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Keywords:
eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound
eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound
Summary:
This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which give at least asymptotically lower bounds of the exact eigenvalues. We present some new numerical experiments for the plate bending problem on a rectangular domain. The main result is that if we know an estimate from below by nonconforming FEM, then by using a postprocessing procedure we can obtain two-sided bounds of the first (essential) eigenvalue. For the other eigenvalues $\lambda _l$, $l = 2,3,\ldots $, we prove and give conditions when this method is applicable. Finally, the numerical results presented and discussed in the paper illustrate the efficiency of our method.
References:
[1] Adini, A., Clough, R.: Analysis of Plate Bending by the Finite Element Method. NSF Report G. 7337 (1961).
[2] Andreev, A. B., Lazarov, R. D., Racheva, M. R.: Postprocessing and higher order convergence of mixed finite element approximations of biharmonic eigenvalue problems. J. Comput. Appl. Math. 182 (2005), 333-349. DOI 10.1016/j.cam.2004.12.015 | MR 2147872
[3] Andreev, A. B., Racheva, M. R.: Superconvergent FE postprocessing for eigenfunctions. C. R. Acad. Bulg. Sci. 55 (2002), 17-22. MR 1885694 | Zbl 1007.65087
[4] Andreev, A. B., Racheva, M. R.: Lower bounds for eigenvalues by nonconforming FEM on convex domain. Application of Mathematics in Technical and Natural Sciences Proceedings of the 2nd international conference, Sozopol, Bulgaria, 2010. AIP Conf. Proc. 1301 Amer. Inst. Phys., Melville (2010), 361-369 M. Todorov et al. (2010), 361-369. MR 2810107 | Zbl 1232.35105
[5] Andreev, A. B., Racheva, M. R.: Properties and estimates of an integral type nonconforming finite element. Large-Scale Scientific Computing 8th international conference, LSSC 2011, Sozopol, Bulgaria, 2011. Lecture Notes in Computer Science 7116, 2012, pp. 252-532 Springer, Berlin I. Lirkov et al. MR 2955161
[6] Andreev, A. B., Racheva, M. R.: Lower bounds for eigenvalues and postprocessing by an integral type nonconforming FEM. Sib. Zh. Vychisl. Mat. 15 235-249 (2012), Russian Numer. Analysis Appl. 5 (2012), 191-203. Zbl 1299.35217
[7] Andreev, A. B., Racheva, M. R., Tsanev, G. S.: A Nonconforming Finite Element with Integral Type Bubble Function. Proceedings of 5th Annual Meeting of the BG. Section of SIAM'10 (2010), 3-6.
[8] Armentano, M. G., Durán, R. G.: Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. ETNA, Electron. Trans. Numer. Anal. (electronic only) 17 (2004), 93-101. MR 2040799 | Zbl 1065.65127
[9] Babuška, I., Kellogg, R. B., Pitkäranta, J.: Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33 (1979), 447-471. DOI 10.1007/BF01399326 | MR 0553353 | Zbl 0423.65057
[10] Babuška, I., Osborn, J.: Eigenvalue problems. Handbook of Numerical Analysis, Vol. II: Finite Element Methods (Part 1) J.-L. Lions, P. G. Ciarlet North-Holland, Amsterdam (1991), 641-787. MR 1115240
[11] Brenner, S. C., Scott, R. L.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15 Springer, New York (1994). DOI 10.1007/978-1-4757-4338-8_7 | MR 1278258 | Zbl 0804.65101
[12] Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications. Vol. 4 North-Holland, Amsterdam (1978). MR 0520174 | Zbl 0383.65058
[13] Ciarlet, P. G.: Basic error estimates for elliptic problems. Handbook of Numerical Analysis. Vol. II: Finite Element Methods (Part 1) P. G. Ciarlet et al. North-Holland, Amsterdam (1991). MR 1115237 | Zbl 0875.65086
[14] Crouzeix, M., Raviart, P. A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Franc. Automat. Inform. Rech. Operat. 7 (1973), 33-75. MR 0343661
[15] Forsythe, G. E.: Asymptotic lower bounds for the fundamental frequency of convex membranes. Pac. J. Math. 5 (1955), 691-702. DOI 10.2140/pjm.1955.5.691 | MR 0073048 | Zbl 0068.10304
[16] Grisvard, P.: Singularities in Boundary Problems. MASSON and Springer, Berlin (1985).
[17] Huang, H. T., Li, Z. C., Lin, Q.: New expansions of numerical eigenvalues by finite elements. J. Comput. Appl. Math. 217 (2008), 9-27. DOI 10.1016/j.cam.2007.06.011 | MR 2427427 | Zbl 1147.65090
[18] Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. Rev. Franc. Automat. Inform. Rech. Operat. {\it 9}, Analyse numer. R-1 9-53 (1975). MR 0423968 | Zbl 0319.73042
[19] Lin, Q., Lin, J. F.: Finite Element Methods: Accuracy and Improvement. Science Press, Beijing (2006).
[20] Lin, Q., Huang, H. T., Li, Z. C.: New expansions of numerical eigenvalues for $-\Delta u=\lambda\rho u$ by nonconforming elements. Math. Comput. 77 (2008), 2061-2084. DOI 10.1090/S0025-5718-08-02098-X | MR 2429874 | Zbl 1198.65228
[21] Lin, Q., Tobiska, L., Zhou, A.: Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25 (2005), 160-181. DOI 10.1093/imanum/drh008 | MR 2110239 | Zbl 1068.65122
[22] Lin, Q., Xie, H.: The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods. Math. Pract. Theory 42 (2012), 219-226 Chinese. MR 3013284 | Zbl 1289.65251
[23] Lin, Q., Xie, H., Luo, F., Li, Y., Yang, Y.: Stokes eigenvalue approximations from below with nonconforming mixed finite element methods. Math. Pract. Theory 40 (2010), 157-168. MR 2768711
[24] Lin, Q., Xie, H., Xu, J.: Lower bounds of the discretization for piecewise polynomials. http://arxiv.org/abs/1106.4395 (2011). MR 3120579
[25] Liu, H. P., Yan, N. N.: Four finite element solutions and comparison of problem for the Poisson equation eigenvalue. Chin. J. Numer. Math. Appl. 27 27-39 (2005), 81-91. MR 2159418 | Zbl 1106.65327
[26] Luo, F., Lin, Q., Xie, H.: Computing the lower and upper bounds of Laplace eigenvalue problem by combining conforming and nonconforming finite element methods. Sci. China, Math. 55 (2012), 1069-1082. DOI 10.1007/s11425-012-4382-2 | MR 2912496 | Zbl 1261.65112
[27] Morley, L. S. D.: The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19 (1968), 149-169. DOI 10.1017/S0001925900004546
[28] Nicaise, S.: A posteriori error estimations of some cell-centered finite volume methods. SIAM J. Numer. Anal. (electronic) 43 (2005), 1481-1503. DOI 10.1137/S0036142903437787 | MR 2182137 | Zbl 1103.65110
[29] Racheva, M. R., Andreev, A. B.: Superconvergence postprocessing for eigenvalues. Comput. Methods Appl. Math. 2 (2002), 171-185. DOI 10.2478/cmam-2002-0011 | MR 1930846 | Zbl 1012.65113
[30] Rannacher, R.: Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33 (1979), 23-42. DOI 10.1007/BF01396493 | MR 0545740 | Zbl 0394.65035
[31] Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equations 8 (1992), 97-111. DOI 10.1002/num.1690080202 | MR 1148797 | Zbl 0742.76051
[32] Shi, Z.-C.: On the error estimates of Morley element. Math. Numer. Sin. Chinese 12 (1990), 113-118 translation in Chinese J. Numer. Math. Appl. 12 (1990), 102-108. MR 1070298 | Zbl 0850.73337
[33] Strang, G., Fix, G. J.: An Analysis of the Finite Element Method. Prentice-Hall Series in Automatic Computation Prentice-Hall, Englewood Cliffs (1973). MR 0443377 | Zbl 0356.65096
[34] Wang, M., Shi, Z.-C., Xu, J.: Some $n$-rectangle nonconforming elements for fourth order elliptic equations. J. Comput. Math. 25 (2007), 408-420. MR 2337403 | Zbl 1142.65451
[35] Wang, L., Wu, Y., Xie, X.: Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems. Numer. Methods Partial Differ. Equations 29 (2013), 721-737. DOI 10.1002/num.21723 | MR 3039784
[36] Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70 (2001), 17-25. DOI 10.1090/S0025-5718-99-01180-1 | MR 1677419 | Zbl 0959.65119
[37] Yang, Y. D.: A posteriori error estimates in Adini finite element for eigenvalue problems. J. Comput. Math. 18 (2000), 413-418. MR 1773912 | Zbl 0957.65092
[38] Yang, Y. D., Zhang, Z. M., Lin, F. B.: Eigenvalue approximation from below using non-conforming finite elements. Sci. China, Math. 53 (2010), 137-150. DOI 10.1007/s11425-009-0198-0 | MR 2594754 | Zbl 1187.65125
[39] Zhang, H. Q., Wang, M.: The Mathematical Theory of Finite Elements. Science Press, Beijing (1991).
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