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

Jiayu Han

doi:10.3934/dcdsb.2018281

Article Contents

2018, Volume 23, Issue 8: 3195-3212. Doi: 10.3934/dcdsb.2018281

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

Jiayu Han^,

School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550001, China

* Corresponding author: hanjiayu126@126.com
* Corresponding author: hanjiayu126@126.com

Received: May 31, 2017

Revised: October 31, 2017

Early access: August 2018

Published: August 2018

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

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Abstract

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.

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Mathematics Subject Classification: 65N25, 65N30.

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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)

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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

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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

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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

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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

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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

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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

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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

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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

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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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