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

Qilong Zhai; Ran Zhang

doi:10.3934/dcdsb.2018091

Article Contents

2019, Volume 24, Issue 1: 403-413. Doi: 10.3934/dcdsb.2018091

This issue Previous Article Two-grid finite element method for the stabilization of mixed Stokes-Darcy model Next Article

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

Qilong Zhai and
Ran Zhang^,

School of Mathematics, Jilin University, Changchun, China

Received: October 31, 2016

Revised: September 30, 2017

Early access: March 2018

Published: January 2019

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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Abstract

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.

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

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Table 1. Numerical results for the eigenvalues with $k = 1$

$h$	1/4	1/8	1/16	1/32	1/64	1/128	$\lambda$
$ \lambda_{1, h}$	1.540722	1.851769	1.958022	1.988587	1.996933	1.999178	2
rate		1.6315	1.8202	1.8790	1.8955	1.9001
$\tilde \lambda_{1, h}$	2.223673	2.068547	2.018267	2.004693	2.001189	2.000299	2
rate		1.7062	1.9079	1.9606	1.9809	1.9905
$ \lambda_{2, h}$	2.821831	4.135190	4.734921	4.926297	4.980094	4.994667	5
rate		1.3327	1.7060	1.8466	1.8885	1.9001
$\tilde \lambda_{2, h}$	6.588245	5.484282	5.125546	5.031824	5.008012	5.002010	5
rate		1.7135	1.9476	1.9800	1.9899	1.9947
$ \lambda_{3, h}$	2.836209	4.142792	4.737301	4.926955	4.980270	4.994713	5
rate		1.3358	1.7062	1.8465	1.8884	1.8999
$\tilde \lambda_{3, h}$	5.909723	5.304362	5.079723	5.020284	5.005121	5.001287	5
rate		1.5796	1.9327	1.9747	1.9859	1.9921
$ \lambda_{4, h}$	3.681649	6.061917	7.368179	7.820465	7.951145	7.986866	8
rate		1.1558	1.6170	1.8152	1.8777	1.8952
$\tilde \lambda_{4, h}$	11.615328	9.158161	8.297756	8.075492	8.019049	8.004789	8
rate		1.6423	1.9596	1.9797	1.9866	1.9920

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

$h$	1/4	1/8	1/16	1/32	1/64	1/128	$\lambda$
$ \lambda_{1, h}$	1.977623	1.998594	1.999907	1.999994	1.9999995	1.99999997	2
rate		3.9922	3.9260	3.9060	3.9006	3.8999
$\tilde \lambda_{1, h}$	2.047797	2.001416	2.000060	2.000003	2.0000002	2.00000001	2
rate		5.0769	4.5640	4.3668	4.2495	4.1640
$ \lambda_{2, h}$	4.562639	4.973361	4.998298	4.999888	4.999992	4.9999994	5
rate		4.0372	3.9683	3.9193	3.9049	3.9011
$\tilde \lambda_{2, h}$	5.965094	5.025528	5.000995	5.000048	5.000003	5.0000001	5
rate		5.2405	4.6807	4.3769	4.2375	4.1526
$ \lambda_{3, h}$	4.656697	4.980159	4.998739	4.999917	4.999994	4.9999996	5
rate		4.1130	3.9757	3.9208	3.9056	3.9016
$\tilde \lambda_{3, h}$	6.352000	5.023431	5.000784	5.000034	5.000002	5.00000008	5
rate		5.8505	4.9017	4.5228	4.3518	4.2405
$ \lambda_{4, h}$	6.423791	7.902345	7.993940	7.999603	7.999974	7.999998	8
rate		4.0126	4.0103	3.9302	3.9073	3.9011
$\tilde \lambda_{4, h}$	14.174453	8.132512	8.004208	8.000190	8.000010	8.000001	8
rate		5.5421	4.9768	4.4660	4.2751	4.1726

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

$h$	1/4	1/8	1/16	1/32	1/64	1/128
$ \lambda_{1, h}$	5.907113	8.154417	9.155869	9.490441	9.593250	9.624793
$\tilde \lambda_{1, h}$	11.656280	10.498323	9.908178	9.722418	9.666084	9.648513
$ \lambda_{2, h}$	8.201188	12.308821	14.293314	14.943779	15.128548	15.178811
$\tilde \lambda_{2, h}$	17.974323	16.396266	15.533543	15.284908	15.219633	15.202913
$ \lambda_{3, h}$	9.444494	15.205676	18.276194	19.324905	19.626571	19.708934
$\tilde \lambda_{3, h}$	26.262083	22.353621	20.453712	19.923453	19.785982	19.750997
$ \lambda_{4, h}$	11.124165	20.258292	26.287193	28.580159	29.263739	29.452134
$\tilde \lambda_{4, h}$	44.270371	34.588644	30.891333	29.870772	29.609766	29.543696

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

$h$	1/4	1/8	1/16	1/32	1/64	1/128
$ \lambda_{1, h}$	9.076541	9.556156	9.615812	9.630832	9.636223	9.638331
$\tilde \lambda_{1, h}$	12.345634	10.075057	9.790403	9.698357	9.662955	9.648961
$ \lambda_{2, h}$	13.584241	15.097385	15.190417	15.196722	15.197204	15.197247
$\tilde \lambda_{2, h}$	22.666873	15.346520	15.202918	15.197605	15.197285	15.197256
$ \lambda_{3, h}$	16.280287	19.517968	19.725325	19.738296	19.739148	19.739205
$\tilde \lambda_{3, h}$	35.044030	20.076880	19.750149	19.739700	19.739234	19.739210
$ \lambda_{4, h}$	19.966348	28.777512	29.476177	29.518504	29.521278	29.521467
$\tilde \lambda_{4, h}$	74.081123	30.746432	29.554518	29.522929	29.521560	29.521486

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