New expansions of numerical eigenvalues for $-\Delta u=\lambda \rho u$ by nonconforming elements
HTML articles powered by AMS MathViewer
- by Qun Lin, Hung-Tsai Huang and Zi-Cai Li PDF
- Math. Comp. 77 (2008), 2061-2084 Request permission
Abstract:
The paper explores new expansions of the eigenvalues for $-\Delta u=\lambda \rho u$ in $S$ with Dirichlet boundary conditions by the bilinear element (denoted $Q_1$) and three nonconforming elements, the rotated bilinear element (denoted $Q_1^{rot}$), the extension of $Q_1^{rot}$ (denoted $EQ_1^{rot}$) and Wilson’s elements. The expansions indicate that $Q_1$ and $Q_1^{rot}$ provide upper bounds of the eigenvalues, and that $EQ_1^{rot}$ and Wilson’s elements provide lower bounds of the eigenvalues. By extrapolation, the $O(h^4)$ convergence rate can be obtained, where $h$ is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.References
- I. Babuška and J. E. Osborn, Estimates for the errors in eigenvalue and eigenvector approximation by Galerkin methods, with particular attention to the case of multiple eigenvalues, SIAM J. Numer. Anal. 24 (1987), no. 6, 1249–1276. MR 917451, DOI 10.1137/0724082
- I. Babuška and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), no. 186, 275–297. MR 962210, DOI 10.1090/S0025-5718-1989-0962210-8
- I. Babuška and J. Osborn, Eigenvalue problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787. MR 1115240
- H. Blum, Q. Lin, and R. Rannacher, Asymptotic error expansion and Richardson extrapolation for linear finite elements, Numer. Math. 49 (1986), no. 1, 11–37. MR 847015, DOI 10.1007/BF01389427
- J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, DOI 10.1137/0707006
- Françoise Chatelin, Convergence of approximation methods to compute eigenelements of linear operations, SIAM J. Numer. Anal. 10 (1973), 939–948. MR 349004, DOI 10.1137/0710080
- Hong Sen Chen and Bo Li, Superconvergence analysis and error expansion for the Wilson nonconforming finite element, Numer. Math. 69 (1994), no. 2, 125–140. MR 1310313, DOI 10.1007/s002110050084
- George E. Forsythe, Asymptotic lower bounds for the frequencies of certain polygonal membranes, Pacific J. Math. 4 (1954), 467–480. MR 63784
- Jun Hu, Pingbing Ming, and Zhongci Shi, Nonconforming quadrilateral rotated $Q_1$ element for Reissner-Mindlin plate, J. Comput. Math. 21 (2003), no. 1, 25–32. Special issue dedicated to the 80th birthday of Professor Zhou Yulin. MR 1974269
- William G. Kolata, Approximation in variationally posed eigenvalue problems, Numer. Math. 29 (1977/78), no. 2, 159–171. MR 482047, DOI 10.1007/BF01390335
- Zi Cai Li, Combined methods for elliptic equations with singularities, interfaces and infinities, Mathematics and its Applications, vol. 444, Kluwer Academic Publishers, Dordrecht, 1998. MR 1639538, DOI 10.1007/978-1-4613-3338-8
- Lin Qun, Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners, Numer. Math. 58 (1991), no. 6, 631–640. MR 1083525, DOI 10.1007/BF01385645
- Q. Lin and J. Lin, Finite Element Methods; Accuracy and Improvement, Science Press, Beijing, 2006.
- Q. Lin and Q. Zhu, Processing and Post processing for the Finite Element Method (in Chinese), Shanghai Scientific & Technical Press., 1994.
- T. Lü, C.B. Liem and T.M.Shih, The Splitting Extrapolation and Combination Techniques $-$ New Techniques of Parallel Solutions for Multi-dimensional Problems (in Chinese), Scientific Publishers, Beijing, 1998.
- Ping Luo and Qun Lin, High accuracy analysis of the Wilson element, J. Comput. Math. 17 (1999), no. 2, 113–124. MR 1688000
- B. Mercier, J. Osborn, J. Rappaz, and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp. 36 (1981), no. 154, 427–453. MR 606505, DOI 10.1090/S0025-5718-1981-0606505-9
- J. G. Pierce and R. S. Varga, Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems. II. Improved error bounds for eigenfunctions, Numer. Math. 19 (1972), 155–169. MR 323133, DOI 10.1007/BF01402526
- Rolf Rannacher, Nonconforming finite element methods for eigenvalue problems in linear plate theory, Numer. Math. 33 (1979), no. 1, 23–42. MR 545740, DOI 10.1007/BF01396493
- Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
- Hans F. Weinberger, Variational methods for eigenvalue approximation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 15, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Based on a series of lectures presented at the NSF-CBMS Regional Conference on Approximation of Eigenvalues of Differential Operators, Vanderbilt University, Nashville, Tenn., June 26–30, 1972. MR 0400004
- Dong-sheng Wu, Convergence and superconvergence of Hermite bicubic element for eigenvalue problem of the biharmonic equation, J. Comput. Math. 19 (2001), no. 2, 139–142. MR 1816677
- Yi Du Yang, Computable error bounds of finite element approximations for eigenvalue problems, Math. Numer. Sinica 16 (1994), no. 3, 286–295 (Chinese, with English and Chinese summaries); English transl., Chinese J. Numer. Math. Appl. 17 (1995), no. 1, 68–77. MR 1392853
- Yi-du Yang, A posteriori error estimates in Adini finite element for eigenvalue problems, J. Comput. Math. 18 (2000), no. 4, 413–418. MR 1773912
Additional Information
- Qun Lin
- Affiliation: Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 1000080, China
- Email: qlin@lsec.cc.ac.cn
- Hung-Tsai Huang
- Affiliation: Department of Applied Mathematics, I-Shou University, Taiwan 840
- Email: huanght@isu.edu.tw
- Zi-Cai Li
- Affiliation: Department of Applied Mathematics, and Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan 80424
- Email: zcli@math.nsysu.edu.tw
- Received by editor(s): February 24, 2006
- Received by editor(s) in revised form: February 14, 2007
- Published electronically: May 29, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 2061-2084
- MSC (2000): Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-08-02098-X
- MathSciNet review: 2429874