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

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.

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

 Citation:

• 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

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

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

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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