Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T05:01:10.460Z Has data issue: false hasContentIssue false
  • English
  • Français

Extremal Eigenvalues of the Sturm-Liouville Problems with Discontinuous Coefficients

Published online by Cambridge University Press:  28 May 2015

Shuangbing Guo ,
Dingfang Li ,
Hui Feng  and
Xiliang Lu
Shuangbing Guo*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
Dingfang Li*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
Hui Feng*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
Xiliang Lu*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
*
Corresponding author.Email address:iceflyingsouth@163.com
Corresponding author.Email address:dfli@whu.edu.en
Corresponding author.Email address:hui_fengl63@tom.com
Corresponding author.Email address:xllv.math@whu.edu.cn
Get access

Abstract

In this paper, an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated. Liouville transformation is applied to change the problem into an equivalent minimization problem. Finite element method is proposed and the convergence for the finite element solution is established. A monotonic decreasing algorithm is presented to solve the extremal eigenvalue problem. A global convergence for the algorithm in the continuous case is proved. A few numerical results are given to depict the efficiency of the method.

Keywords

65L6065L1534B09Extremal eigenvalue problemSturm-Liouville problemfinite element methodconvergence analysis
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 4 , November 2013 , pp. 657 - 684
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Allaire, G., Aubry, S. and Jouve, F., Eigenfrequency optimization in optimal design, Comput. Methods Appl. Mech. Eng., 190(28) (2001), pp. 3565–3579.CrossRefGoogle Scholar
[2]Alvino, A., Trombetti, G. and Lions, P. L., On optimization problems with prescribed rearrangements, Nonlinear Anal. Theory Methods Appl., 13(2) (1989), pp. 185–220.CrossRefGoogle Scholar
[3]Babuska, I. and Osborn, J., Numerical treatment of eigenvalue problems for differential equations with discontinuous coefficients, Math. Comput., 32(144) (1978), pp. 991–1023.CrossRefGoogle Scholar
[4]Bajalinov, E. B., Linear-Fractional Programming: Theory, Methods, Applications and Software, Kluwer Academic, 2003.CrossRefGoogle Scholar
[5]Bendsøe, M., Optimization of Structural Topology, Shape and Material, Springer, New York, 1995.CrossRefGoogle Scholar
[6]Bielak, J., Some remarks on bounds to eigenvalues of Sturm-Liouville problems with discontinuous coefficients, Zeitschrift Angewandte Mathematik und Physik (ZAMP), 32 (1981), pp. 647–657.Google Scholar
[7]Birkhoff, G., De Boor, C., Swartz, B. and Wendroff, B., Rayleigh-Ritz approximation by piece-wise cubic polynomials, SIAM J. Numer. Anal., 3(2) (1966), pp. 188–203.CrossRefGoogle Scholar
[8]Ciarlet, P., The Finite Element Method for Elliptic Problems, North Holland, 1978.Google Scholar
[9]Conca, C., Mahadevan, R. and Sanz, L., An extremal eigenvalue problem for a two-phase conductor in a ball, Appl. Math. Optim., 60(2) (2009), pp. 173–184.CrossRefGoogle Scholar
[10]Conca, C., Mahadevan, R., and Sanz, L., Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball, ESAIM: Proceedings, 27 (2009), pp. 311–321.Google Scholar
[11]Courant, R. and Hilbert, D., Methods of Mathematical Physics, Wiley-Interscience, New York, 1962.Google Scholar
[12]Cox, S. and Lipton, R., Extremal eigenvalue problems for two-phase conductors, Archive Rational Mech. Anal., 136(2) (1996), pp. 101–117.CrossRefGoogle Scholar
[13]Cox, S., The two phase drum with the deepest bass note, Japan J. Indus. Appl. Math., 8(3) (1991), pp. 345–355.CrossRefGoogle Scholar
[14]Cox, S. and Mclaughlin, J., Extremal eigenvalue problems for composite membranes, I, Appl. Math. Optim., 22(1) (1990), pp. 153–167.Google Scholar
[15]Cox, S. and Mclaughlin, J., Extremal eigenvalue problems for composite membranes, II, Appl. Math. Optim., 22(2) (1990), pp. 169–187.Google Scholar
[16]Egorov, Y. and Kondratiev, V., On Spectral Theory of Elliptic Operators, Birkhauser, New York, 1996.CrossRefGoogle Scholar
[17]Friedland, S., Extremal eigenvalue problems defined for certain classes of functions, Archive Rational Mech. Anal., 67(1) (1977), pp. 73–81.CrossRefGoogle Scholar
[18]Henrot, A., Extremum Problems for Eigenvalues of Elliptic Operators, Birkhauser, Basel, 2006.CrossRefGoogle Scholar
[19]Horgan, C. and Nemat-Nasser, S., Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients, Zeitschrift Angewandte Mathematik und Physik (ZAMP), 30 (1979), pp. 77–86.Google Scholar
[20]Jouron, C., Sur un probléme d’optimisation ou la contrainte porte sur la fréquence fondamentale, RAIRO Anal. Numér., 12 (1978), pp. 349–375.CrossRefGoogle Scholar
[21]Kao, C. Y. and Su, S., Efficient rearrangement algorithms for shape optimization on elliptic eigenvalue problems, J. Sci. Comput., July 2012, pp. 1–21.Google Scholar
[22]Kesavan, S., Topics in Functional Analysis and Applications, Wiley, New York, 1989.Google Scholar
[23]Krein, M. G., On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, American Mathematical Society Translations, 1 (1955), pp. 163–187.Google Scholar
[24]Liang, K., Lu, X. and Yang, J. Z., Finite element approximation to the extremal eigenvalue problem for inhomogenous materials, subbmitted.Google Scholar
[25]Mcnabb, A., Anderssen, R. and Lapwood, E., Asymptotic behavior of the eigenvalues of a Sturm-Liouville system with discontinuous coefficients, J. Math. Anal. Appl., 54(3) (1976), pp. 741– 751.CrossRefGoogle Scholar
[26]Murat, F. and Tartar, L., Calculus of variations and homogenization, Topics in the MathEmatical Modelling Of Composite Materials, Progr. Nonlinear Differential Equations Appl., 31 (1997), Birkhauser, Boston, pp. 139–173.Google Scholar
[27]Nemat-Nasser, S. and Fu, C., Harmonic waves in layered composites: bounds on frequencies, J. Appl. Mech., 41(1) (1974), pp. 288–290.CrossRefGoogle Scholar
[28]Nemat-Nasser, S. and Minagawa, S., Harmonic waves in layered composites: comparison among several schemes, J. Appl. Mech., 42(3) (1975), pp. 699–704.CrossRefGoogle Scholar
[29]Osher, S. and Santosa, F., Level set methods for optimization problems involving geometry and constraints: I, frequencies of a two-density inhomogeneous drum, J. Comput. Phys., 171(1) (2001), pp. 272–288.CrossRefGoogle Scholar
[30]Vainikko, G., On the rate of convergence of certain approximation methods of Galerkin type in an eigenvalue problem, Amer. Math. Soc. Transl., 86 (1970), pp. 249–258.Google Scholar