Advanced Search
Article Contents
Article Contents

Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes

The research of this author was supported in part by China Natural National Science Foundation (U1530116,91630201,11471141), and by the Program for Cheung Kong Scholars of Ministry of Education of China, Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, 130012, P.R. China.
Abstract Full Text(HTML) Figure(0) / Table(4) Related Papers Cited by
  • In this paper, we investigate the weak Galerkin method for the Laplacian eigenvalue problem. We use the weak Galerkin method to obtain lower bounds of Laplacian eigenvalues, and apply a postprocessing technique to get upper bounds. Thus, we can verify the accurate intervals which the exact eigenvalues lie in. This postprocessing technique is efficient and does not need to solve any auxiliary problem. Both theoretical analysis and numerical experiments are presented in this paper.

    Mathematics Subject Classification: Primary: 65N30, 65N15, 65N12, 74N20; Secondary: 35B45, 35J50, 35J35.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Table 1.  Numerical results for the eigenvalues with $k = 1$

    $h$1/41/81/161/321/641/128 $\lambda$
    $ \lambda_{1, h}$1.5407221.8517691.9580221.9885871.9969331.9991782
    $\tilde \lambda_{1, h}$2.2236732.0685472.0182672.0046932.0011892.0002992
    $ \lambda_{2, h}$2.8218314.1351904.7349214.9262974.9800944.9946675
    $\tilde \lambda_{2, h}$6.5882455.4842825.1255465.0318245.0080125.0020105
    $ \lambda_{3, h}$2.8362094.1427924.7373014.9269554.9802704.9947135
    $\tilde \lambda_{3, h}$5.9097235.3043625.0797235.0202845.0051215.0012875
    $ \lambda_{4, h}$3.6816496.0619177.3681797.8204657.9511457.9868668
    $\tilde \lambda_{4, h}$11.6153289.1581618.2977568.0754928.0190498.0047898
     | Show Table
    DownLoad: CSV

    Table 2.  Numerical results for the eigenvalues with $k = 2$

    $h$1/41/81/161/321/641/128 $\lambda$
    $ \lambda_{1, h}$1.9776231.9985941.9999071.9999941.99999951.999999972
    $\tilde \lambda_{1, h}$2.0477972.0014162.0000602.0000032.00000022.000000012
    $ \lambda_{2, h}$4.5626394.9733614.9982984.9998884.9999924.99999945
    $\tilde \lambda_{2, h}$5.9650945.0255285.0009955.0000485.0000035.00000015
    $ \lambda_{3, h}$4.6566974.9801594.9987394.9999174.9999944.99999965
    $\tilde \lambda_{3, h}$6.3520005.0234315.0007845.0000345.0000025.000000085
    $ \lambda_{4, h}$6.4237917.9023457.9939407.9996037.9999747.9999988
    $\tilde \lambda_{4, h}$14.1744538.1325128.0042088.0001908.0000108.0000018
     | Show Table
    DownLoad: CSV

    Table 3.  Numerical results for the Lshape domain with $k = 1$

    $ \lambda_{1, h}$5.9071138.1544179.1558699.4904419.5932509.624793
    $\tilde \lambda_{1, h}$11.65628010.4983239.9081789.7224189.6660849.648513
    $ \lambda_{2, h}$8.20118812.30882114.29331414.94377915.12854815.178811
    $\tilde \lambda_{2, h}$17.97432316.39626615.53354315.28490815.21963315.202913
    $ \lambda_{3, h}$9.44449415.20567618.27619419.32490519.62657119.708934
    $\tilde \lambda_{3, h}$26.26208322.35362120.45371219.92345319.78598219.750997
    $ \lambda_{4, h}$11.12416520.25829226.28719328.58015929.26373929.452134
    $\tilde \lambda_{4, h}$44.27037134.58864430.89133329.87077229.60976629.543696
     | Show Table
    DownLoad: CSV

    Table 4.  Numerical results for the Lshape domain with $k = 2$

    $ \lambda_{1, h}$9.0765419.5561569.6158129.6308329.6362239.638331
    $\tilde \lambda_{1, h}$12.34563410.0750579.7904039.6983579.6629559.648961
    $ \lambda_{2, h}$13.58424115.09738515.19041715.19672215.19720415.197247
    $\tilde \lambda_{2, h}$22.66687315.34652015.20291815.19760515.19728515.197256
    $ \lambda_{3, h}$16.28028719.51796819.72532519.73829619.73914819.739205
    $\tilde \lambda_{3, h}$35.04403020.07688019.75014919.73970019.73923419.739210
    $ \lambda_{4, h}$19.96634828.77751229.47617729.51850429.52127829.521467
    $\tilde \lambda_{4, h}$74.08112330.74643229.55451829.52292929.52156029.521486
     | Show Table
    DownLoad: CSV
  • [1] M. G. Armentano and R. G. Duran, Asymptotic lower bounds for eigenvalues by nonconforming finite element methods, ETNA, Electron. Trans. Numer. Anal., 17 (2004), 93-101. 
    [2] I. Babuska and J. Osborn, Handbook of Numerical Analysis, Vol II, Part1, Elsevier Science Publishers, North-Holland, 1991.
    [3] I. Babuska and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comput., 52 (1989), 275-297.  doi: 10.1090/S0025-5718-1989-0962210-8.
    [4] C. CarstensenD. Gallistl and M. Schedensack, Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems, Math. Comput., 84 (2014), 1061-1087. 
    [5] C. Carstensen and J. Gedicke, Guaranteed lower bounds for eigenvalues, Math. Comput., 83 (2014), 2605-2629.  doi: 10.1090/S0025-5718-2014-02833-0.
    [6] L. ChenJ. Wang and X. Ye, A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J. Sci. Comput., 59 (2014), 496-511.  doi: 10.1007/s10915-013-9771-3.
    [7] D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55 (2013), 601-667.  doi: 10.1137/120880173.
    [8] J. HuY. Huang and Q. Lin, Guaranteed lower bounds for eigenvalues of elliptic operators, J. Sci. Comput., 67 (2016), 1181-1197.  doi: 10.1007/s10915-015-0126-0.
    [9] _____, 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.
    [10] J. HuY. Huang and Q. Shen, Constructing both lower and upper bounds for the eigenvalues of elliptic operators by nonconforming finite element methods, Numer. Math., 131 (2015), 273-302.  doi: 10.1007/s00211-014-0688-z.
    [11] _____, The lower/upper bound property of approximate eigenvalues by nonconforming finite element methods for elliptic operators, J. Sci. Comput., 58 (2014), 574-591.  doi: 10.1007/s10915-013-9744-6.
    [12] J. R. Kuttler, Direct methods for computing eigenvalues of the finite-difference Laplacian, SIAM J. Numer. Anal., 11 (1974), 732-740.  doi: 10.1137/0711059.
    [13] M. G. Larson, A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems, SIAM J. Numer. Anal., 38 (2000), 608-625.  doi: 10.1137/S0036142997320164.
    [14] Q. LinH. Huang and Z. Li, New expansions of numerical eigenvalues by nonconforming elements, Math. Comput., 77 (2008), 2061-2084.  doi: 10.1090/S0025-5718-08-02098-X.
    [15] Q. LinH. Xie and J. Xu, Lower bounds of the discretization error for piecewise polynomials, Math. Comput., 83 (2014), 1-13. 
    [16] X. Liu and S. I. Oishi, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal., 51 (2013), 1634-1654.  doi: 10.1137/120878446.
    [17] F. LuoQ. Lin and 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.
    [18] L. MuJ. WangG. WeiX. Ye and S. Zhao, Weak Galerkin methods for second order elliptic interface problems, J. Comput. Phys., 250 (2013), 106-125.  doi: 10.1016/j.jcp.2013.04.042.
    [19] L. MuJ. Wang and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numer. Methods Partial Differ. Equ., 30 (2014), 1003-1029.  doi: 10.1002/num.21855.
    [20] L. MuJ. WangX. Ye and S. Zhang, A $C^0$ -weak Galerkin finite element method for the biharmonic equation, J. Sci. Comput., 59 (2014), 473-495.  doi: 10.1007/s10915-013-9770-4.
    [21] _____, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363-386.  doi: 10.1007/s10915-014-9964-4.
    [22] L. MuX. Wang and X. Ye, A modified weak Galerkin finite element method for the Stokes equations, J. Comput. Appl. Math., 275 (2015), 79-90.  doi: 10.1016/j.cam.2014.08.006.
    [23] J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.
    [24] _____, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83 (2014), 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.
    [25] R. WangX. WangQ. Zhai and R. Zhang, A weak Galerkin finite element scheme for solving the stationary Stokes equations, J. Comput. Appl. Math., 302 (2016), 171-185.  doi: 10.1016/j.cam.2016.01.025.
    [26] X. WangQ. Zhai and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow, J. Comput. Appl. Math., 307 (2016), 13-24.  doi: 10.1016/j.cam.2016.04.031.
    [27] H. Xie, Q. Zhai and R. Zhang, The weak Galerkin method for eigenvalue problems, arXiv: 1508.05304, (2015).
    [28] Q. ZhaiR. Zhang and X. Wang, A hybridized weak Galerkin finite element scheme for the Stokes equations, Sci. China Math., 58 (2015), 2455-2472.  doi: 10.1007/s11425-015-5030-4.
    [29] R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order, J. Sci. Comput., 64 (2015), 559-585.  doi: 10.1007/s10915-014-9945-7.
  • 加载中



Article Metrics

HTML views(1190) PDF downloads(579) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint