Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T19:37:57.175Z Has data issue: false hasContentIssue false
  • English
  • Français

C0IPG for a Fourth Order Eigenvalue Problem

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  01 February 2016

Xia Ji ,
Hongrui Geng ,
Jiguang Sun  and
Liwei Xu
Xia Ji*
Affiliation:
LSEC, Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing 100190, P.R. China
Hongrui Geng
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China
Jiguang Sun
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, United States
Liwei Xu
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China Institute of Computing and Data Sciences, Chongqing University, Chongqing 400044, P.R. China
*
*Corresponding author. Email addresses:[email protected] (X. Ji), [email protected] (H. Geng), [email protected] (J. Sun), [email protected] (L. Xu)
Get access

Abstract

This paper concerns numerical computation of a fourth order eigenvalue problem. We first show the well-posedness of the source problem. An interior penalty discontinuous Galerkin method (C0IPG) using Lagrange elements is proposed and its convergence is studied. The method is then used to compute the eigenvalues. We show that the method is spectrally correct and prove the optimal convergence. Numerical examples are presented to validate the theory.

Keywords

Fourth order eigenvalue problemdiscontinuous Galerkin methodspectrum approximation

MSC classification

Secondary: 65N25: Eigenvalue problems 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N22: Solution of discretized equations
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 2 , February 2016 , pp. 393 - 410
Copyright
Copyright © Global-Science Press 2016 

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]Antonietti, P. F., Buffa, A., and Perugia, I., Discontinuous Galerkin approximation of the Laplace eigenproblem, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3483–3503.CrossRefGoogle Scholar
[2]Argyris, J. H., Fried, I., and Scharpf, D. W., The TUBA family of plate elements for the matrix displacement method, Aero. J. Roy. Aero. Soc., 72 (1968), 701–709.Google Scholar
[3]Babuška, I. and Osborn, J., Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, Finite Element Methods (Part 1), Edited by Ciarlet, P.G. and Lions, J.L., Elsevier Science Publishers B.V. (North-Holland), 1991.Google Scholar
[4]Boffi, D., Finite element approximation of eigenvalue problems, Acta Numer., 19 (2010), 1–120.Google Scholar
[5]Brenner, S. C., C0 interior penalty methods, in Frontiers in Numerical Analysis - Durham 2010, Lecture Notes in Computational Science and Engineering 85, Springer-Verlag, 2012, 79–147.Google Scholar
[6]Brenner, S. C. and Sung, L., C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput., 22/23 (2005), 83–118.Google Scholar
[7]Brenner, S. C., Monk, P. B., and Sun, J., C0IPG Method for Biharmonic Eigenvalue Problems, Lecture Notes in Computational Science and Engineering, 106, Springer, 2015, 3–15.Google Scholar
[8]Cakoni, F. and Haddar, H., On the existence of transmission eigenvalues in an inhomogenous medium, Appl. Anal., 88 (2009), 475–493.CrossRefGoogle Scholar
[9]Davis, C. B., A partition of unity method with penalty for fourth order problems, J. Sci. Comput., 60 (2014), 228–248.Google Scholar
[10]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, 40, SIAM, Philadelphia, 2002.CrossRefGoogle Scholar
[11]Descloux, J., Nassif, N., and Rappaz, J., On spectral approximation Part 1. The problem of convergence, RAIRO Anal. Numér., 12 (1978), 97–112.CrossRefGoogle Scholar
[12]Descloux, J., Nassif, N., and Rappaz, J., On spectral approximation Part 2. Error estimates for the Galerkin method convergence, RAIRO Anal. Numér., 12 (1978), 113–119.Google Scholar
[13]Engel, G., Garikipati, K., Hughes, T. J. R., Larson, M. G., Mazzei, L., and Taylor, R. L., Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg., 191 (2002), 3669–3750.Google Scholar
[14]Georgoulis, E. and Houston, P., Discontinuous Galerkin methods for the biharmonic problem, IMA J. Numer. Anal., 29 (2009), 573–594.Google Scholar
[15]Grisvard, P., Elliptic Problems in Nonsmooth Domains, Pitman Advanced Publishing Program, Boston, 1985.Google Scholar
[16]Ishihara, I., A mixed finite element method for the biharmonic eigenvalue problems of plate bending, Publ. Res. Inst. Math. Sci., Kyoto Univ., 14 (1978), 399–414.Google Scholar
[17]Ji, X., Sun, J., and Turner, T., A mixed finite element method for Helmholtz transmission eigenvalues, ACM Transaction on Math. Soft., 38 (2012), Algorithm 922.Google Scholar
[18]Ji, X., Sun, J., and Yang, Y., Optimal penalty parameter for C0 IPDG, Applied Mathematics Letters, 37 (2014), 112–117.Google Scholar
[19]Kato, T., Perturbation Theory of Linear Operators, Springer-Verlag, 1966.Google Scholar
[20]Morley, L., The triangular equilibrium problem in the solution of plate bending problems, Aero. Quart., 19 (1968), 149–169.CrossRefGoogle Scholar
[21]Sun, J., Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860–1874.Google Scholar
[22]Sun, J., A new family of high regularity elements, Numer. Meth. P. D. E., 28 (2012), 1–16.Google Scholar
[23]Wells, G. N. and Dung, N. T., A C0 discontinuous Galerkin formulation for Kirchhoff plates, Comput. Meth. Appl. Mech. Engrg., 196 (2007), 3370–3380.Google Scholar