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

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January 2019, 24(1): 403-413. doi: 10.3934/dcdsb.2018091

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 November 2016 Revised October 2017 Published March 2018

Fund Project: 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

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.

Keywords: Weak Galerkin, eigenvalue problem, lower bound, upper bound.

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

Citation: Qilong Zhai, Ran Zhang. Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 403-413. doi: 10.3934/dcdsb.2018091

References:

[1]	M. G. Armentano and R. G. Duran, Asymptotic lower bounds for eigenvalues by nonconforming finite element methods, ETNA, Electron. Trans. Numer. Anal., 17 (2004), 93-101.
[2]	I. Babuska and J. Osborn, Handbook of Numerical Analysis, Vol II, Part1, Elsevier Science Publishers, North-Holland, 1991.
[3]	I. Babuska and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comput., 52 (1989), 275-297. doi: 10.1090/S0025-5718-1989-0962210-8.
[4]	C. Carstensen, D. Gallistl and M. Schedensack, Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems, Math. Comput., 84 (2014), 1061-1087.
[5]	C. Carstensen and J. Gedicke, Guaranteed lower bounds for eigenvalues, Math. Comput., 83 (2014), 2605-2629. doi: 10.1090/S0025-5718-2014-02833-0.
[6]	L. Chen, J. Wang and X. Ye, A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J. Sci. Comput., 59 (2014), 496-511. doi: 10.1007/s10915-013-9771-3.
[7]	D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55 (2013), 601-667. doi: 10.1137/120880173.
[8]	J. Hu, Y. Huang and Q. Lin, Guaranteed lower bounds for eigenvalues of elliptic operators, J. Sci. Comput., 67 (2016), 1181-1197. doi: 10.1007/s10915-015-0126-0.
[9]	_____, Lower bounds for eigenvalues of elliptic operators: By nonconforming finite element methods, J. Sci. Comput., 61 (2014), 196-221. doi: 10.1007/s10915-014-9821-5.
[10]	J. Hu, Y. Huang and Q. Shen, Constructing both lower and upper bounds for the eigenvalues of elliptic operators by nonconforming finite element methods, Numer. Math., 131 (2015), 273-302. doi: 10.1007/s00211-014-0688-z.
[11]	_____, The lower/upper bound property of approximate eigenvalues by nonconforming finite element methods for elliptic operators, J. Sci. Comput., 58 (2014), 574-591. doi: 10.1007/s10915-013-9744-6.
[12]	J. R. Kuttler, Direct methods for computing eigenvalues of the finite-difference Laplacian, SIAM J. Numer. Anal., 11 (1974), 732-740. doi: 10.1137/0711059.
[13]	M. G. Larson, A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems, SIAM J. Numer. Anal., 38 (2000), 608-625. doi: 10.1137/S0036142997320164.
[14]	Q. Lin, H. Huang and Z. Li, New expansions of numerical eigenvalues by nonconforming elements, Math. Comput., 77 (2008), 2061-2084. doi: 10.1090/S0025-5718-08-02098-X.
[15]	Q. Lin, H. Xie and J. Xu, Lower bounds of the discretization error for piecewise polynomials, Math. Comput., 83 (2014), 1-13.
[16]	X. Liu and S. I. Oishi, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal., 51 (2013), 1634-1654. doi: 10.1137/120878446.
[17]	F. Luo, Q. Lin and H. Xie, Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods, Sci. China Math., 55 (2012), 1069-1082. doi: 10.1007/s11425-012-4382-2.
[18]	L. Mu, J. Wang, G. Wei, X. Ye and S. Zhao, Weak Galerkin methods for second order elliptic interface problems, J. Comput. Phys., 250 (2013), 106-125. doi: 10.1016/j.jcp.2013.04.042.
[19]	L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numer. Methods Partial Differ. Equ., 30 (2014), 1003-1029. doi: 10.1002/num.21855.
[20]	L. Mu, J. Wang, X. Ye and S. Zhang, A $C^0$ -weak Galerkin finite element method for the biharmonic equation, J. Sci. Comput., 59 (2014), 473-495. doi: 10.1007/s10915-013-9770-4.
[21]	_____, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363-386. doi: 10.1007/s10915-014-9964-4.
[22]	L. Mu, X. Wang and X. Ye, A modified weak Galerkin finite element method for the Stokes equations, J. Comput. Appl. Math., 275 (2015), 79-90. doi: 10.1016/j.cam.2014.08.006.
[23]	J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115. doi: 10.1016/j.cam.2012.10.003.
[24]	_____, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83 (2014), 2101-2126. doi: 10.1090/S0025-5718-2014-02852-4.
[25]	R. Wang, X. Wang, Q. Zhai and R. Zhang, A weak Galerkin finite element scheme for solving the stationary Stokes equations, J. Comput. Appl. Math., 302 (2016), 171-185. doi: 10.1016/j.cam.2016.01.025.
[26]	X. Wang, Q. Zhai and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow, J. Comput. Appl. Math., 307 (2016), 13-24. doi: 10.1016/j.cam.2016.04.031.
[27]	H. Xie, Q. Zhai and R. Zhang, The weak Galerkin method for eigenvalue problems, arXiv: 1508.05304, (2015).
[28]	Q. Zhai, R. Zhang and X. Wang, A hybridized weak Galerkin finite element scheme for the Stokes equations, Sci. China Math., 58 (2015), 2455-2472. doi: 10.1007/s11425-015-5030-4.
[29]	R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order, J. Sci. Comput., 64 (2015), 559-585. doi: 10.1007/s10915-014-9945-7.

show all references

References:

[1]	M. G. Armentano and R. G. Duran, Asymptotic lower bounds for eigenvalues by nonconforming finite element methods, ETNA, Electron. Trans. Numer. Anal., 17 (2004), 93-101.
[2]	I. Babuska and J. Osborn, Handbook of Numerical Analysis, Vol II, Part1, Elsevier Science Publishers, North-Holland, 1991.
[3]	I. Babuska and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comput., 52 (1989), 275-297. doi: 10.1090/S0025-5718-1989-0962210-8.
[4]	C. Carstensen, D. Gallistl and M. Schedensack, Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems, Math. Comput., 84 (2014), 1061-1087.
[5]	C. Carstensen and J. Gedicke, Guaranteed lower bounds for eigenvalues, Math. Comput., 83 (2014), 2605-2629. doi: 10.1090/S0025-5718-2014-02833-0.
[6]	L. Chen, J. Wang and X. Ye, A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J. Sci. Comput., 59 (2014), 496-511. doi: 10.1007/s10915-013-9771-3.
[7]	D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55 (2013), 601-667. doi: 10.1137/120880173.
[8]	J. Hu, Y. Huang and Q. Lin, Guaranteed lower bounds for eigenvalues of elliptic operators, J. Sci. Comput., 67 (2016), 1181-1197. doi: 10.1007/s10915-015-0126-0.
[9]	_____, Lower bounds for eigenvalues of elliptic operators: By nonconforming finite element methods, J. Sci. Comput., 61 (2014), 196-221. doi: 10.1007/s10915-014-9821-5.
[10]	J. Hu, Y. Huang and Q. Shen, Constructing both lower and upper bounds for the eigenvalues of elliptic operators by nonconforming finite element methods, Numer. Math., 131 (2015), 273-302. doi: 10.1007/s00211-014-0688-z.
[11]	_____, The lower/upper bound property of approximate eigenvalues by nonconforming finite element methods for elliptic operators, J. Sci. Comput., 58 (2014), 574-591. doi: 10.1007/s10915-013-9744-6.
[12]	J. R. Kuttler, Direct methods for computing eigenvalues of the finite-difference Laplacian, SIAM J. Numer. Anal., 11 (1974), 732-740. doi: 10.1137/0711059.
[13]	M. G. Larson, A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems, SIAM J. Numer. Anal., 38 (2000), 608-625. doi: 10.1137/S0036142997320164.
[14]	Q. Lin, H. Huang and Z. Li, New expansions of numerical eigenvalues by nonconforming elements, Math. Comput., 77 (2008), 2061-2084. doi: 10.1090/S0025-5718-08-02098-X.
[15]	Q. Lin, H. Xie and J. Xu, Lower bounds of the discretization error for piecewise polynomials, Math. Comput., 83 (2014), 1-13.
[16]	X. Liu and S. I. Oishi, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal., 51 (2013), 1634-1654. doi: 10.1137/120878446.
[17]	F. Luo, Q. Lin and H. Xie, Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods, Sci. China Math., 55 (2012), 1069-1082. doi: 10.1007/s11425-012-4382-2.
[18]	L. Mu, J. Wang, G. Wei, X. Ye and S. Zhao, Weak Galerkin methods for second order elliptic interface problems, J. Comput. Phys., 250 (2013), 106-125. doi: 10.1016/j.jcp.2013.04.042.
[19]	L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numer. Methods Partial Differ. Equ., 30 (2014), 1003-1029. doi: 10.1002/num.21855.
[20]	L. Mu, J. Wang, X. Ye and S. Zhang, A $C^0$ -weak Galerkin finite element method for the biharmonic equation, J. Sci. Comput., 59 (2014), 473-495. doi: 10.1007/s10915-013-9770-4.
[21]	_____, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363-386. doi: 10.1007/s10915-014-9964-4.
[22]	L. Mu, X. Wang and X. Ye, A modified weak Galerkin finite element method for the Stokes equations, J. Comput. Appl. Math., 275 (2015), 79-90. doi: 10.1016/j.cam.2014.08.006.
[23]	J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115. doi: 10.1016/j.cam.2012.10.003.
[24]	_____, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83 (2014), 2101-2126. doi: 10.1090/S0025-5718-2014-02852-4.
[25]	R. Wang, X. Wang, Q. Zhai and R. Zhang, A weak Galerkin finite element scheme for solving the stationary Stokes equations, J. Comput. Appl. Math., 302 (2016), 171-185. doi: 10.1016/j.cam.2016.01.025.
[26]	X. Wang, Q. Zhai and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow, J. Comput. Appl. Math., 307 (2016), 13-24. doi: 10.1016/j.cam.2016.04.031.
[27]	H. Xie, Q. Zhai and R. Zhang, The weak Galerkin method for eigenvalue problems, arXiv: 1508.05304, (2015).
[28]	Q. Zhai, R. Zhang and X. Wang, A hybridized weak Galerkin finite element scheme for the Stokes equations, Sci. China Math., 58 (2015), 2455-2472. doi: 10.1007/s11425-015-5030-4.
[29]	R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order, J. Sci. Comput., 64 (2015), 559-585. doi: 10.1007/s10915-014-9945-7.

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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$h$	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

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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$h$	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

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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$h$	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

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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$h$	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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2018 Impact Factor: 1.008

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2018 Impact Factor: 1.008

American Institute of Mathematical Sciences