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Numerical treatment of eigenvalue problems for differential equations with discontinuous coefficients


Authors: I. Babuška and J. E. Osborn
Journal: Math. Comp. 32 (1978), 991-1023
MSC: Primary 65L15
DOI: https://doi.org/10.1090/S0025-5718-1978-0501962-0
MathSciNet review: 0501962
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Abstract: The eigenvalues of a second order differential equation are approximated by "factoring" the second order equations into a first order system and then applying the Ritz-Galerkin method to this system. Convergence results and error estimates are derived. These error estimates are based on the application of Sobolev spaces with variable order.


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  • [1] I. BABUŠKA, "Homogenization and its applications. Mathematical and computational problems," Numerical Solution of Partial Differential Equations-III (B. Hubbard, Editor), Academic Press, New York, 1976, pp. 89-116. MR 0502025 (58:19215)
  • [2] I. BABUŠKA & A. K. AZIZ, "Survey lectures on the mathematical foundations of the finite element method," The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Editor), Academic Press, New York, 1973, pp. 5-359.
  • [3] I. BABUŠKA, J. T. ODEN & J. K. LEE, "Mixed-hybrid finite element approximations of second order elliptic boundary-value problems," Comput. Methods Appl. Mech. Engrg., v. 11, 1977, pp. 175-206. MR 0451771 (56:10053)
  • [4] C. de BOOR, "A bound on the $ {L_\infty }$-norm of $ {L_2}$-approximation by splines in terms of a global mesh ratio," Math. Comp., v. 30, 1976, pp. 765-771. MR 0425432 (54:13387)
  • [5] J. H. BRAMBLE & J. E. OSBORN, "Rate of convergence estimates for non-selfadjoint eigenvalue approximations," Math. Comp., v. 27, 1973, pp. 525-549. MR 0366029 (51:2280)
  • [6] C. CONUTO, "Eigenvalue approximation by mixed methods," Rev. Française Automat. Informat. Recherche Opérationelle Sér. Rouge Anal. Numér. (To appear.)
  • [7] F. CHATELIN, "La méthode de Galerkin. Ordre de convergence des éléments propres, C. R. Acad. Sci. Paris Sér. A, v. 278, 1974, pp. 1213-1215. MR 0343592 (49:8332)
  • [8] J. DOUGLAS, T. DUPONT & L. WAHLBIN, "Optimal $ {L_\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems," Math. Comp., v. 29, 1975, pp. 475-483. MR 0371077 (51:7298)
  • [9] G. J. FIX, "Eigenvalue approximation by the finite element method," Advances in Math., v. 10, 1973, pp. 300-316. MR 0341900 (49:6646)
  • [10] L. HERRMANN, "Finite element bending analysis for plates," J. Engrg. Mech. Div. ASCE, v. 93, 1967, pp. 13-26.
  • [11] W. G. KOLATA, "Approximation in variationally posed eigenvalue problems," Numer. Math., v. 29, 1978, pp. 159-171. MR 482047 (80a:49077)
  • [12] S. NEMAT-NASSER, "General variational methods for elastic waves in composites," J. Elasticity, v. 2, 1972, pp. 73-90.
  • [13] S. NEMAT-NASSER, "Harmonic waves in layered composites," J. Appl. Mech., v. 39, 1972, pp. 850-852.
  • [14] S. NEMAT-NASSER, "General variational principles in nonlinear and linear elasticity with applications," Mechanics Today, vol. 1, Pergamon Press, New York, 1974, pp. 214-261.
  • [15] S. NEMAT-NASSER & F. FU, "Harmonic waves in layered composites: bounds on frequencies," J. Appl. Mech., v. 41, 1974, pp. 288-290.
  • [16] S. NEMAT-NASSER & S. MINOGAWA, "Harmonic waves in layered composites: comparison among several schemes." (Preprint.) MR 0495382 (58:14094)
  • [17] J. A. NITSCHE, private communication.
  • [18] J. S. NITSCHE & A. H. SCHATZ, "On local approximation properties of $ {L_2}$-projections on spline subspaces," Applicable Anal., v. 2, 1972, pp. 161-168. MR 0397268 (53:1127)
  • [19] J. A. NITSCHE & A. H. SCHATZ, "Interior estimates for Ritz-Galerkin methods," Math. Comp., v. 28, 1974, pp. 937-958. MR 0373325 (51:9525)
  • [20] D. NORRIE & G. de VRIES, Finite Element Bibliography, Plenum Press, New York, 1976. MR 0445868 (56:4201)
  • [21] J. T. ODEN & J. N. REDDY, "Mathematical theory of mixed finite element approximations," Quart. Appl. Math., v. 33, 1975, pp. 255-280. MR 0451782 (56:10064)
  • [22] J. E. OSBORN, "Spectral approximation for compact operators," Math. Comp., v. 29, 1975, pp. 712-725. MR 0383117 (52:3998)
  • [23] R. STRICHARTZ, "Multipliers on fractional Sobolev spaces," J. Math. Mech., v. 16, 1967, pp. 1031-1060. MR 0215084 (35:5927)
  • [24] P. A. RAVIART & J. M. THOMAS, "Primal hybrid finite element methods for 2nd order elliptic equations," Math. Comp., v. 31, 1977, pp. 391-413. MR 0431752 (55:4747)
  • [25] M. I. VISHIK, "The Sobolev-Slobodetski spaces of changing order with weighted norms and applications to elliptic boundary value problems of mixed type," Partial Differential Equations, "Nauka", Moscow, 1970, pp. 71-76.
  • [26] M. I. VISHIK & G. I. ESKIN, "Equations in convolutions in a bounded region," Russian Math. Surveys, v. 20, 1965, pp. 85-151.
  • [27] M. I. VISHIK & G. I. ESKIN, "Elliptic equations in convolution in a bounded domain and their applications," Russian Math. Surveys, v. 22, 1967, pp. 13-75.
  • [28] M. I. VISHIK & G. I. ESKIN, "Sobolev-Slobodecky spaces of variable order with weighted norms and their applications to mixed boundary value problems," Sibirsk. Mat. Ž., v. 9, 1968, pp. 973-997. MR 0236696 (38:4991)
  • [29] A. UNTERBERGER, "Sobolev spaces of variable order and problems of convexity for partial differential operators with constant coefficients," Colloq. Internat. CNRS sur les Équations aux Dérivées Partielles Linéaires (University Paris-Sud, Orsay, 1972), pp. 325-341. MR 0393774 (52:14583)
  • [30] A. UNTERBERGER, "Espaces de Sobolev d'ordre variable et applications," Séminaire, Goulaouic-Schwartz 1970-1971, Equations aux Dérivées Partielles et Analyse Fonctionnelle, Exposé no. 5, Centre de Math., École Polytéch., Paris, 1971. MR 0394180 (52:14984)

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DOI: https://doi.org/10.1090/S0025-5718-1978-0501962-0
Article copyright: © Copyright 1978 American Mathematical Society

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