Applications of Mathematics, Vol. 62, No. 3, pp. 243-267, 2017

A posteriori error estimates for a discontinuous Galerkin approximation of Steklov eigenvalue problems

Yuping Zeng, Feng Wang

Received April 12, 2016.   First published April 30, 2017.

Yuping Zeng, School of Mathematics, Jiaying University, Meizhou 514015, China, e-mail: zeng$_-$yuping@163.com; Feng Wang, Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China, e-mail: fwang@njnu.edu.cn

Abstract: We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings.

Keywords: discontinuous Galerkin method; Steklov eigenvalue problem; a posteriori error estimate

Classification (MSC 2010): 65N15, 65N25, 65N30

DOI: 10.21136/AM.2017.0115-16

Full text available as PDF.


References:
  [1] H. J. Ahn: Vibrations of a pendulum consisting of a bob suspended from a wire: the method of integral equations. Q. Appl. Math. 39 (1981), 109-117. DOI 10.1090/qam/613954 | MR 0613954 | Zbl 0458.70018
  [2] M. Ainsworth: A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal. 45 (2007), 1777-1798. DOI 10.1137/060665993 | MR 2338409 | Zbl 1151.65083
  [3] M. Ainsworth, R. Rankin: Fully computable error bounds for discontinuous Galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes. SIAM J. Numer. Anal. 47 (2010), 4112-4141. DOI 10.1137/080725945 | MR 2585181 | Zbl 1208.65155
  [4] A. Alonso, A. D. 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
  [5] A. B. Andreev, T. D. Todorov: 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
  [6] P. F. Antonietti, A. Buffa, I. Perugia: Discontinuous Galerkin approximation of the Laplace eigenproblem. Comput. Methods Appl. Mech. Eng. 195 (2006), 3483-3503. DOI 10.1016/j.cma.2005.06.023 | MR 2220929 | Zbl 1168.65410
  [7] M. G. Armentano: The effect of reduced integration in the Steklov eigenvalue problem. M2AN, Math. Model. Numer. Anal. 38 (2004), 27-36. DOI 10.1051/m2an:2004002 | MR 2073929 | Zbl 1077.65115
  [8] M. G. Armentano, C. Padra: 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
  [9] D. N. Arnold, F. Brezzi, B. Cockburn, L. D. Marini: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002), 1749-1779. DOI 10.1137/S0036142901384162 | MR 1885715 | Zbl 1008.65080
  [10] I. Babuška, J. Osborn: Eigenvalue problems. Handbook of Numerical Analysis. Volume II: Finite Element Methods (Part 1) (P. G. Ciarlet et al., eds.) North-Holland, Amsterdam (1991), 641-787. DOI 10.1016/S1570-8659(05)80042-0 | MR 1115240 | Zbl 0875.65087
  [11] R. Becker, P. Hansbo, M. G. Larson: Energy norm a posteriori error estimation for discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 192 (2003), 723-733. DOI 10.1016/S0045-7825(02)00593-5 | MR 1952357 | Zbl 1042.65083
  [12] S. Bergman, M. Schiffer: Kernel Functions and Elliptic Differential Equations in Mathematical Physics. Pure and Applied Mathematics 4, Academic Press, New York (1953). MR 0054140 | Zbl 0053.39003
  [13] 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. DOI 10.1007/s002110000175 | MR 1804656 | Zbl 0998.76046
  [14] H. Bi, H. Li, Y. Yang: An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem. Appl. Numer. Math. 105 (2016), 64-81. DOI 10.1016/j.apnum.2016.02.003 | MR 3488074 | Zbl 06576298
  [15] A. Bonito, R. H. Nochetto: Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method. SIAM J. Numer. Anal. 48 (2010), 734-771. DOI 10.1137/08072838X | MR 2670003 | Zbl 1254.65120
  [16] D. Braess, T. Fraunholz, R. H. W. Hoppe: An equilibrated a posteriori error estimator for the interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 52 (2014), 2121-2136. DOI 10.1137/130916540 | MR 3249368 | Zbl 1302.65239
  [17] 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 Proc. Sympos. Univ. Maryland, Baltimore, 1972, Academic Press, New York (1972), 387-408. DOI 10.1016/b978-0-12-068650-6.50019-8 | MR 0431740 | Zbl 0264.35055
  [18] S. C. Brenner: Poincaré-Friedrichs inequalities for piecewise $H^1$ functions. SIAM J. Numer. Anal. 41 (2003), 306-324. DOI 10.1137/S0036142902401311 | MR 1974504 | Zbl 1045.65100
  [19] L. Chen, C. ZhangAFEM@matlab: a Matlab package of adaptive finite element methods. Technical report, University of Maryland at College Park (2006).
  [20] P. Clément: Approximation by finite element functions using local regularization. Rev. Franc. Automat. Inform. Rech. Operat. 9, Analyse numer., No. R-2 (1975), 77-84. DOI 10.1051/m2an/197509r200771 | MR 0400739 | Zbl 0368.65008
  [21] C. Conca, J. Planchard, M. Vanninathan: Fluids and Periodic Structures. Research in Applied Mathematics, Wiley, Chichester; Masson, Paris (1995). MR 1652238 | Zbl 0910.76002
  [22] V. Dolejší, I. Šebestová, M. Vohralík: Algebraic and discretization error estimation by equilibrated fluxes for discontinuous Galerkin methods on nonmatching grids. J. Sci. Comput. 64 (2015), 1-34. DOI 10.1007/s10915-014-9921-2 | MR 3353932 | Zbl 1326.65147
  [23] W. Dörfler: A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996), 1106-1124. DOI 10.1137/0733054 | MR 1393904 | Zbl 0854.65090
  [24] A. Ern, J. Proft: A posteriori discontinuous Galerkin error estimates for transient convection-diffusion equations. Appl. Math. Lett. 18 (2005), 833-841. DOI 10.1016/j.aml.2004.05.019 | MR 2145454 | Zbl 1084.65092
  [25] 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
  [26] S. Giani, E. J. C. Hall: An a posteriori error estimator for $hp$-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems. Math. Models Methods Appl. Sci. 22 (2012), 1250030, 35 pages. DOI 10.1142/S0218202512500303 | MR 2974168 | Zbl 1257.65062
  [27] X. Han, Y. Li, H. Xie: A multilevel correction method for Steklov eigenvalue problem by nonconforming finite element methods. Numer. Math. Theory Methods Appl. 8 (2015), 383-405. DOI 10.4208/nmtma.2015.m1334 | MR 3395398 | Zbl 1349.65603
  [28] D. B. Hinton, J. K. Shaw: Differential operators with spectral parameter incompletely in the boundary conditions. Funkc. Ekvacioj, Ser. Int. 33 (1990), 363-385. MR 1086767 | Zbl 0715.34133
  [29] R. H. W. Hoppe, G. Kanschat, T. Warburton: Convergence analysis of an adaptive interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 47 (2008), 534-550. DOI 10.1137/070704599 | MR 2475951 | Zbl 1189.65274
  [30] P. Houston, I. Perugia, D. Schötzau: An a posteriori error indicator for discontinuous Galerkin discretizations of $H$(curl)-elliptic partial differential equations. IMA J. Numer. Anal. 27 (2007), 122-150. DOI 10.1093/imanum/drl012 | MR 2289274 | Zbl 1148.65088
  [31] P. Houston, D. Schötzau, T. P. Wihler: Energy norm a posteriori error estimation of $hp$-adaptive discontinuous Galerkin methods for elliptic problems. Math. Models Methods Appl. Sci. 17 (2007), 33-62. DOI 10.1142/S0218202507001826 | MR 2290408 | Zbl 1116.65115
  [32] O. A. Karakashian, F. Pascal: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003), 2374-2399. DOI 10.1137/S0036142902405217 | MR 2034620 | Zbl 1058.65120
  [33] O. A. Karakashian, F. Pascal: Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems. SIAM J. Numer. Anal. 45 (2007), 641-665. DOI 10.1137/05063979X | MR 2300291 | Zbl 1140.65083
  [34] 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
  [35] Q. Li, Y. Yang: A two-grid discretization scheme for the Steklov eigenvalue problem. J. Appl. Math. Comput. 36 (2011), 129-139. DOI 10.1007/s12190-010-0392-9 | MR 2794136 | Zbl 1220.65160
  [36] Q. Lin, H. Xie: A multilevel correction type of adaptive finite element method for Steklov eigenvalue problems. Proceedings of the International Conference Applications of Mathematics, Praha (J. Brandts et al., eds.) Academy of Sciences of the Czech Republic, Institute of Mathematics, Praha (2012), 134-143. MR 3204407 | Zbl 1313.65298
  [37] I. Perugia, D. Schötzau: The $hp$-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comput. 72 (2003), 1179-1214. DOI 10.1090/S0025-5718-02-01471-0 | MR 1972732 | Zbl 1084.78007
  [38] B. Rivière, M. F. Wheeler: A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Log number: R74. Comput. Math. Appl. 46 (2003), 141-163. DOI 10.1016/S0898-1221(03)90086-1 | MR 2015276 | Zbl 1059.65098
  [39] A. Romkes, S. Prudhomme, J. T. Oden: A posteriori error estimation for a new stabilized discontinuous Galerkin method. Appl. Math. Lett. 16 (2003), 447-452. DOI 10.1016/S0893-9659(03)00018-1 | MR 1983711 | Zbl 1046.65089
  [40] A. D. 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
  [41] R. Schneider, Y. Xu, A. Zhou: An analysis of discontinuous Galerkin methods for elliptic problems. Adv. Comput. Math. 25 (2006), 259-286. DOI 10.1007/s10444-004-7619-y | MR 2231704 | Zbl 1099.65116
  [42] L. R. Scott, S. Zhang: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990), 483-493. DOI 10.2307/2008497 | MR 1011446 | Zbl 0696.65007
  [43] S. Sun, M. F. Wheeler: $L^2(H^1)$ norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems. J. Sci. Comput. 22 (2005), 501-530. DOI 10.1007/s10915-004-4148-2 | MR 2142207 | Zbl 1066.76037
  [44] S. K. Tomar, S. I. Repin: Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems. J. Comput. Appl. Math. 226 (2009), 358-369. DOI 10.1016/j.cam.2008.08.015 | MR 2502931 | Zbl 1163.65077
  [45] R. Verfürth: A Review of a Posteriori Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley, Chichester (1996).
  [46] H. Xie: A type of multilevel method for the Steklov eigenvalue problem. IMA J. Numer. Anal. 34 (2014), 592-608. DOI 10.1093/imanum/drt009 | MR 3194801 | Zbl 1312.65178
  [47] J. Yang, Y. Chen: A unified a posteriori error analysis for discontinuous Galerkin approximations of reactive transport equations. J. Comput. Math. 24 (2006), 425-434. MR 2229721 | Zbl 1142.76034
  [48] 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
  [49] Y. Zeng, J. Chen, F. Wang: A posteriori error estimates of a weakly over-penalized symmetric interior penalty method for elliptic eigenvalue problems. East Asian J. Appl. Math. 5 (2015), 327-341. DOI 10.4208/eajam.060415.230915a | MR 3421807


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is https://link.springer.com/journal/10492.

[Previous Article] [Next Article] [Contents of This Number] [Contents of Applications of Mathematics]