Extrapolation and superconvergence of the Steklov eigenvalue problem. (English) Zbl 1213.65141

Summary: On the basis of a transform lemma, an asymptotic expansion of the bilinear finite element is derived over graded meshes for the Steklov eigenvalue problem, such that the Richardson extrapolation can be applied to increase the accuracy of the approximation, from which the approximation of \(O(h ^{3.5})\) is obtained. In addition, by means of the Rayleigh quotient acceleration technique and an interpolation postprocessing method, the superconvergence of the bilinear finite element is presented over graded meshes for the Steklov eigenvalue problem, and the approximation of \(O(h ^{3})\) is gained. Finally, numerical experiments are provided to demonstrate the theoretical results.


65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65B05 Extrapolation to the limit, deferred corrections
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[1] Armentano, M.G.: The effect of reduced integration in the Steklov eigenvalue problem. Math. Model. Numer. Anal. (M2AN) 38, 27–36 (2004) · Zbl 1077.65115
[2] Babuška, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, Finite Element Methods (Part 1), vol. II, pp. 641–787. North-Holland, Amsterdam (1991)
[3] Bergmann, S., Schiffer, M.: Kernel Function and Elliptic Differential Equations in Mathematical Physics. Academic, New York (1953) · Zbl 0053.39003
[4] Bermúdez, A., Rodríguez, R., Santamarina, D.: A finite element solution of an added mass formulation for coupled fluid-solid vibrations. Numer. Math. 87, 201–227 (2000) · Zbl 0998.76046
[5] Blum, H., Lin, Q., Rannacher, R.: Asymptotic error expansion and Richardson extrapolation for linear finite elements. Numer. Math. 49, 11–38 (1986) · Zbl 0594.65082
[6] Brandts, J.: Superconvergence phenomena in finite element methods. Ph.D. Thesis, Utrecht Univ. (1995)
[7] Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994) · Zbl 0804.65101
[8] Brunner, H., Lin, Y., Zhang, S.: Higher accuracy methods for second-kind Volterra integral equations based on asymptotic expansions of iterated Galerkin methods. J. Integral Equations Appl. 10, 375–396 (1998) · Zbl 0944.65140
[9] Chen, C., Huang, Y.: High Accuracy Theory for Finite Element Methods. Hunan Scientific and Technology, Hunan (1995)
[10] Chen, W.: The analysis of mixed finite element methods for eigenvalue problems. Postdoc. Thesis, Chinese Academy of Sciences (2003)
[11] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) · Zbl 0383.65058
[12] Conca, C., Planchard, J., Vanninathan, M.: Fluid and Periodic Structures. Wiley, New York (1995) · Zbl 0910.76002
[13] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985) · Zbl 0695.35060
[14] Helfrich, P.: Asymptotic expansion for the finite element approximations of parabolic problems. Bonner Math. Schriften 158, 11–30 (1983)
[15] Křížek, M., Neittaanmäki, P.: Bibliography on superconvergence. In: Proc. Conf. Finite Element Methods: Superconvergence, Post-processing and A Posteriori Estimates. Lecture Notes in Pure and Appl. Math., vol. 196, pp. 315–348. Marcel Dekker, New York (1998) · Zbl 0884.00048
[16] Liem, C., Lu, T., Shih, T.: Splitting Extrapoltion Method. World Scientific, River Edge (1995)
[17] Lin, Q.: Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners. Numer. Math. 58, 631–640 (1991) · Zbl 0695.65064
[18] Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement. Science, Beijing (2006)
[19] Lin, Q., Sloan, I.H., Xie, R.: Extrapolation of the iterated-collocation method for integral equations of the second kind. SIAM J. Numer. Anal. 27, 1535–1541 (1990) · Zbl 0724.65128
[20] Lin, Q., Yan, N.: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Publishers, Hebei (1996)
[21] Lin, Q., Zhang, S., Yan, N.: Asymptotic error expansion and defect correction for Sobolev and viscoelasticity type equations. J. Comput. Math. 16, 57–62 (1998) · Zbl 0889.65098
[22] Lin, Q., Zhang, S., Yan, N.: High accuracy analysis for integrodifferential equations. Acta Math. Appl. Sinica 14, 202–211 (1998) · Zbl 0943.65159
[23] Lin, Q., Zhang, S., Yan, N.: An acceleration method for integral equations by using interpolation post-processing. Adv. Comput. Math. 9, 117–128 (1998) · Zbl 0920.65087
[24] Lin, T., Lin, Y., Rao, M., Zhang, S.: Petrov-Galerkin methods for linear Volterra integro-differential equations. SIAM J. Numer. Anal. 38(3), 937–963 (2000) · Zbl 0983.65138
[25] Marchuk, G., Shaidurov, V.: Difference Methods and their Extrapolation. Springer, New York (1983) · Zbl 0511.65076
[26] Strang, G., Fix, G.: An Analysis of the Finite Element Method. Prentice-Hall, New York (1972) · Zbl 0356.65096
[27] Wang, J.: Superconvergence and extrapolation for mixed finite element method on rectangular domains. Math. Comput. 56, 447–503 (1991) · Zbl 0729.65084
[28] Weinberger, H.: Variational Methods for Eigenvalue Approximation. SIAM, Philadelphia (1974) · Zbl 0296.49033
[29] Xu, Y., Zhao, Y.: An extrapolation method for a class of boundary integral equations. Math. Comput. 65, 587–610 (1996) · Zbl 0851.65077
[30] Yan, N., Li, K.: An extrapolation method for BEM. J. Comput. Math. 2, 217–224 (1989) · Zbl 0673.65072
[31] Yang, Y.: An Analysis of the Finite Element Method for Eigenvalue Problems. Guizhou People’s Publishing House, Guizhou (1994)
[32] Zhou, A.: Multi-parameter asymptotic error resolution of the mixed finite element method for the Stokes problem. Math. Model. Numer. Anal. (M2AN) 33, 89–97 (1999) · Zbl 0917.76042
[33] Zhu, Q., Lin, Q.: Superconvergence Theory of Finite Element Methods. Hunan Science, Hunan (1989)
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