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 Free Access

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]**Ivo Babuška,*Homogenization and its application. Mathematical and computational problems*, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 89–116. MR**0502025****[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, and J. K. Lee,*Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems*, Comput. Methods Appl. Mech. Engrg.**11**(1977), no. 2, 175–206. MR**0451771**, https://doi.org/10.1016/0045-7825(77)90058-5**[4]**Carl de Boor,*A bound on the 𝐿_{∞}-norm of 𝐿₂-approximation by splines in terms of a global mesh ratio*, Math. Comp.**30**(1976), no. 136, 765–771. MR**0425432**, https://doi.org/10.1090/S0025-5718-1976-0425432-1**[5]**J. H. Bramble and J. E. Osborn,*Rate of convergence estimates for nonselfadjoint eigenvalue approximations*, Math. Comp.**27**(1973), 525–549. MR**0366029**, https://doi.org/10.1090/S0025-5718-1973-0366029-9**[6]**C. CONUTO, "Eigenvalue approximation by mixed methods,"*Rev. Française Automat. Informat. Recherche Opérationelle Sér. Rouge Anal. Numér*. (To appear.)**[7]**Françoise Chatelin,*La méthode de Galerkin. Ordre de convergence des éléments propres*, C. R. Acad. Sci. Paris Sér. A**278**(1974), 1213–1215 (French). MR**0343592****[8]**Jim Douglas Jr., Todd Dupont, and Lars Wahlbin,*Optimal 𝐿_{∞} error estimates for Galerkin approximations to solutions of two-point boundary value problems*, Math. Comp.**29**(1975), 475–483. MR**0371077**, https://doi.org/10.1090/S0025-5718-1975-0371077-0**[9]**George J. Fix,*Eigenvalue approximation by the finite element method*, Advances in Math.**10**(1973), 300–316. MR**0341900**, https://doi.org/10.1016/0001-8708(73)90113-8**[10]**L. HERRMANN, "Finite element bending analysis for plates,"*J. Engrg. Mech. Div. ASCE*, v. 93, 1967, pp. 13-26.**[11]**William G. Kolata,*Approximation in variationally posed eigenvalue problems*, Numer. Math.**29**(1977/78), no. 2, 159–171. MR**482047**, https://doi.org/10.1007/BF01390335**[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,*Stability of a system of interacting cracks*, Internat. J. Engrg. Sci.**16**(1978), no. 4, 277–285. MR**0495382**, https://doi.org/10.1016/0020-7225(78)90094-0**[17]**J. A. NITSCHE, private communication.**[18]**J. Nitsche and A. Schatz,*On local approximation properties of 𝐿₂-projection on spline-subspaces*, Applicable Anal.**2**(1972), 161–168. Collection of articles dedicated to Wolfgang Haack on the occasion of his 70th birthday. MR**0397268**, https://doi.org/10.1080/00036817208839035**[19]**Joachim A. Nitsche and Alfred H. Schatz,*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937–958. MR**0373325**, https://doi.org/10.1090/S0025-5718-1974-0373325-9**[20]***Finite element bibliography*, IFI/Plenum, New York-London, 1976. Compiled by Douglas Norrie and Gerard de Vries. MR**0445868****[21]**J. N. Reddy and J. T. Oden,*Mixed finite-element approximations of linear boundary-value problems*, Quart. Appl. Math.**33**(1975/76), no. 3, 255–280. MR**0451782**, https://doi.org/10.1090/S0033-569X-1975-0451782-0**[22]**John E. Osborn,*Spectral approximation for compact operators*, Math. Comput.**29**(1975), 712–725. MR**0383117**, https://doi.org/10.1090/S0025-5718-1975-0383117-3**[23]**Robert S. Strichartz,*Multipliers on fractional Sobolev spaces*, J. Math. Mech.**16**(1967), 1031–1060. MR**0215084****[24]**P.-A. Raviart and J. M. Thomas,*Primal hybrid finite element methods for 2nd order elliptic equations*, Math. Comp.**31**(1977), no. 138, 391–413. MR**0431752**, https://doi.org/10.1090/S0025-5718-1977-0431752-8**[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. Višik and G. I. Èskin,*Sobolev-Slobodeckiĭ spaces of variable order with weighted norms, and their applications to mixed boundary value problems*, Sibirsk. Mat. Ž.**9**(1968), 973–997 (Russian). MR**0236696****[29]**André Unterberger,*Sobolev spaces of variable order and problems of convexity for partial differential operators with constant coefficients*, Colloque International, C.N.R.S. sur les Équations aux Dérivées Partielles Linéaires (Univ. Paris-Sud, Orsay, 1972) Soc. Math. France, Paris, 1973, pp. 325–341. Astérisque, 2 et 3. MR**0393774****[30]**André Unterberger,*Espaces de Sobolev d’ordre variable et applications*, Séminaire Goulaouic-Schwartz (1970/1971), Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 5, Centre de Math., École Polytech., Paris, 1971, pp. 19 pp. (unpaged errata following Annex. No. 1) (French). MR**0394180**

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