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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.
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  • 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.

    Citation:

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  • 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
    rate1.63151.82021.87901.89551.9001
    $\tilde \lambda_{1, h}$2.2236732.0685472.0182672.0046932.0011892.0002992
    rate1.70621.90791.96061.98091.9905
    $ \lambda_{2, h}$2.8218314.1351904.7349214.9262974.9800944.9946675
    rate1.33271.70601.84661.88851.9001
    $\tilde \lambda_{2, h}$6.5882455.4842825.1255465.0318245.0080125.0020105
    rate1.71351.94761.98001.98991.9947
    $ \lambda_{3, h}$2.8362094.1427924.7373014.9269554.9802704.9947135
    rate1.33581.70621.84651.88841.8999
    $\tilde \lambda_{3, h}$5.9097235.3043625.0797235.0202845.0051215.0012875
    rate1.57961.93271.97471.98591.9921
    $ \lambda_{4, h}$3.6816496.0619177.3681797.8204657.9511457.9868668
    rate1.15581.61701.81521.87771.8952
    $\tilde \lambda_{4, h}$11.6153289.1581618.2977568.0754928.0190498.0047898
    rate1.64231.95961.97971.98661.9920
     | 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
    rate3.99223.92603.90603.90063.8999
    $\tilde \lambda_{1, h}$2.0477972.0014162.0000602.0000032.00000022.000000012
    rate5.07694.56404.36684.24954.1640
    $ \lambda_{2, h}$4.5626394.9733614.9982984.9998884.9999924.99999945
    rate4.03723.96833.91933.90493.9011
    $\tilde \lambda_{2, h}$5.9650945.0255285.0009955.0000485.0000035.00000015
    rate5.24054.68074.37694.23754.1526
    $ \lambda_{3, h}$4.6566974.9801594.9987394.9999174.9999944.99999965
    rate4.11303.97573.92083.90563.9016
    $\tilde \lambda_{3, h}$6.3520005.0234315.0007845.0000345.0000025.000000085
    rate5.85054.90174.52284.35184.2405
    $ \lambda_{4, h}$6.4237917.9023457.9939407.9996037.9999747.9999988
    rate4.01264.01033.93023.90733.9011
    $\tilde \lambda_{4, h}$14.1744538.1325128.0042088.0001908.0000108.0000018
    rate5.54214.97684.46604.27514.1726
     | Show Table
    DownLoad: CSV

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

    $h$1/41/81/161/321/641/128
    $ \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 $k = 2$

    $h$1/41/81/161/321/641/128
    $ \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
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