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A two-grid discretization scheme for the Steklov eigenvalue problem. (English) Zbl 1220.65160
The authors discuss a two-grid discretization scheme for the Steklov eigenvalue problem. The solution of the Steklov eigenvalue problem on a fine grid is reduced to the solution of the Steklov eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on a fine grid. Numerical experiments are performed to confirm the theoretical results.

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
65N15 Error bounds for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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[1] Alonso, A., Russo, A.D.: Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods. J. Comput. Appl. Math. 223, 177–197 (2009) · Zbl 1156.65094
[2] Andreev, A.B., Todorov, T.D.: Isoparametric finite-element approximation of a Steklov eigenvalue problem. IMA J. Numer. Anal. 24, 309–322 (2004) · Zbl 1069.65120
[3] Armentano, M.G.: The effect of reduced integration in the Steklov eigenvalue problem. Math. Mod. Numer. Anal. 38, 27–36 (2004) · Zbl 1077.65115
[4] Armentano, M.G., Padra, C.: A posteriori error estimates for the Steklov eigenvalue problem. Appl. Numer. Math. 58, 93–601 (2008) · Zbl 1140.65078
[5] Babuska, I., Osborn, J.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52, 275–297 (1989) · Zbl 0675.65108
[6] Babuska, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Finite Element Methods (Part 1). Handbook of Numerical Analysis, vol. 2, pp. 640–787. Elsevier Science Publishers, North-Holand (1991)
[7] Bergamann, S., Schiffer, M.: Kernel Functions and Elliptic Differential Equations in Mathematics Physics. Academic Press, San Diego (1953)
[8] Bermudez, A., Rodriguez, 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
[9] Bramble, J.H., Osborn, J.E.: Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators. In: Aziz, A.K. (ed.) Math. Foundations of the Finite Element Method with Applications to PDE, pp. 387–408. Academic Press, San Diego (1972) · Zbl 0264.35055
[10] Chen, H.J., Liu, F., Zhou, A.H.: A two-scale higher-order finite element discretization for Schrödinger Equation. J. Comput. Math. 27, 315–337 (2009) · Zbl 1212.65432
[11] Chien, C.S., Jeng, B.W.: A two-grid finite element discretization scheme for nonlinear eigenvalue problems. Comput. Methods 1951–1955 (2006)
[12] Chien, C.S., Jeng, B.W.: A two-grid finite element discretization scheme for semilinear elliptic eigenvalue problems. SIAM J. Sci. Comput. 27, 1287–1304 (2006) · Zbl 1095.65100
[13] Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Finite Element Methods (Part 1). Handbook of Numerical Analysis, vol. 2, pp. 21–343. Elsevier Science Publishers, North-Holand (1991)
[14] Conca, C., Planchard, J., Vanninathanm, M.: Fluid and Periodic Structures. Wiley, New York (1995)
[15] Gong, X.G., Shen, L.H., Zhang, D., Zhou, A.H.: Finite element approximations for Schrödinger Equations with applications to electronic structure computations. J. Comput. Math. 26, 310–323 (2008) · Zbl 1174.65047
[16] He, Y.N., Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms for the stokes problem. Numer. Math. 109, 415–434 (2008) · Zbl 1145.65097
[17] Huang, J., Lü, T.: The mechanical quadrature methods and their extrapolation for solving BIE of Steklov eigenvalue problems. J. Comput. Math. 22(5), 719–726 (2004) · Zbl 1069.65123
[18] Li, M.X., Lin, Q., Zhang, S.: Extrapolation and superconvergence of the Steklov eigenvalue problem. Adv. Comput. Math. (2009). doi: 10.1007/s10444-009-9118-7 · Zbl 1213.65141
[19] Tang, W., Guan, Z., Han, H.: Boundary element approximation of Steklov eigenvalue problem for Helmholtz equation. J. Comput. Math. 2, 165–178 (1998) · Zbl 0977.65100
[20] Wang, C., Huang, Z.P., Li, L.K.: Two-grid nonconforming finite element method for second order elliptic problems. Appl. Math. Comput. 177, 211–219 (2006) · Zbl 1101.65109
[21] Xu, J.C.: A new class of iterative methods for nonselfadjoint or indefinite problems. SIAM J. Numer. Anal. 29, 303–319 (1992) · Zbl 0756.65050
[22] Xu, J.C.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996) · Zbl 0860.65119
[23] Xu, J.C., Zhou, A.H.: A two-grid discretization scheme for eigenvalue problem. Math. Comput. 70, 17–25 (2001) · Zbl 0959.65119
[24] Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms for eigenvalue problems. Acta Math. Appl. Sin. Engl. Ser. 18, 80–200 (2002) · Zbl 1015.65060
[25] Yang, Y.D., Fan, X.Y.: Generalized Rayleigh quotient and finite element two-grid discretization schemes. Sci. China Ser. A 52, 1955–1972 (2009) · Zbl 1188.65151
[26] Yang, Y.D., Li, Q., Li, S.R.: Nonconforming finite element approximations of the Steklov eigenvalue problem. Appl. Numer. Math. 59, 2388–2401 (2009) · Zbl 1190.65168
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