Applications of Mathematics, first online, pp. 1-19


Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients

Yu Zhang, Hai Bi, Yidu Yang

Received April 26, 2019.   Published online September 9, 2020.

Abstract:  In this paper, using a new correction to the Crouzeix-Raviart finite element eigenvalue approximations, we obtain asymptotic lower bounds of eigenvalues for the Steklov eigenvalue problem with variable coefficients on $d$-dimensional domains ($d=2, 3$). In addition, we prove that the corrected eigenvalues converge to the exact ones from below. The new result removes the conditions of eigenfunction being singular and eigenvalue being large enough, which are usually required in the existing arguments about asymptotic lower bounds. Further, we prove that the corrected eigenvalues still maintain the same convergence order as uncorrected eigenvalues. Finally, numerical experiments validate our theoretical results.
Keywords:  correction; Steklov eigenvalue problem; Crouzeix-Raviart finite element; asymptotic lower bounds; convergence order
Classification MSC:  65N25, 65N30
DOI:  10.21136/AM.2020.0108-19

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References:
[1] A. Alonso, A. Dello Russo: 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 | Zbl 1156.65094
[2] M. G. Armentano, R. G. Durán: Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. ETNA, Electron. Trans. Numer. Anal. 17 (2004), 93-101. MR 2040799 | Zbl 1065.65127
[3] I. Babuška, J. Osborn: Eigenvalue problems. Finite Element Methods (Part 1). Handbook of Numererical Analysis II. North-Holland, Amsterdam, 1991, 641-787. MR 1115240 | Zbl 0875.65087
[4] D. Boffi: Finite element approximation of eigenvalue problems. Acta Numerica 19 (2010), 1-120. DOI 10.1017/S0962492910000012 | MR 2652780 | Zbl 1242.65110
[5] J. H. Bramble, J. E. Osborn: Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Azis, ed.). Academic Press, New York, 1972, 387-408. DOI 10.1016/B978-0-12-068650-6.50019-8 | MR 0431740 | Zbl 0264.35055
[6] S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15. Springer, Berlin (2002). DOI 10.1007/978-1-4757-3658-8 | MR 1894376 | Zbl 1012.65115
[7] C. Carstensen, D. Gallistl: Guaranteed lower eigenvalue bounds for the biharmonic equation. Numer. Math. 126 (2014), 33-51. DOI 10.1007/s00211-013-0559-z | MR 3149071 | Zbl 1298.65165
[8] C. Carstensen, J. Gedicke: Guaranteed lower bounds for eigenvalues. Math. Comput. 83 (2014), 2605-2629. DOI 10.1090/S0025-5718-2014-02833-0 | MR 3246802 | Zbl 1320.65162
[9] C. Carstensen, J. Gedicke, D. Rim: Explicit error estimates for Courant, Crouzeix-Raviart and Raviart-Thomas finite element methods. J. Comput. Math. 30 (2012), 337-353. DOI 10.4208/jcm.1108-m3677 | MR 2965987 | Zbl 1274.65290
[10] I. Chavel, E. A. Feldman: An optimal Poincaré inequality for convex domains of non-negative curvature. Arch. Ration. Mech. Anal. 65 (1977), 263-273. DOI 10.1007/BF00280444 | MR 0448457 | Zbl 0362.35059
[11] L. Chen: iFEM: an innovative finite element methods package in MATLAB. Technical Report, University of California, Irvine (2008). Available at https://pdfs.semanticscholar.org/b841/653da0c77051e91f411d4363afe3727f5cc5.pdf.
[12] 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), 33-75. DOI 10.1051/m2an/197307R300331 | MR 0343661 | Zbl 0302.65087
[13] A. Dello Russo, A. E. Alonso: A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems. Comput. Math. Appl. 62 (2011), 4100-4117. DOI 10.1016/j.camwa.2011.09.061 | MR 2859966 | Zbl 1236.65142
[14] E. M. Garau, P. Morin: Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems. IMA J. Numer. Anal. 31 (2011), 914-946. DOI 10.1093/imanum/drp055 | MR 2832785 | Zbl 1225.65107
[15] J. Hu, Y. Huang: Lower bounds for eigenvalues of the Stokes operator. Adv. Appl. Math. Mech. 5 (2013), 1-18. DOI 10.4208/aamm.11-m11103 | MR 3021142 | Zbl 1262.65171
[16] J. Hu, Y. Huang, Q. Lin: Lower bounds for eigenvalues of elliptic operators: by nonconforming finite element methods. J. Sci. Comput. 61 (2014), 196-221. DOI 10.1007/s10915-014-9821-5 | MR 3254372 | Zbl 1335.65089
[17] J. Hu, Y. Huang, R. Ma: Guaranteed lower bounds for eigenvalues of elliptic operators. J. Sci. Comput. 67 (2016), 1181-1197. DOI 10.1007/s10915-015-0126-0 | MR 3493499 | Zbl 1343.65131
[18] Y. Li: Lower approximation of eigenvalues by the nonconforming finite element method. Math. Numer. Sin. 30 (2008), 195-200. (In Chinese.) MR 2437993 | Zbl 1174.65514
[19] Q. Li, Q. Lin, H. Xie: Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations. Appl. Math., Praha 58 (2013), 129-151. DOI 10.1007/s10492-013-0007-5 | MR 3034819 | Zbl 1274.65296
[20] Q. Li, X. Liu: Explicit finite element error estimates for nonhomogeneous Neumann problems. Appl. Math., Praha 63 (2018), 367-379. DOI 10.21136/AM.2018.0095-18 | MR 3833665 | Zbl 06945737
[21] Q. Lin, H.-T. Huang, Z.-C. Li: 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
[22] Q. Lin, H. Xie: Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Probl. Imaging 7 (2013), 795-811. DOI 10.3934/ipi.2013.7.795 | MR 3105355 | Zbl 1273.65178
[23] 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. MR 2768711
[24] X. Liu: A framework of verified eigenvalue bounds for self-adjoint differential operators. Appl. Math. Comput. 267 (2015), 341-355. DOI 10.1016/j.amc.2015.03.048 | MR 3399052 | Zbl 1410.35088
[25] F. Luo, Q. Lin, H. Xie: 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
[26] J. T. Oden, J. N. Reddy: An Introduction to the Mathematical Theory of Finite Elements. Pure and Applied Mathematics. Wiley-Interscience, New York (1976). MR 0461950 | Zbl 0336.35001
[27] G. Savaré: Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1998), 176-201. DOI 10.1006/jfan.1997.3158 | MR 1600081 | Zbl 0889.35018
[28] I. Šebestová, T. Vejchodský: Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs, Poincaré, trace, and similar constants. SIAM J. Numer. Anal. 52 (2014), 308-329. DOI 10.1137/13091467X | MR 3163245 | Zbl 1287.35050
[29] M. Xie, H. Xie, X. Liu: Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements. Japan J. Ind. Appl. Math. 35 (2018), 335-354. DOI 10.1007/s13160-017-0291-7 | MR 3768250 | Zbl 06859028
[30] Y. Yang, J. Han, H. Bi, Y. Yu: The lower/upper bound property of the Crouzeix-Raviart element eigenvalues on adaptive meshes. J. Sci. Comput. 62 (2015), 284-299. DOI 10.1007/s10915-014-9855-8 | MR 3295037 | Zbl 1320.65163
[31] Y. Yang, Q. Li, S. Li: 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 1190.65168
[32] Y. Yang, Q. Lin, H. Bi, Q. Li: Eigenvalue approximations from below using Morley elements. Adv. Comput. Math. 36 (2012), 443-450. DOI 10.1007/s10444-011-9185-4 | MR 2893474 | Zbl 1253.65181
[33] Y. Yang, Y. Zhang, H. Bi: A type of adaptive $C^0$ non-conforming finite element method for the Helmholtz transmission eigenvalue problem. Comput. Methods Appl. Mech. Eng. 360 (2020), Article ID 112697, 20 pages. DOI 10.1016/j.cma.2019.112697 | MR 4049892 | Zbl 07194504
[34] Y. Yang, Z. Zhang, F. Lin: 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
[35] C. You, H. Xie, X. Liu: Guaranteed eigenvalue bounds for the Steklov eigenvalue problem. SIAM J. Numer. Anal. 57 (2019), 1395-1410. DOI 10.1137/18M1189592 | MR 3961991 | Zbl 1427.65384
[36] Z. Zhang, Y. Yang, Z. Chen: Eigenvalue approximation from below by Wilson's element. Math. Numer. Sin. 29 (2007), 319-321. (In Chinese.) MR 2370469 | Zbl 1142.65435

Affiliations:   Yu Zhang, School of Mathematical Sciences, Guizhou Normal University, No. 116 Baoshan Road (N), Guiyang 550001, China; School of Mathematics & Statistics, Guizhou University of Finance and Economics, Huayan Rd, Huaxi District, Guiyang 550025, China, e-mail: zhang_hello_hi@126.com; Hai Bi, Yidu Yang (corresponding author), School of Mathematical Sciences, Guizhou Normal University, No. 116 Baoshan Road (N), Guiyang 550001, China, e-mail: bihaimath@gznu.edu.cn, ydyang@gznu.edu.cn


 
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