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Nonconforming elements of class $L^2$ for Helmholtz transmission eigenvalue problems

Supported by National Natural Science Foundation of China (grant Nos. 11561014 and 11761022)

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  • For solving the Helmholtz transmission eigenvalue problem, we use the mixed formulation of Cakoni et al. to construct a new nonconforming element discretization. Based on the discretization, this paper first discuss the nonconforming element methods of class $ L^2 $, and prove the error estimates of the discrete eigenvalues obtained by the cubic tetrahedron element, incomplete cubic tetrahedral element and Morley element et al. We report some numerical examples using the nonconforming elements mixed with linear Lagrange element to show that our discretization can obtain the transmission eigenvalues of higher accuracy in 3D domains than the nonconforming element discretization in the existing literature.

    Mathematics Subject Classification: 65N25, 65N30.

    Citation:

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  • Figure 1.  Error curves computed by MZ element mixed with linear element for $n = 16$ on the unit square (left top) and on the L-shaped (right top), for $n = 8+x_1-x_2$ on the unit square (left bottom) and on the L-shaped (right bottom)

    Table 1.  The eigenvalues obtained by MZ element mixed with linear element in 2D domains

    $n=16$$n=8+x_1-x_2$
    $h$$j$ $k^S_{j, h}$$k^L_{j, h}$$j$$k^S_{j, h}$$k^L_{j, h}$
    $\frac{\sqrt2}{32}$11.8790834921.477307712.8194074342.302531586
    $\frac{\sqrt2}{64}$11.8794479181.476703812.8214373582.302554268
    $\frac{\sqrt2}{128}$11.8795541831.476394412.8219969852.302391353
    $\frac{\sqrt2}{32}$22.4431682931.569436123.5344520842.394060079
    $\frac{\sqrt2}{64}$22.4439294691.569651123.5375246312.395270328
    $\frac{\sqrt2}{128}$22.4441564671.569707123.5383948532.395585637
    $\frac{\sqrt2}{32}$32.4431682931.704972554.498450-0.871213i2.928086-0.563389i
    $\frac{\sqrt2}{64}$32.4439294691.705104554.496942-0.871413i2.925438-0.564575i
    $\frac{\sqrt2}{128}$32.4441564671.705097854.496644-0.871466i2.924644-0.564720i
    $\frac{\sqrt2}{32}$42.8652215841.782706564.498450+0.871213i2.928086+0.563389i
    $\frac{\sqrt2}{64}$42.8660327771.783002664.496942+0.871413i2.925438+0.564575i
    $\frac{\sqrt2}{128}$42.8663297341.783086964.496644+0.871466i2.924644+0.564720i
     | Show Table
    DownLoad: CSV

    Table 2.  The eigenvalues obtained by CT element mixed with linear element on the cube

    $Dof$$j$ $k^C_{j, h}$($n=16$)$Dof$ $j$$k^C_{j, h}$($n=8+x_1-x_2$)
    $1895$12.0449189512.9579
    $16179$12.06041617913.0051
    $55903$12.06415590313.0162
    $1895$2, 3, 42.5461189523.6098
    $16179$2, 3, 42.57141617923.6853
    $55903$2, 3, 42.57855590323.7049
    $1895$5, 6, 72.940218953, 43.6119, 3.6121
    $16179$5, 6, 72.9683161793, 43.6878, 3.6879
    $55903$5, 6, 72.9781559033, 43.7074, 3.7075
     | Show Table
    DownLoad: CSV

    Table 3.  The eigenvalues obtained by MZ element on the cube

    $Dof$$j$ $ k^C_{j, h}$($n=16$)$Dof$ $j$$ k^C_{j, h}$($n=8+x_1-x_2$)
    $3608$12.1651360813.2181
    $30648$12.09353064813.0747
    $105688$12.079110568813.0474
    $3608$2, 3, 42.7794, 2.7803, 2.7807360824.0750
    $30648$2, 3, 42.6350, 2.6350, 2.63513064823.8098
    $105688$2, 3, 42.606710568823.7596
    $3608$5, 6, 73.2927, 3.2959, 3.297336083, 44.0790, 4.0800
    $30648$5, 6, 73.0680, 3.0682, 3.0684306483, 43.8126, 3.8126
    $105688$5, 6, 73.02341056883, 43.7622, 3.7623
     | Show Table
    DownLoad: CSV

    Table 4.  The eigenvalues obtained by CT element mixed with linear element on the tetrahedron, $n = 16$

    $Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$
    $1071$2.75603.3119, 3.3119, 3.36173.9131, 3.9623, 3.9738
    $9955$2.76503.3220, 3.3239, 3.32393.9144, 3.9151, 3.9151
    $85963$2.77583.3371, 3.3396, 3.33963.9320, 3.9320, 3.9342
     | Show Table
    DownLoad: CSV

    Table 5.  The eigenvalues obtained by MZ element on the tetrahedron, $n = 16$

    $Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$
    $2072$3.23334.0873, 4.0873, 4.19295.1997, 5.1997, 5.2041
    $19000$2.88703.5128, 3.5128, 3.56664.2089, 4.2528, 4.2557
    $162936$2.80513.3852, 3.3852, 3.39804.0036, 4.0185, 4.0185
     | Show Table
    DownLoad: CSV

    Table 6.  The eigenvalues obtained by CT element mixed with linear element on the thick L-shaped

    $Dof$$j$ $k^{TL}_{j, h}$($n=16$) $j$$k^{TL}_{j, h}$($n=8+x_1-x_2$)
    $257$12.460313.3865
    $2575$11.975612.8223
    $23171$11.820912.6248
    $257$22.524423.4304
    $2575$22.035922.8372
    $23171$21.880122.6415
    $257$32.531133.5598
    $2575$32.097433.0510
    $23171$31.943132.8327
     | Show Table
    DownLoad: CSV

    Table 7.  The eigenvalues obtained by MZ element on the thick L-shaped

    $Dof$$j$ $k^{TL}_{j, h}$($n=16$) $j$$k^{TL}_{j, h}$($n=8+x_1-x_2$)
    $504$13.627214.5809 - 2.0747i
    $4952$12.504113.9589
    $44088$11.930512.8511
    $504$23.984224.5809 + 2.0747i
    $4952$22.761124.5904
    $44088$21.994322.8939
    $504$34.684435.2114 - 2.2381i
    $4952$32.837334.9593
    $44088$32.085133.1157
     | Show Table
    DownLoad: CSV

    Table 8.  The eigenvalues obtained by CT element mixed with linear element on the sphere, $n = 16$

    $Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$
    16082.34312.9533, 2.9550, 2.95643.5575, 3.5597, 3.5600
    166432.30752.9250, 2.9252, 2.92543.5257, 3.5260, 3.5261
    512772.30352.9229, 2.9229, 2.92303.5261, 3.5262, 3.5263
     | Show Table
    DownLoad: CSV

    Table 9.  The eigenvalues obtained by MZ element on the sphere, $n = 16$

    $Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$
    30762.48603.3110, 3.3259, 3.34414.2252, 4.2297, 4.2574
    315682.33062.9875, 2.9886, 2.99053.6451, 3.6471, 3.6520
    970482.31472.9520, 2.9521, 2.95293.5817, 3.5825, 3.5848
     | Show Table
    DownLoad: CSV
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