A domain decomposition method for solving a Helmholtz-like problem in elasticity based on the Wilson nonconforming element
Bennethum, Lynn Schreyer ; Feng, Xiaobing
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 31 (1997), p. 1-25 / Harvested from Numdam
Publié le : 1997-01-01
@article{M2AN_1997__31_1_1_0,
     author = {Bennethum, Lynn Schreyer and Feng, Xiaobing},
     title = {A domain decomposition method for solving a Helmholtz-like problem in elasticity based on the Wilson nonconforming element},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {31},
     year = {1997},
     pages = {1-25},
     mrnumber = {1432850},
     zbl = {0877.73061},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_1997__31_1_1_0}
}
Bennethum, Lynn Schreyer; Feng, Xiaobing. A domain decomposition method for solving a Helmholtz-like problem in elasticity based on the Wilson nonconforming element. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 31 (1997) pp. 1-25. http://gdmltest.u-ga.fr/item/M2AN_1997__31_1_1_0/

[1] R. A. Adams, 1975, Sobolev Spaces, Academic Press, New York. | MR 450957 | Zbl 0314.46030

[2] L. Bers, F. John and M. Schechter, 1964, Partial Differential Equations, John Wiley & Sons, New York. | MR 162045 | Zbl 0126.00207

[3] P. G. Ciarlet, 1978, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam. | MR 520174 | 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. | MR 975122 | 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.

[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. | 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. | MR 1159757 | Zbl 0785.65117

[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. | MR 1217441 | 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. | MR 667620 | Zbl 0482.65057

[10] G. Duvaut and J. L. Lions, 1976, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin. | MR 521262 | 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. | MR 1173209 | Zbl 0766.35001

[15] F. John, 1982, Partial Differential Equations, Fourth Edition, Springer-Verlag, New York. | MR 831655

[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. | MR 1282720 | Zbl 0812.35001

[17] S. Kim, 1994, A parallelizable iterative procedure for the Helmholtz problem, Appl. Numer. Math., 14, pp. 435-449. | MR 1285471 | Zbl 0805.65100

[18] V. D. Kupradze, 1965, Potential Methods in the Theory of Elasticity, Israel Program for Scientiflc Translations, Jerusalem. | MR 223128 | 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. | MR 455479 | 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. | MR 89978 | Zbl 0075.10103

[21] J. L. Lions and E. Magenes, 1972, Nonhomogeneous Boundary Value Problems and Applications, Vol I, Springer-Verlag, New York. | 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. | MR 972510

[23] L. D. Marini and A. Quarteroni, 1989, A relaxation procedure for domain decomposition methods using finite elements, Numer. Math., 55, pp. 575-598. | MR 998911 | Zbl 0661.65111

[24] J. A. Nitsche, 1981, On Korn's second inequality, R.A.I.R.O Anal. Numér., 15, pp. 237-248. | Numdam | MR 631678 | 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. | MR 373326 | 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. | MR 1193013 | Zbl 0788.65037

[31] K. Yosida, 1980, Functional Analysis, Springer-Verlag, Berlin-New York. | MR 617913 | Zbl 0435.46002