American Institute of Mathematical Sciences

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

 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.

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.

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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.
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
 $h$ 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
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
 $h$ 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
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
 $h$ 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
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
 $h$ 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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