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Keywords:
Steklov eigenvalue problem; nonconforming finite element; error estimate; lower bound of the eigenvalues
Summary:
The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, $Q_{1}^{\rm rot}$, $EQ_{1}^{\rm rot}$ and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results.
References:
[1] Ahn, H. J.: Vibration of a pendulum consisiting of a bob suspended from a wire: the method of integral equations. Quart. Appl. Math. 39 (1981), 109-117. MR 0613954
[2] Russo, A. Alonso A. D.: Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods. J. Comput. Appl. Math. 223 (2009), 177-197. DOI 10.1016/j.cam.2008.01.008 | MR 2463110
[3] Andreev, A. B., Todorov, T. D.: Isoparametric finite-element approximation of a Steklov eigenvalue problem. IMA J. Numer. Anal. 24 (2004), 309-322. DOI 10.1093/imanum/24.2.309 | MR 2046179 | Zbl 1069.65120
[4] Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comput. 64 (1995), 943-972. MR 1303084 | Zbl 0829.65127
[5] Armentano, M. G., Durán, R. G.: Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. ETNA, Electron. Trans. Numer. Anal. 17 (2004), 93-101. MR 2040799 | Zbl 1065.65127
[6] Armentano, M. G., Padra, C.: A posteriori error estimates for the Steklov eigenvalue problem. Appl. Numer. Math. 58 (2008), 593-601. DOI 10.1016/j.apnum.2007.01.011 | MR 2407734 | Zbl 1140.65078
[7] Babuška, I., Osborn, J.: Eigenvalue problems. In: Finite Element Methods (Part 1). Handbook of Numerical Analysis, Vol. 2 North-Holland Amsterdam (1991), 641-787. MR 1115240
[8] Beattie, C., Goerisch, F.: Methods for computing lower bounds to eigenvalues of self-adjoint operators. Numer. Math. 72 (1995), 143-172. DOI 10.1007/s002110050164 | MR 1362258 | Zbl 0857.65063
[9] Bergman, S., Schiffer, M.: Kernel Functions and Elliptic Differential Equations in Mathematical Physics. Academic Press New York (1953). MR 0054140 | Zbl 0053.39003
[10] Bermúdez, A., Rodríguez, R., Santamarina, D.: A finite element solution of an added mass formulation for coupled fluid-solid vibrations. Numer. Math. 87 (2000), 201-227. DOI 10.1007/s002110000175 | MR 1804656 | Zbl 0998.76046
[11] Bernardi, C., Hecht, F.: Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comput. 71 (2002), 1371-1403. DOI 10.1090/S0025-5718-01-01401-6 | MR 1933036 | Zbl 1012.65108
[12] Bramble, J. H., Osborn, J. E.: Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators. Math. Found. Finite Element Method Applications PDE A. Aziz Academic Press New York (1972), 387-408. MR 0431740 | Zbl 0264.35055
[13] Bucur, D., Ionescu, I. R.: Asymptotic analysis and scaling of friction parameters. Z. Angew. Math. Phys. 57 (2006), 1042-1056. DOI 10.1007/s00033-006-0070-9 | MR 2279256 | Zbl 1106.35038
[14] Cai, Z., Ye, X., Zhang, S.: Discontinuous Galerkin finite element methods for interface problems: A priori and a posteriori error estimations. SIAM J. Numer. Anal. 49 (2011), 1761-1787. DOI 10.1137/100805133 | MR 2837483 | Zbl 1232.65142
[15] Ciarlet, P. G.: Basic error estimates for elliptic problems. In: Part 1. Finite Element Methods. Handbook of Numerical Analysis, Vol. 2 P. Ciarlet, J.-L. Lions North-Holland (1991), 21-343. DOI 10.1016/S1570-8659(05)80039-0 | MR 1115237 | Zbl 0875.65086
[16] Conca, C., Planchard, J., Vanninathan, M.: Fluid and Periodic Structures. John Wiley & Sons Chichester (1995). MR 1652238
[17] 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), 33-76. MR 0343661
[18] Dunford, N., Schwartz, J. T.: Linear Operators, Part II: Spectral Theory. Selfadjoint Operators in Hilbert Space. Interscience Publishers/John Wiley & Sons New York/London (1963). MR 1009163
[19] Goerisch, F., Albrecht, J.: The Convergence of a New Method for Calculating Lower Bounds to Eigenvalues, Equadiff 6 (Brno, 1985). Lecture Notes in Math. Vol. 1192. Springer Berlin (1986). MR 0877140
[20] Goerisch, F., He, Z.: The Determination of Guaranteed Bounds to Eigenvalues with the Use of Variational Methods. I. Computer Arithmetic and Self-validating Numerical Methods (Basel, 1989), Notes Rep. Math. Sci. Engrg., 7. Academic Press Boston (1990). MR 1104000
[21] Han, H. D., Guan, Z.: An analysis of the boundary element approximation of Steklov eigenvalue problems. In: Numerical Methods for Partial Differential Equations World Scientific River Edge (1992), 35-51. MR 1160822
[22] Han, H. D., Guan, Z., He, B.: Boundary element approximation of Steklov eigenvalue problem. Gaoxiao Yingyong Shuxue Xuebao Ser. A 9 (1994), 128-135 Chinese. MR 1293212
[23] Hinton, D. B., Shaw, J. K.: Differential operators with spectral parameter incompletely in the boundary conditions. Funkc. Ekvacioj, Ser. Int. 33 (1990), 363-385. MR 1086767 | Zbl 0715.34133
[24] Hu, J., Huang, Y., Lin, Q.: The analysis of the lower approximation of eigenvalues by nonconforming elements. (to appear).
[25] Huang, J., Lü, T.: The mechanical quadrature methods and their extrapolation for solving BIE of Steklov eigenvalue problems. J. Comput. Math. 22 (2004), 719-726. MR 2080438 | Zbl 1069.65123
[26] Křížek, M., Roos, H.-G., Chen, W.: Two-sided bounds of the discretization error for finite elements. ESAIM, Math. Model. Numer. Anal. 45 (2011), 915-924 (2011). DOI 10.1051/m2an/2011003 | MR 2817550 | Zbl 1269.65113
[27] Li, M., Lin, Q., Zhang, S.: Extrapolation and superconvergence of the Steklov eigenvalue problem. Adv. Comput. Math. 33 (2010), 25-44. DOI 10.1007/s10444-009-9118-7 | MR 2645290 | Zbl 1213.65141
[28] Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement. Science Press Beijing (2006).
[29] 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
[30] 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
[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] Tang, W., Guan, Z., Han, H.: Boundary element approximation of Steklov eigenvalue problem for Helmholtz equation. J. Comput. Math. 16 (1998), 165-178. MR 1610674 | Zbl 0977.65100
[33] Wang, L., Xu, X.: Foundation of Mathematics in Finite Element Methods. Scientific and Technical Publishers Beijing (2004).
[34] Yang, Y.: A posteriori error estimates in Adini finite element for eigenvalue problems. J. Comput. Math. 18 (2000), 413-418. MR 1773912 | Zbl 0957.65092
[35] Yang, Y., Chen, Z.: The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators. Sci. China, Ser. A 51 (2008), 1232-1242. DOI 10.1007/s11425-008-0002-6 | MR 2417491 | Zbl 1153.65055
[36] Yang, Y., Li, Q., Li, S.: Nonconforming finite element approximations of the Steklov eigenvalue problem. Appl. Numer. Math. 59 (2009), 2388-2401. DOI 10.1016/j.apnum.2009.04.005 | MR 2553141 | Zbl 1212.65435
[37] Yang, Y., Bi, H.: Lower spectral bounds by Wilson's brick discretization. Appl. Numer. Math. 60 (2010), 782-787. DOI 10.1016/j.apnum.2010.03.019 | MR 2647432 | Zbl 1198.65220
[38] Yang, Y., Zhang, Z., Lin, F.: 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, Z., Yang, Y., Chen, Z.: Eigenvalue approximation from below by Wilson's element. Math. Numer. Sin. 29 (2007), 319-321 Chinese. MR 2370469 | Zbl 1142.65435
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