×

Eigenvalue approximation from below using non-conforming finite elements. (English) Zbl 1187.65125

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: 6mm
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 \(EQ^{\text{rot}}_1\) element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements;
3)
the explanation of the phenomena that numerical eigenvalues \(\lambda _{1,h }\) and \(\lambda _{3,h }\) of the non-conforming \(Q^{\text{rot}}_1\) element approximate the true eigenvalues from below for the L-shaped domain.
Finally, several unsolved problems are listed.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
65N15 Error bounds for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Armentano M G, Duran R G. Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. Electron Trans Numer Anal, 2004, 17: 92–101 · Zbl 1065.65127
[2] 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
[3] Chen C M, Huang Y Q. High Accuracy Theory of Finite Element Methods. Changsha: Hunan Science & Technology Publisher, 1995
[4] Chen Z, Yang Y D. The global stress superconvergence of Wilson’s brick. Numer Math Chinese Univ, 2005, 27: 301–305
[5] 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 · Zbl 0875.65086
[6] Lin Q, Lin J F. Finite Element Methods: Accuracy and Improvement. Beijing: Science Press, 2006
[7] 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 · Zbl 1068.65122
[8] 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 · Zbl 1106.65327
[9] Rannacher R. Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer Math, 1979, 33: 23–42 · Zbl 0394.65035
[10] Rannacher R, Turek S. Simple nonconforming quadrilateral Stokes element. Numer Methods Partial Differential Equations, 1992, 8: 97–111 · Zbl 0742.76051
[11] Shi Z C. A remark on the optimal order of convergence of Wilson’s nonconforming element. Math Numer Sin, 1986, 2: 159–163 · Zbl 0607.65071
[12] Strang G, Fix G J. An Analysis of the Finite Element Method. Englewood Cliffs, NJ: Prentice-Hall, 1973 · Zbl 0356.65096
[13] Weinstein A, Chien W Z. On the vibrations of a clamped plate under tension. Quart Appl Math, 1943, 1: 61–68 · Zbl 0063.08203
[14] Yang Y D. A posteriori error estimates in Adini finite element for eigenvalue problems. J Comput Math, 2000, 18: 413–418 · Zbl 0957.65092
[15] 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 · Zbl 1153.65055
[16] 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 · Zbl 1142.65435
[17] Zhu Q D, Lin Q. Superconvergence Theory of Finite Element Methods. Changsha: Hunan Science & Technology Publisher, 1989
[18] Zienkiewicz O C, Cheung Y K. The Finite Element Method in Structrural and Continuum Mechanics. New York: McGraw-Hill, 1967 · Zbl 0189.24902
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.