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Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations. (English) Zbl 1274.65296

The authors consider the Steklov eigenvalue problem given by a second-order elliptic equation in a two-dimensional bounded polygonal domain and a Neumann-type boundary condition containing the eigenvalue \(\lambda\). They use four kinds of nonconforming finite elements for its numerical approximation. The main theorem establishes an optimal rate of convergence of the associated normed eigenfunctions on convex and also non-convex domains. Guaranteed lower bounds of particular eigenvalues are derived as well. Numerical tests nicely illustrate the theoretical rate of convergence of the eigenvalues on the L-shape domain.
Reviewer: Michal Krizek

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
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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[1] H. J. Ahn: Vibration of a pendulum consisiting of a bob suspended from a wire: the method of integral equations. Quart. Appl. Math. 39 (1981), 109-117. · Zbl 0458.70018
[2] A. Alonso A.D. Russo: Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods. J. Comput. Appl. Math. 223 (2009), 177-197. · Zbl 1156.65094
[3] A.B. Andreev, T.D. Todorov: Isoparametric finite-element approximation of a Steklov eigenvalue problem. IMA J. Numer. Anal. 24 (2004), 309-322. · Zbl 1069.65120
[4] T. Arbogast, Z. Chen: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comput. 64 (1995), 943-972. · Zbl 0829.65127
[5] M. G. Armentano, R.G. Durán: Asymptotic lower bounds for eigenvalues by non-conforming finite element methods. ETNA, Electron. Trans. Numer. Anal. 17 (2004), 93-101. · Zbl 1065.65127
[6] M. G. Armentano, C. Padra: A posteriori error estimates for the Steklov eigenvalue problem. Appl. Numer. Math. 58 (2008), 593-601. · Zbl 1140.65078
[7] Babuška, I.; Osborn, J., Eigenvalue problems, 641-787 (1991), Amsterdam · Zbl 0875.65087
[8] C. Beattie, F. Goerisch: Methods for computing lower bounds to eigenvalues of self-adjoint operators. Numer. Math. 72 (1995), 143-172. · Zbl 0857.65063
[9] S. Bergman, M. Schiffer: Kernel Functions and Elliptic Differential Equations in Math-ematical Physics. Academic Press, New York, 1953.
[10] A. Bermúdez, R. Rodríguez, D. Santamarina: A finite element solution of an added mass formulation for coupled fluid-solid vibrations. Numer. Math. 87 (2000), 201-227. · Zbl 0998.76046
[11] C. Bernardi, F. Hecht: Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comput. 71 (2002), 1371-1403. · Zbl 1012.65108
[12] J. H. Bramble, J. E. Osborn: Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators. Math. Found. Finite Element Method Applications PDE (A. Aziz, ed.). Academic Press, New York, 1972, pp. 387-408. · Zbl 0264.35055
[13] D. Bucur, I. R. Ionescu: Asymptotic analysis and scaling of friction parameters. Z. Angew. Math. Phys. 57 (2006), 1042-1056. · Zbl 1106.35038
[14] Z. Cai, X. Ye, S. Zhang: Discontinuous Galerkin finite element methods for interface problems: A priori and a posteriori error estimations. SIAM J. Numer. Anal. 49 (2011), 1761-1787. · Zbl 1232.65142
[15] P. G. Ciarlet: Basic error estimates for elliptic problems. In: Part 1. Finite Element Methods. Handbook of Numerical Analysis, Vol. 2 (P. Ciarlet, J.-L. Lions, eds.). North-Holland, 1991, pp. 21-343. · Zbl 0875.65086
[16] C. Conca, J. Planchard, M. Vanninathan: Fluid and Periodic Structures. John Wiley & Sons, Chichester, 1995. · Zbl 0910.76002
[17] M. Crouzeix, P.-A. Raviart: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. Rev. Franc. Automat. Inform. Rech. Operat. 7 (1973), 33-76. · Zbl 0302.65087
[18] N. Dunford, J. T. Schwartz: Linear Operators, Part II: Spectral Theory. Selfadjoint Operators in Hilbert Space. Interscience Publishers/JohnWiley & Sons, New York/London, 1963. · Zbl 0128.34803
[19] F. Goerisch, J. Albrecht: 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. · Zbl 0609.65072
[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) (1990), Boston
[21] Han, H. D.; Guan, Z., An analysis of the boundary element approximation of Steklov eigenvalue problems, 35-51 (1992), River Edge
[22] H. D. Han, Z. Guan, B. He: Boundary element approximation of Steklov eigenvalue problem. Gaoxiao Yingyong Shuxue Xuebao Ser. A 9 (1994), 128-135. (In Chinese.) · Zbl 0812.65092
[23] D. B. Hinton, J. K. Shaw: Differential operators with spectral parameter incompletely in the boundary conditions. Funkc. Ekvacioj, Ser. Int. 33 (1990), 363-385. · Zbl 0715.34133
[24] J. Hu, Y. Huang, Q. Lin: The analysis of the lower approximation of eigenvalues by nonconforming elements. To appear.
[25] J. Huang, T. Lü: The mechanical quadrature methods and their extrapolation for solving BIE of Steklov eigenvalue problems. J. Comput. Math. 22 (2004), 719-726. · Zbl 1069.65123
[26] M. Křížek, H.-G. Roos, W. Chen: Two-sided bounds of the discretization error for finite elements. ESAIM, Math. Model. Numer. Anal. 45 (2011), 915-924. · Zbl 1269.65113
[27] M. Li, Q. Lin, S. Zhang: Extrapolation and superconvergence of the Steklov eigenvalue problem. Adv. Comput. Math. 33 (2010), 25-44. · Zbl 1213.65141
[28] Q. Lin, J. Lin: Finite Element Methods: Accuracy and Improvement. Science Press, Beijing, 2006.
[29] Q. Lin, L. Tobiska, A. Zhou: Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25 (2005), 160-181. · Zbl 1068.65122
[30] Q. Lin, H. Xie, F. Luo, Y. Li, Y. Yang: Stokes eigenvalue approximations from below with nonconforming mixed finite element methods. Math. Pract. Theory 40 (2010), 157-168. · Zbl 1493.65214
[31] R. Rannacher, S. Turek: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equations 8 (1992), 97-111. · Zbl 0742.76051
[32] W. Tang, Z. Guan, H. Han: Boundary element approximation of Steklov eigenvalue problem for Helmholtz equation. J. Comput. Math. 16 (1998), 165-178. · Zbl 0977.65100
[33] L. Wang, X. Xu: Foundation of Mathematics in Finite Element Methods. Scientific and Technical Publishers, Beijing, 2004.
[34] Y. Yang: A posteriori error estimates in Adini finite element for eigenvalue problems. J. Comput. Math. 18 (2000), 413-418. · Zbl 0957.65092
[35] Y. Yang, Z. Chen: The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators. Sci. China, Ser. A 51 (2008), 1232-1242. · Zbl 1153.65055
[36] Y. Yang, Q. Li, S. Li: Nonconforming finite element approximations of the Steklov eigenvalue problem. Appl. Numer. Math. 59 (2009), 2388-2401. · Zbl 1190.65168
[37] Y. Yang, H. Bi: Lower spectral bounds by Wilson’s brick discretization. Appl. Numer. Math. 60 (2010), 782-787. · Zbl 1198.65220
[38] Y. Yang, Z. Zhang, F. Lin: Eigenvalue approximation from below using non-conforming finite elements. Sci. China Math. 53 (2010), 137-150. · Zbl 1187.65125
[39] Z. Zhang, Y. Yang, Z. Chen: Eigenvalue approximation from below byWilson’s element. Math. Numer. Sin. 29 (2007), 319-321. (In Chinese.) · Zbl 1142.65435
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