# zbMATH — the first resource for mathematics

An operator method for a numerical quadrature finite element approximation for a class of second-order elliptic eigenvalue problems in composite structures. (English) Zbl 0836.65113
The authors consider a second-order elliptic eigenvalue problem on a two- dimensional convex polygonal domain, divided in $$M$$ non-overlapping subdomains. The conormal derivative of the unknown function is continuous on the interfaces, while the function itself is discontinuous. The authors study the finite element approximation without and with numerical quadrature.
The method developed by the authors in an earlier paper is refined and extended to the multicomponent structure with discontinuities on the interfaces. Rectangular mesh is admitted, as well as finite elements of higher degree. A nonstandard variational formulation of the eigenvalue problem is used. The authors emphasize the error analysis of the approximate eigenpairs.
Reviewer: P.Burda (Praha)

##### MSC:
 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35P15 Estimates of eigenvalues in context of PDEs
Full Text:
##### References:
 [1] A. B. ANDREEV, V. A. KASCIEVA & M. VANMAELE, Some results in lumped, mass finite-element approximation of eigenvalue problems using numerical quadrature, J. Comp. Appl. Math., 43, 1992, 291-311. Zbl0762.65056 MR1193808 · Zbl 0762.65056 · doi:10.1016/0377-0427(92)90016-Q [2] I. BABUŠKA & J. E. OSBORN, Eigenvalue Problems. In : Handbook of Numerical Analysis, Vol. II, Finite Element Methods (Part 1) (P. G. Ciarlet, J. L. Lions, eds.). Amsterdam: North-Holland, 1991, 641-787. Zbl0875.65087 MR1115240 · Zbl 0875.65087 [3] U. BANERJEE, A note on the effect of numerical quadrature in finite element eigenvalue approximation, Numer. Math., 61, 1992, 145-152. Zbl0748.65078 MR1147574 · Zbl 0748.65078 · doi:10.1007/BF01385502 · eudml:133615 [4] U. BANERJEE & J. E. OSBORN, Estimation of the Effect of Numerical Integration in Finite Element Eigenvalue Approximation, Numer. Math., 56, 1990, 735-762. Zbl0693.65071 MR1035176 · Zbl 0693.65071 · doi:10.1007/BF01405286 · eudml:133425 [5] F. CHATELIN, Spectral Approximation of Linear Operators, New York : Academic Press, 1983. Zbl0517.65036 MR716134 · Zbl 0517.65036 [6] P. G. CIARLET, The Finite Element Method for Elliptic Problems, Amsterdam : North-Holland, 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058 [7] R. DAUTRAY & J.-L. LIONS, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Tome 2. Paris : Masson, 1985. Zbl0749.35005 MR902801 · Zbl 0749.35005 [8] R J. DAVIS & P. RABINOWITZ, Methods of numerical integration, New York : Academic Press, 1975. Zbl0304.65016 MR448814 · Zbl 0304.65016 [9] J. DESCLOUX, N. NASSIF & J. RAPPAZ, On Spectral Approximation. Problem of Convergence, R.A.I.R.O. Numerical Analysis, 12, 1978, 97-112. Zbl0393.65024 MR483400 · Zbl 0393.65024 · eudml:193319 [10] G. J. FIX, Eigenvalue Approximation by the Finite Element Method, Advances in Mathematics, 10, 1973, 300-316. Zbl0257.65086 MR341900 · Zbl 0257.65086 · doi:10.1016/0001-8708(73)90113-8 [11] J. KAČUR & R. VAN KEER, On the Numerical Solution of some Heat Transfer Problems in Multi-component Structures with Non-perfect Thermal Contacts. In : Numerical Methods for Thermal Problems VII (R. W. Lewis, ed.). Swansea : Pineridge Press, 1991, 1378-1388. MR1132506 [12] H. KARDESTUNCER & D. H. NORRIE, Finite Element Handbook, New York : McGraw-Hill Book Comp, 1987. Zbl0638.65076 MR900813 · Zbl 0638.65076 [13] T. KATO, Perturbation Theory for Linear Operators, Berlin : Springer-Verlag, 1976. Zbl0342.47009 MR407617 · Zbl 0342.47009 [14] B. MERCIER, Lectures on Topics in Finite Element Solution of Elliptic Problems, Berlin : Springer-Verlag, 1976. Zbl0445.65100 · Zbl 0445.65100 [15] M. N. ÖZISIK, Heat Conduction, New York John Wiley & Sons, 1980. [16] M. VANMAELE, On optimal and nearly optimal error estimates of a numerical quadrature finite element method for 2nd-order eigenvalue problems with Dirichlet boundary conditions, Simon Stevin, 67, 1992, 121-132. Zbl0802.65106 MR1249049 · Zbl 0802.65106 [17] M. VANMAELE, A numerical quadrature finite element method for 2nd-order eigenvalue problems with Dirichlet-Robin boundary conditions. Proceedings ISNA ’92. Prague, 1994, 269-292. [18] M. VANMAELE & R. VAN KEER, Error estimates for a finite element method with numerical quadrature for a class of elliptic eigenvalue problems. In : Numerical Methods (D. Greenspan, R Rósza, eds.). Colloq. Math. Soc. János Bolyai, 59, 1990, Amsterdam, North-Holland, 267-282. Zbl0760.65096 MR1161236 · Zbl 0760.65096 [19] M. VANMAELE & R. VAN KEER, On a numerical quadrature finite element method for a class of elliptic eigenvalue problems in composite structures, Math. Comp.(submitted). · Zbl 0836.65113 [20] M. VANMAELE & A. ŽENIŠEK, External finite element approximations of eigen-functions in case of multiple eigenvalues. J. Comp. Appl. Math. 50 (to appear). Zbl0811.65090 MR1284251 · Zbl 0811.65090 · doi:10.1016/0377-0427(94)90289-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.