Article Contents
Article Contents

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

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

• 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:

• 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

 $n=16$ $n=8+x_1-x_2$ $h$ $j$ $k^S_{j, h}$ $k^L_{j, h}$ $j$ $k^S_{j, h}$ $k^L_{j, h}$ $\frac{\sqrt2}{32}$ 1 1.879083492 1.4773077 1 2.819407434 2.302531586 $\frac{\sqrt2}{64}$ 1 1.879447918 1.4767038 1 2.821437358 2.302554268 $\frac{\sqrt2}{128}$ 1 1.879554183 1.4763944 1 2.821996985 2.302391353 $\frac{\sqrt2}{32}$ 2 2.443168293 1.5694361 2 3.534452084 2.394060079 $\frac{\sqrt2}{64}$ 2 2.443929469 1.5696511 2 3.537524631 2.395270328 $\frac{\sqrt2}{128}$ 2 2.444156467 1.5697071 2 3.538394853 2.395585637 $\frac{\sqrt2}{32}$ 3 2.443168293 1.7049725 5 4.498450-0.871213i 2.928086-0.563389i $\frac{\sqrt2}{64}$ 3 2.443929469 1.7051045 5 4.496942-0.871413i 2.925438-0.564575i $\frac{\sqrt2}{128}$ 3 2.444156467 1.7050978 5 4.496644-0.871466i 2.924644-0.564720i $\frac{\sqrt2}{32}$ 4 2.865221584 1.7827065 6 4.498450+0.871213i 2.928086+0.563389i $\frac{\sqrt2}{64}$ 4 2.866032777 1.7830026 6 4.496942+0.871413i 2.925438+0.564575i $\frac{\sqrt2}{128}$ 4 2.866329734 1.7830869 6 4.496644+0.871466i 2.924644+0.564720i

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

 $Dof$ $j$ $k^C_{j, h}$($n=16$) $Dof$ $j$ $k^C_{j, h}$($n=8+x_1-x_2$) $1895$ 1 2.0449 1895 1 2.9579 $16179$ 1 2.0604 16179 1 3.0051 $55903$ 1 2.0641 55903 1 3.0162 $1895$ 2, 3, 4 2.5461 1895 2 3.6098 $16179$ 2, 3, 4 2.5714 16179 2 3.6853 $55903$ 2, 3, 4 2.5785 55903 2 3.7049 $1895$ 5, 6, 7 2.9402 1895 3, 4 3.6119, 3.6121 $16179$ 5, 6, 7 2.9683 16179 3, 4 3.6878, 3.6879 $55903$ 5, 6, 7 2.9781 55903 3, 4 3.7074, 3.7075

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

 $Dof$ $j$ $k^C_{j, h}$($n=16$) $Dof$ $j$ $k^C_{j, h}$($n=8+x_1-x_2$) $3608$ 1 2.1651 3608 1 3.2181 $30648$ 1 2.0935 30648 1 3.0747 $105688$ 1 2.0791 105688 1 3.0474 $3608$ 2, 3, 4 2.7794, 2.7803, 2.7807 3608 2 4.0750 $30648$ 2, 3, 4 2.6350, 2.6350, 2.6351 30648 2 3.8098 $105688$ 2, 3, 4 2.6067 105688 2 3.7596 $3608$ 5, 6, 7 3.2927, 3.2959, 3.2973 3608 3, 4 4.0790, 4.0800 $30648$ 5, 6, 7 3.0680, 3.0682, 3.0684 30648 3, 4 3.8126, 3.8126 $105688$ 5, 6, 7 3.0234 105688 3, 4 3.7622, 3.7623

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

 $Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$ $1071$ 2.7560 3.3119, 3.3119, 3.3617 3.9131, 3.9623, 3.9738 $9955$ 2.7650 3.3220, 3.3239, 3.3239 3.9144, 3.9151, 3.9151 $85963$ 2.7758 3.3371, 3.3396, 3.3396 3.9320, 3.9320, 3.9342

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

 $Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$ $2072$ 3.2333 4.0873, 4.0873, 4.1929 5.1997, 5.1997, 5.2041 $19000$ 2.8870 3.5128, 3.5128, 3.5666 4.2089, 4.2528, 4.2557 $162936$ 2.8051 3.3852, 3.3852, 3.3980 4.0036, 4.0185, 4.0185

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

 $Dof$ $j$ $k^{TL}_{j, h}$($n=16$) $j$ $k^{TL}_{j, h}$($n=8+x_1-x_2$) $257$ 1 2.4603 1 3.3865 $2575$ 1 1.9756 1 2.8223 $23171$ 1 1.8209 1 2.6248 $257$ 2 2.5244 2 3.4304 $2575$ 2 2.0359 2 2.8372 $23171$ 2 1.8801 2 2.6415 $257$ 3 2.5311 3 3.5598 $2575$ 3 2.0974 3 3.0510 $23171$ 3 1.9431 3 2.8327

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

 $Dof$ $j$ $k^{TL}_{j, h}$($n=16$) $j$ $k^{TL}_{j, h}$($n=8+x_1-x_2$) $504$ 1 3.6272 1 4.5809 - 2.0747i $4952$ 1 2.5041 1 3.9589 $44088$ 1 1.9305 1 2.8511 $504$ 2 3.9842 2 4.5809 + 2.0747i $4952$ 2 2.7611 2 4.5904 $44088$ 2 1.9943 2 2.8939 $504$ 3 4.6844 3 5.2114 - 2.2381i $4952$ 3 2.8373 3 4.9593 $44088$ 3 2.0851 3 3.1157

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

 $Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$ 1608 2.3431 2.9533, 2.9550, 2.9564 3.5575, 3.5597, 3.5600 16643 2.3075 2.9250, 2.9252, 2.9254 3.5257, 3.5260, 3.5261 51277 2.3035 2.9229, 2.9229, 2.9230 3.5261, 3.5262, 3.5263

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

 $Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$ 3076 2.4860 3.3110, 3.3259, 3.3441 4.2252, 4.2297, 4.2574 31568 2.3306 2.9875, 2.9886, 2.9905 3.6451, 3.6471, 3.6520 97048 2.3147 2.9520, 2.9521, 2.9529 3.5817, 3.5825, 3.5848
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