American Institute of Mathematical Sciences

Advanced
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  January 2019 Early access  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 and 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$1/41/81/161/321/641/128 $\lambda$
$ \lambda_{1, h}$1.5407221.8517691.9580221.9885871.9969331.9991782
rate1.63151.82021.87901.89551.9001
$\tilde \lambda_{1, h}$2.2236732.0685472.0182672.0046932.0011892.0002992
rate1.70621.90791.96061.98091.9905
$ \lambda_{2, h}$2.8218314.1351904.7349214.9262974.9800944.9946675
rate1.33271.70601.84661.88851.9001
$\tilde \lambda_{2, h}$6.5882455.4842825.1255465.0318245.0080125.0020105
rate1.71351.94761.98001.98991.9947
$ \lambda_{3, h}$2.8362094.1427924.7373014.9269554.9802704.9947135
rate1.33581.70621.84651.88841.8999
$\tilde \lambda_{3, h}$5.9097235.3043625.0797235.0202845.0051215.0012875
rate1.57961.93271.97471.98591.9921
$ \lambda_{4, h}$3.6816496.0619177.3681797.8204657.9511457.9868668
rate1.15581.61701.81521.87771.8952
$\tilde \lambda_{4, h}$11.6153289.1581618.2977568.0754928.0190498.0047898
rate1.64231.95961.97971.98661.9920
Table Options

Download as excel

$h$1/41/81/161/321/641/128 $\lambda$
$ \lambda_{1, h}$1.5407221.8517691.9580221.9885871.9969331.9991782
rate1.63151.82021.87901.89551.9001
$\tilde \lambda_{1, h}$2.2236732.0685472.0182672.0046932.0011892.0002992
rate1.70621.90791.96061.98091.9905
$ \lambda_{2, h}$2.8218314.1351904.7349214.9262974.9800944.9946675
rate1.33271.70601.84661.88851.9001
$\tilde \lambda_{2, h}$6.5882455.4842825.1255465.0318245.0080125.0020105
rate1.71351.94761.98001.98991.9947
$ \lambda_{3, h}$2.8362094.1427924.7373014.9269554.9802704.9947135
rate1.33581.70621.84651.88841.8999
$\tilde \lambda_{3, h}$5.9097235.3043625.0797235.0202845.0051215.0012875
rate1.57961.93271.97471.98591.9921
$ \lambda_{4, h}$3.6816496.0619177.3681797.8204657.9511457.9868668
rate1.15581.61701.81521.87771.8952
$\tilde \lambda_{4, h}$11.6153289.1581618.2977568.0754928.0190498.0047898
rate1.64231.95961.97971.98661.9920
Table 2.  Numerical results for the eigenvalues with $k = 2$
$h$1/41/81/161/321/641/128 $\lambda$
$ \lambda_{1, h}$1.9776231.9985941.9999071.9999941.99999951.999999972
rate3.99223.92603.90603.90063.8999
$\tilde \lambda_{1, h}$2.0477972.0014162.0000602.0000032.00000022.000000012
rate5.07694.56404.36684.24954.1640
$ \lambda_{2, h}$4.5626394.9733614.9982984.9998884.9999924.99999945
rate4.03723.96833.91933.90493.9011
$\tilde \lambda_{2, h}$5.9650945.0255285.0009955.0000485.0000035.00000015
rate5.24054.68074.37694.23754.1526
$ \lambda_{3, h}$4.6566974.9801594.9987394.9999174.9999944.99999965
rate4.11303.97573.92083.90563.9016
$\tilde \lambda_{3, h}$6.3520005.0234315.0007845.0000345.0000025.000000085
rate5.85054.90174.52284.35184.2405
$ \lambda_{4, h}$6.4237917.9023457.9939407.9996037.9999747.9999988
rate4.01264.01033.93023.90733.9011
$\tilde \lambda_{4, h}$14.1744538.1325128.0042088.0001908.0000108.0000018
rate5.54214.97684.46604.27514.1726
Table Options

Download as excel

$h$1/41/81/161/321/641/128 $\lambda$
$ \lambda_{1, h}$1.9776231.9985941.9999071.9999941.99999951.999999972
rate3.99223.92603.90603.90063.8999
$\tilde \lambda_{1, h}$2.0477972.0014162.0000602.0000032.00000022.000000012
rate5.07694.56404.36684.24954.1640
$ \lambda_{2, h}$4.5626394.9733614.9982984.9998884.9999924.99999945
rate4.03723.96833.91933.90493.9011
$\tilde \lambda_{2, h}$5.9650945.0255285.0009955.0000485.0000035.00000015
rate5.24054.68074.37694.23754.1526
$ \lambda_{3, h}$4.6566974.9801594.9987394.9999174.9999944.99999965
rate4.11303.97573.92083.90563.9016
$\tilde \lambda_{3, h}$6.3520005.0234315.0007845.0000345.0000025.000000085
rate5.85054.90174.52284.35184.2405
$ \lambda_{4, h}$6.4237917.9023457.9939407.9996037.9999747.9999988
rate4.01264.01033.93023.90733.9011
$\tilde \lambda_{4, h}$14.1744538.1325128.0042088.0001908.0000108.0000018
rate5.54214.97684.46604.27514.1726
Table 3.  Numerical results for the Lshape domain with $k = 1$
$h$1/41/81/161/321/641/128
$ \lambda_{1, h}$5.9071138.1544179.1558699.4904419.5932509.624793
$\tilde \lambda_{1, h}$11.65628010.4983239.9081789.7224189.6660849.648513
$ \lambda_{2, h}$8.20118812.30882114.29331414.94377915.12854815.178811
$\tilde \lambda_{2, h}$17.97432316.39626615.53354315.28490815.21963315.202913
$ \lambda_{3, h}$9.44449415.20567618.27619419.32490519.62657119.708934
$\tilde \lambda_{3, h}$26.26208322.35362120.45371219.92345319.78598219.750997
$ \lambda_{4, h}$11.12416520.25829226.28719328.58015929.26373929.452134
$\tilde \lambda_{4, h}$44.27037134.58864430.89133329.87077229.60976629.543696
Table Options

Download as excel

$h$1/41/81/161/321/641/128
$ \lambda_{1, h}$5.9071138.1544179.1558699.4904419.5932509.624793
$\tilde \lambda_{1, h}$11.65628010.4983239.9081789.7224189.6660849.648513
$ \lambda_{2, h}$8.20118812.30882114.29331414.94377915.12854815.178811
$\tilde \lambda_{2, h}$17.97432316.39626615.53354315.28490815.21963315.202913
$ \lambda_{3, h}$9.44449415.20567618.27619419.32490519.62657119.708934
$\tilde \lambda_{3, h}$26.26208322.35362120.45371219.92345319.78598219.750997
$ \lambda_{4, h}$11.12416520.25829226.28719328.58015929.26373929.452134
$\tilde \lambda_{4, h}$44.27037134.58864430.89133329.87077229.60976629.543696
Table 4.  Numerical results for the Lshape domain with $k = 2$
$h$1/41/81/161/321/641/128
$ \lambda_{1, h}$9.0765419.5561569.6158129.6308329.6362239.638331
$\tilde \lambda_{1, h}$12.34563410.0750579.7904039.6983579.6629559.648961
$ \lambda_{2, h}$13.58424115.09738515.19041715.19672215.19720415.197247
$\tilde \lambda_{2, h}$22.66687315.34652015.20291815.19760515.19728515.197256
$ \lambda_{3, h}$16.28028719.51796819.72532519.73829619.73914819.739205
$\tilde \lambda_{3, h}$35.04403020.07688019.75014919.73970019.73923419.739210
$ \lambda_{4, h}$19.96634828.77751229.47617729.51850429.52127829.521467
$\tilde \lambda_{4, h}$74.08112330.74643229.55451829.52292929.52156029.521486
Table Options

Download as excel

$h$1/41/81/161/321/641/128
$ \lambda_{1, h}$9.0765419.5561569.6158129.6308329.6362239.638331
$\tilde \lambda_{1, h}$12.34563410.0750579.7904039.6983579.6629559.648961
$ \lambda_{2, h}$13.58424115.09738515.19041715.19672215.19720415.197247
$\tilde \lambda_{2, h}$22.66687315.34652015.20291815.19760515.19728515.197256
$ \lambda_{3, h}$16.28028719.51796819.72532519.73829619.73914819.739205
$\tilde \lambda_{3, h}$35.04403020.07688019.75014919.73970019.73923419.739210
$ \lambda_{4, h}$19.96634828.77751229.47617729.51850429.52127829.521467
$\tilde \lambda_{4, h}$74.08112330.74643229.55451829.52292929.52156029.521486
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