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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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 -norm of -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 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 -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)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L15

Retrieve articles in all journals with MSC: 65L15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0501962-0

Article copyright:
© Copyright 1978
American Mathematical Society