and Numencal Methods for Science and Technology, I, Springer-Verlag, New York. Zbl0683.35001 · Zbl 0683.35001
[6] B. DESPRÉS, 1991, Méthodes de décomposition de domaines pour les problèmes de propagation d’ondes en régime harmonique, Ph. D. Thesis, Université Paris IX Dauphine, UER Mathématiques de la Decision. Zbl0849.65085 · Zbl 0849.65085
[7] B. DESPRÉS, P. JOLY and J. E. ROBERTS, A domain decomposition method for the harmonie Maxwell equations, Itérative Methods in Linear Algebra, Elsevier Science Publishers B. V. (North-Holland), Amsterdam, pp. 475-484, R. Beauwens and P. de Groen, eds. Zbl0785.65117 MR1159757 · Zbl 0785.65117
[8] [8] J. DOUGLAS Jr, P. J. S. PAES LEME, J. E. ROBERTS and J. WANG, 1993, A parallel iterative procedure applicable to the approximate solution of second order partial differential e-quations by mixed finite element methods, Numer. Math., 65, pp.95-108. Zbl0813.65122 MR1217441 · Zbl 0813.65122
[9] J. DOUGLAS Jr and J. E. ROBERTS, 1982, Mixed finite element methods for second order elliptic problems, Matemática Aplicada e Computacional, 1, pp. 91-103. Zbl0482.65057 MR667620 · Zbl 0482.65057
[10] G. DUVAUT and J. L. LIONS, 1976, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin. Zbl0331.35002 MR521262 · Zbl 0331.35002
[11] X. FENG, 1992, On miscible displacement in porous media and absorbing boundary conditions for electromagnetic wave propagation and on elastic and nearly elastic waves in the frequency domam, Ph. D. Thesis, Purdue University, 1992.
[12] X. FENG, A mixed finite element domam decomposition method for nearly elastic waves in the frequency domain (submitted).
[13] X. FENG, A domain decomposition method for convection-dominated convection-diffusion equations, preprint.
[14] P. GRISVARD, 1992, Singularities in Boundary Value Problems, Research Notes in Applied Mathematics, Vol. 22, Springer-Verlag and Masson. Zbl0766.35001 MR1173209 · Zbl 0766.35001
[15] F. JOHN, 1982, Partial Differential Equations, Fourth Edition, Springer-Verlag, New York. MR831655
[16] C. E. KENIG, 1994, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conference Series in Mathematics, No. 83, American Mathematical Society. Zbl0812.35001 MR1282720 · Zbl 0812.35001
[17] S. KIM, 1994, A parallelizable iterative procedure for the Helmholtz problem, Appl. Numer. Math., 14, pp. 435-449. Zbl0805.65100 MR1285471 · Zbl 0805.65100
[18] V. D. KUPRADZE, 1965, Potential Methods in the Theory of Elasticity, Israel Program for Scientiflc Translations, Jerusalem. Zbl0188.56901 MR223128 · Zbl 0188.56901
[19] P. LESAINT, 1976, On the convergence of Wilson’s nonconforing element for solving the elastic problem, Comput. Methods Appl. Mech. Engrg, 7, pp. 1-16. Zbl0345.65058 MR455479 · Zbl 0345.65058
[20] J. L LIONS, 1955, Contributions à un problème de M. M. Picone, Ann. Mat. Pura e Appl., 41, pp. 201-215. Zbl0075.10103 MR89978 · Zbl 0075.10103
[21] J. L. LIONS and E. MAGENES, 1972, Nonhomogeneous Boundary Value Problems and Applications, Vol I, Springer-Verlag, New York. Zbl0223.35039 · Zbl 0223.35039
[22] P. L. LIONS, 1988, 1988, On the Schwartz alternatmg method I, III, First and Third International Symposium on Domain Decomposition Method for Partial Differential Equations, SIAM, Philadelphia. MR972510
[23] [23] L. D. MARINI and A. QUARTERONI, 1989, A relaxation procedure for domain decomposition methods using finite elements, Numer. Math., 55, pp. 575-598. Zbl0661.65111 MR998911 · Zbl 0661.65111
[24] [24] J. A. NITSCHE, 1981, On Korn’s second inequality, R.A.I.R.O Anal. Numér., 15, pp. 237-248. Zbl0467.35019 MR631678 · Zbl 0467.35019
[25] C. L. RAVAZZOLI, J. DOUGLAS Jr, J. E. SANTOS and D. SHEEN, 1992, On the solution of the equations of motion for nearly elastic solids in the frequency domain, Proceedings of the IV Reunion de Trabajo en Procesamiento de la Información y Control, Centro de Cálculo Cientifico, Comisión Nacional de Energía Atómica, Buenos Aires, Argentina, November 1991, or Technical Report #164, Center for Applied Mathematics, Purdue University.
[26] A. SCHATZ, 1974, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp, 28, pp. 959-962. Zbl0321.65059 MR373326 · Zbl 0321.65059
[27] J.-M. THOMAS, 1977, Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes, Thèse d’État, Université Pierre et Marie Curie, Paris.
[28] J. E. WHITE, 1965, Seismic Waves, Radiation, Transmission and Attenuation, McGraw-Hill.
[29] E. L. WILSON, R. L. TAYLOR, W. P. DOHERTY and J. GHABOUSSI, 1971, Incompatible displacement models, Symposium on Numerical and Computer Methods in Structural Engineering, O.N.R., University of Illinois.
[30] J. XU, 1992, Iterative methods by space decomposition and subspace correction, SIAM Review, 34, pp, 581-613. Zbl0788.65037 MR1193013 · Zbl 0788.65037
[31] K. YOSIDA, 1980, Functional Analysis, Springer-Verlag, Berlin-New York. Zbl0435.46002 MR617913 · Zbl 0435.46002
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