Hettich, R.; Haaren, E.; Ries, M.; Still, G. Accurate numerical approximations of eigenfrequencies and eigenfunctions of elliptic membranes. (English) Zbl 0626.73098 Z. Angew. Math. Mech. 67, 589-597 (1987). In earlier papers [e.g. the first author and P. Zencke, Lect. Notes Control Inf. Sci. 66, 132-155 (1985; Zbl 0561.90084)] a defect- minimization method was proposed to compute approximate eigenfunctions of membranes by means of a parametric semi-infinite optimization problem. The method gives approximations and error bounds of eigenvalues and eigenfunctions. An algorithm based on this method has been implemented for the special case of elliptic membranes with variable eccentricity. In this paper we show that by this method we obtain very accurate results even for eccentricities near one. In addition the method is very appropriate to study the behavior of eigenfunctions. Cited in 5 Documents MSC: 74P99 Optimization problems in solid mechanics 65K10 Numerical optimization and variational techniques 74K15 Membranes 65K05 Numerical mathematical programming methods 49R50 Variational methods for eigenvalues of operators (MSC2000) Keywords:Mathieu equation; defect-minimization method; parametric semi-infinite optimization problem; error bounds of eigenvalues; eigenfunctions; elliptic membranes; variable eccentricity Citations:Zbl 0561.90084 PDFBibTeX XMLCite \textit{R. Hettich} et al., Z. Angew. Math. Mech. 67, 589--597 (1987; Zbl 0626.73098) Full Text: DOI References: [1] ; , Handbook of Mathematical Functions, Nat. Bur. Standards, Washington, DC 1964. [2] On the computation of membrane-eigenvalues by semi-infinite programming, in: ; (eds.), Infinite Programming, Springer Lecture Notes in Economics and Mathem. Sciences, Berlin-Heidelberg-New York 1985, pp. 79–89. [3] ; , Local aspects of a method for solving membrane-eigenvalue problems by parametric semi-infinite programming, to appear. · Zbl 0641.90078 [4] ; , Two case-studies in parametric semi-infinite programming, in: ; , (eds.), Systems and Optimization, Springer Lecture Notes in Control and Information Sciences, Berlin-Heidelberg-New York 1984, pp. 132 to 155. [5] Mifflin, Math. Programming 28 pp 50– (1984) [6] Moler, SIAM J. Numer. Anal. 5 pp 64– (1968) [7] Tables Relating to Mathieu Functions, Nat. Bur. Standards, Appl. Math. Ser. 59, Washington, DC 1967. [8] The Group, J. Engineering Math. 7 pp 39– (1973) [9] Troesch, Math. Comput. 27 pp 755– (1973) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.