Abstract
This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators, with emphasis on obtaining lower bounds. In addition, this article also contains some new materials for eigenvalue approximations of the Laplace operator, which include: 1) the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below; 2) the proof of the fact that the non-conforming EQ rot1 element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements; 3) the explanation of the phenomena that numerical eigenvalues λ 1,h and λ 3,h of the non-conforming Q rot1 element approximate the true eigenvalues from below for the L-shaped domain. Finally, we list several unsolved problems.
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References
Armentano M G, Duran R G. Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. Electron Trans Numer Anal, 2004, 17: 92–101
Babuška I, Osborn J. Eigenvalue Problems. In: Finite Element Methods (Part 1), Ciarlet P G, Lions J L, eds. Handbook of Numerical Analysis, Vol. 2. North-Holand: Elsevier Science Publishers, 1991, 640–787
Chen C M, Huang Y Q. High Accuracy Theory of Finite Element Methods. Changsha: Hunan Science & Technology Publisher, 1995
Chen Z, Yang Y D. The global stress superconvergence of Wilson’s brick. Numer Math Chinese Univ, 2005, 27: 301–305
Ciarlet P G. Basic Error Estimates for Elliptic Problems. In: Finite Element Methods (Part 1), Ciarlet P G, Lions J L, eds. Handbook of Numerical Analysis, Vol. 2. North-Holand: Elsevier Science Publishers, 1991, 21–343
Lin Q, Lin J F. Finite Element Methods: Accuracy and Improvement. Beijing: Science Press, 2006
Lin Q, Tobiska L, Zhou A. Superconvergence and extrapolation of nonconforming low order finite elements applied to the poisson equation. IMA J Numer Anal, 2005, 25: 160–181
Liu H P, Yan N N. Four finite element solutions and comparison of problem for the poisson equation eigenvalue. Chinese J Numer Meth Comput Appl, 2005, 2: 81–91
Rannacher R. Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer Math, 1979, 33: 23–42
Rannacher R, Turek S. Simple nonconforming quadrilateral Stokes element. Numer Methods Partial Differential Equations, 1992, 8: 97–111
Shi Z C. A remark on the optimal order of convergence of Wilson’s nonconforming element. Math Numer Sin, 1986, 2: 159–163
Strang G, Fix G J. An Analysis of the Finite Element Method. Englewood Cliffs, NJ: Prentice-Hall, 1973
Weinstein A, Chien W Z. On the vibrations of a clamped plate under tension. Quart Appl Math, 1943, 1: 61–68
Yang Y D. A posteriori error estimates in Adini finite element for eigenvalue problems. J Comput Math, 2000, 18: 413–418
Yang Y D, Chen Z. The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators. Sci China Ser A, 2008, 51: 1232–1242
Zhang Z M, Yang Y D, Chen Z. Eigenvalue approximation from below by Wilson’s element. Chinese J Numer Math Appl, 2007, 29: 81–84
Zhu Q D, Lin Q. Superconvergence Theory of Finite Element Methods. Changsha: Hunan Science & Technology Publisher, 1989
Zienkiewicz O C, Cheung Y K. The Finite Element Method in Structrural and Continuum Mechanics. New York: McGraw-Hill, 1967
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This work was supported by National Natural Science Foundation of China (Grant No. 10761003) and the US National Science Foundation (Grant No. DMS-0612908).
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Yang, Y., Zhang, Z. & Lin, F. Eigenvalue approximation from below using non-conforming finite elements. Sci. China Ser. A-Math. 53, 137–150 (2010). https://doi.org/10.1007/s11425-009-0198-0
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DOI: https://doi.org/10.1007/s11425-009-0198-0