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doi: 10.3934/dcdsb.2022126
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An adaptive finite element method for the elastic transmission eigenvalue problem

Xuqing Zhang 1,2, , Jiayu Han 1, and  Yidu Yang 1,,

1. 

School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China
2. 

School of Biology & Engineering, Guizhou Medical University, Guiyang 550025, China

* Corresponding author: ydyang@gznu.edu.cn (Yidu Yang)

Received  September 2021 Revised  May 2022 Early access July 2022

Fund Project: The first author is supported by the National Natural Science Foundation of China (No.11561014, Grants.12001130), the Science and Technology Foundation of Guizhou Province (No.ZK[2021]012), the Project of Postgraduate for Scientific Research of Education Department of Guizhou Province (No.YJSCXJH[2020]108)

We discuss the a posteriori error estimates of the H$ ^{2} $-conforming finite element for the elastic transmission eigenvalue problem, which is of fourth order and non-selfadjoint, with vector-valued eigenfunctions. We first introduce the error indicators for primal eigenfunction, dual eigenfunction and eigenvalue, respectively. Then the reliability and efficiency of the indicators for both eigenfunctions are proved and an adaptive algorithm of residual type is designed. Finally, we implement numerical experiments to validate our theoretical analysis and show the robustness of the algorithm.

Keywords: Elastic transmission eigenvalues, a posterior error estimates, adaptive algorithm.
Mathematics Subject Classification: Primary: 65N25, 65N30.
Citation: Xuqing Zhang, Jiayu Han, Yidu Yang. An adaptive finite element method for the elastic transmission eigenvalue problem. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022126
References:
[1]

M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience, New York, 2000. doi: 10.1002/9781118032824.

[2]

I. Babuska and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), 736-754.  doi: 10.1137/0715049.

[3]

G. Bao, G. Hu, J. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007.

[4]

C. Bellis, F. Cakoni and B. B. Guzina, Nature of the transmission eigenvalue spectrum for elastic bodies, IMA J. Appl. Math., 78 (2013), 895-923.  doi: 10.1093/imamat/hxr070.

[5]

Ju. M. Berezanski$\breve{i}$, Expansions in Eigenfunctions of Selfadjoint Operators, Naukova Dumka, Kiev, 1965; English transl., Transl. Math. Monos., 17, Amer. Math. Soc., Providence, R.I., 1968.

[6]

L. Chen, IFEM: An Innovative Finite Element Methods Package in MATLAB, Technical Report, University of California at Irvine, 2009.

[7]

Z. Chen and R. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math., 84 (2000), 527-548.  doi: 10.1007/s002110050009.

[8]

P. G. Ciarlet, Basic error estimates for elliptic problems, in: P. G. Ciarlet, J. L. Lions, (Ed.), Finite Element Methods (Part1), Handbook of Numerical Analysis, 2, Elsevier Science Publishers, North-Holand, 1991, 21–343.

[9]

X. Dai, J. Xu and A. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. Math., 110 (2008), 313-355.  doi: 10.1007/s00211-008-0169-3.

[10]

S. Du and Z. Zhang, A robust residual-type a posteriori error estimator for convection-diffusion equations, J. Sci. Comput., 65 (2015), 138-170.  doi: 10.1007/s10915-014-9972-4.

[11]

B. Gong, J. Han, J. Sun and Z. Zhang, A shifted-inverse adaptive multigrid method for the elastic eigenvalue problem, Commun. Comput. Phys., 27 (2020), 251-273.  doi: 10.4208/cicp.OA-2018-0293.

[12]

J. Han and Y. Yang, An adaptive finite element method for the transmission eigenvalue problem, J. Sci. Comput., 69 (2016), 1279-1300.  doi: 10.1007/s10915-016-0234-5.

[13]

X. Ji, J. Sun and P. Li, Computation of interior elastic transmission eigenvalues using a conforming finite element and the secant method, Results Appli. Math., 5 (2020), Paper No. 100083, 12 pp. doi: 10.1016/j.rinam.2019.100083.

[14]

H. Li and Y. Yang, An adaptive C$^{0}$IPG method for the Helmholtz transmission eigenvalue problem, Science China Mathematics, 61 (2018), 1519-1542.  doi: 10.1007/s11425-017-9334-9.

[15]

J. Maubach, Local bisection refinement for n-simplicial grids generated by reflection, SIAM J. Sci. Comput., 16 (1995), 210-227.  doi: 10.1137/0916014.

[16]

P. Morin, R.H. Nochetto and K. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), 466-488.  doi: 10.1137/S0036142999360044.

[17]

Z. Shi and M. Wang, Finite Element Methods, Scientific Publishers, Beijing, 2013.

[18]

J. Sun and A. Zhou, Finite Element Methods for Eigenvalue Problems, Boca Raton, London, New York: CRC Press, Taylor & Francis, Group, 2016.

[19]

R. Verf$\ddot{u}$rth, A posteriori error estimators for the Stokes equations, Numer. Math., 55 (1989), 309-325.  doi: 10.1007/BF01390056.

[20]

R. Verf$\ddot{u}$rth, A posteriori error estimators for convection-diffusion equations, Numer. Math., 80 (1998), 641-663.  doi: 10.1007/s002110050381.

[21]

R. Verf$\ddot{u}$rth, A Review of a Posteriori Error Estimates and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, 1996.

[22]

Y. Xi, H. Geng and X. Ji, A $\rm {C^0IP}$ method of transmission eigenvalues for elastic waves, J. Comput. Phys., 374 (2018), 237-248.  doi: 10.1016/j.jcp.2018.07.053.

[23]

J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators formildly structured grids, Math. Comput., 73 (2004), 1139-1152.  doi: 10.1090/S0025-5718-03-01600-4.

[24]

Y. Yang, J. Han and H. Bi, Error estimates and a two grid scheme for approximating transmission eigenvalues, preprint, arXiv: 1506.06486v2 [math.NA].

[25]

Y. Yang, J. Han and H. Bi, H$^{2}$-conforming methods and two-grid discretizations for the elastic transmission eigenvalue problem, Commun. Comput. Phys., 28 (2020), 1366-1388.  doi: 10.4208/cicp.OA-2019-0171.

[26]

Y. Yang, J. Han, H. Bi, H. Li and Y. Zhang, Mixed methods for the elastic transmission eigenvalue problem, Appli. Math. Comput., 374 (2020), 125081, 15 pp. doi: 10.1016/j.amc.2020.125081.

[27]

Y. Yang, Y. Zhang and H. Bi, A type of adaptive C$^{0}$ non-conforming finite element method for the Helmholtz transmission eigenvalue problem, Comput. Methods Appli. Mech. Engrg., 360 (2020), 112697, 20 pp. doi: 10.1016/j.cma.2019.112697.

[28]

O. Zienkiewicz and J. Zhu, The superconvergent patch recovery and a posteriori error estimates, Part 1: The recovery technique, Int. J. Numer. Methods Eng., 33 (1992), 1331-1364.  doi: 10.1002/nme.1620330702.

show all references

References:
[1]

M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience, New York, 2000. doi: 10.1002/9781118032824.

[2]

I. Babuska and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), 736-754.  doi: 10.1137/0715049.

[3]

G. Bao, G. Hu, J. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007.

[4]

C. Bellis, F. Cakoni and B. B. Guzina, Nature of the transmission eigenvalue spectrum for elastic bodies, IMA J. Appl. Math., 78 (2013), 895-923.  doi: 10.1093/imamat/hxr070.

[5]

Ju. M. Berezanski$\breve{i}$, Expansions in Eigenfunctions of Selfadjoint Operators, Naukova Dumka, Kiev, 1965; English transl., Transl. Math. Monos., 17, Amer. Math. Soc., Providence, R.I., 1968.

[6]

L. Chen, IFEM: An Innovative Finite Element Methods Package in MATLAB, Technical Report, University of California at Irvine, 2009.

[7]

Z. Chen and R. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math., 84 (2000), 527-548.  doi: 10.1007/s002110050009.

[8]

P. G. Ciarlet, Basic error estimates for elliptic problems, in: P. G. Ciarlet, J. L. Lions, (Ed.), Finite Element Methods (Part1), Handbook of Numerical Analysis, 2, Elsevier Science Publishers, North-Holand, 1991, 21–343.

[9]

X. Dai, J. Xu and A. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. Math., 110 (2008), 313-355.  doi: 10.1007/s00211-008-0169-3.

[10]

S. Du and Z. Zhang, A robust residual-type a posteriori error estimator for convection-diffusion equations, J. Sci. Comput., 65 (2015), 138-170.  doi: 10.1007/s10915-014-9972-4.

[11]

B. Gong, J. Han, J. Sun and Z. Zhang, A shifted-inverse adaptive multigrid method for the elastic eigenvalue problem, Commun. Comput. Phys., 27 (2020), 251-273.  doi: 10.4208/cicp.OA-2018-0293.

[12]

J. Han and Y. Yang, An adaptive finite element method for the transmission eigenvalue problem, J. Sci. Comput., 69 (2016), 1279-1300.  doi: 10.1007/s10915-016-0234-5.

[13]

X. Ji, J. Sun and P. Li, Computation of interior elastic transmission eigenvalues using a conforming finite element and the secant method, Results Appli. Math., 5 (2020), Paper No. 100083, 12 pp. doi: 10.1016/j.rinam.2019.100083.

[14]

H. Li and Y. Yang, An adaptive C$^{0}$IPG method for the Helmholtz transmission eigenvalue problem, Science China Mathematics, 61 (2018), 1519-1542.  doi: 10.1007/s11425-017-9334-9.

[15]

J. Maubach, Local bisection refinement for n-simplicial grids generated by reflection, SIAM J. Sci. Comput., 16 (1995), 210-227.  doi: 10.1137/0916014.

[16]

P. Morin, R.H. Nochetto and K. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), 466-488.  doi: 10.1137/S0036142999360044.

[17]

Z. Shi and M. Wang, Finite Element Methods, Scientific Publishers, Beijing, 2013.

[18]

J. Sun and A. Zhou, Finite Element Methods for Eigenvalue Problems, Boca Raton, London, New York: CRC Press, Taylor & Francis, Group, 2016.

[19]

R. Verf$\ddot{u}$rth, A posteriori error estimators for the Stokes equations, Numer. Math., 55 (1989), 309-325.  doi: 10.1007/BF01390056.

[20]

R. Verf$\ddot{u}$rth, A posteriori error estimators for convection-diffusion equations, Numer. Math., 80 (1998), 641-663.  doi: 10.1007/s002110050381.

[21]

R. Verf$\ddot{u}$rth, A Review of a Posteriori Error Estimates and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, 1996.

[22]

Y. Xi, H. Geng and X. Ji, A $\rm {C^0IP}$ method of transmission eigenvalues for elastic waves, J. Comput. Phys., 374 (2018), 237-248.  doi: 10.1016/j.jcp.2018.07.053.

[23]

J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators formildly structured grids, Math. Comput., 73 (2004), 1139-1152.  doi: 10.1090/S0025-5718-03-01600-4.

[24]

Y. Yang, J. Han and H. Bi, Error estimates and a two grid scheme for approximating transmission eigenvalues, preprint, arXiv: 1506.06486v2 [math.NA].

[25]

Y. Yang, J. Han and H. Bi, H$^{2}$-conforming methods and two-grid discretizations for the elastic transmission eigenvalue problem, Commun. Comput. Phys., 28 (2020), 1366-1388.  doi: 10.4208/cicp.OA-2019-0171.

[26]

Y. Yang, J. Han, H. Bi, H. Li and Y. Zhang, Mixed methods for the elastic transmission eigenvalue problem, Appli. Math. Comput., 374 (2020), 125081, 15 pp. doi: 10.1016/j.amc.2020.125081.

[27]

Y. Yang, Y. Zhang and H. Bi, A type of adaptive C$^{0}$ non-conforming finite element method for the Helmholtz transmission eigenvalue problem, Comput. Methods Appli. Mech. Engrg., 360 (2020), 112697, 20 pp. doi: 10.1016/j.cma.2019.112697.

[28]

O. Zienkiewicz and J. Zhu, The superconvergent patch recovery and a posteriori error estimates, Part 1: The recovery technique, Int. J. Numer. Methods Eng., 33 (1992), 1331-1364.  doi: 10.1002/nme.1620330702.

Figure 1.  Error curves for the 1st (left top) and 5th (right top) eigenvalues on $ D_{L} $ and the 1st (left bottom) and 5th (right bottom) eigenvalues on $ D_{S} $ in Case 1
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Figure 2.  Error curves for the 1st (left top) and 3rd (right top) eigenvalues on $ D_{L} $ and the 1st (left bottom) and 3rd (right bottom) eigenvalues on $ D_{S} $ in Case 2
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Figure 3.  Error curves of the 1st eigenvalue on $ D_{L} $ (left) and 1st eigenvalue on $ D_{S} $ (right) in Case 3
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Figure 4.  Error curves of the 1st eigenvalue on $ D_{L} $ (left) and 1st eigenvalue on $ D_{S} $ (right) in Case 4
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Figure 5.  Error curves of the 1st eigenvalue on $ D_{L} $ (left) and 1st eigenvalue on $ D_{S} $ (right) in Case 5
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Figure 6.  Adaptive meshes obtained by Algorithm 4.1 at 35 iteration for the 1st eigenvalue on $ D_{L} $ (left) and 1st eigenvalue on $ D_{S} $ (right) in Case 5
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Table 1.  The eigenvalues obtained by the Argyris element on uniform meshes in Case 1
$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{5} $ dof $ \gamma_{1} $ $ \gamma_{5} $
$ \frac{\sqrt{2}}{32} $ 26368 13.614085 24.446563 35256 3.059038 4.867139
$ \frac{\sqrt{2}}{64} $ 108032 13.582410 24.426026 144248 3.028473 4.857275
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$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{5} $ dof $ \gamma_{1} $ $ \gamma_{5} $
$ \frac{\sqrt{2}}{32} $ 26368 13.614085 24.446563 35256 3.059038 4.867139
$ \frac{\sqrt{2}}{64} $ 108032 13.582410 24.426026 144248 3.028473 4.857275
Table 2.  The eigenvalues obtained by Algorithm 4.1 in Case 1
$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ dof $ \gamma_{5}^{A} $ dof $ \gamma_{1}^{A} $ dof $ \gamma_{5}^{A} $
3 6524 13.634260 6532 24.458026 9020 3.062634 8768 4.870775
4 6648 13.608574 6648 24.442271 9432 3.045809 9056 4.863849
6 6988 13.580336 7060 24.424625 10344 3.022103 9680 4.855441
7 7132 13.573047 7348 24.420141 10848 3.014457 9984 4.852901
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$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ dof $ \gamma_{5}^{A} $ dof $ \gamma_{1}^{A} $ dof $ \gamma_{5}^{A} $
3 6524 13.634260 6532 24.458026 9020 3.062634 8768 4.870775
4 6648 13.608574 6648 24.442271 9432 3.045809 9056 4.863849
6 6988 13.580336 7060 24.424625 10344 3.022103 9680 4.855441
7 7132 13.573047 7348 24.420141 10848 3.014457 9984 4.852901
Table 3.  The eigenvalues obtained by the Argyris element on uniform meshes in Case 2
$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{3} $ dof $ \gamma_{1} $ $ \gamma_{3} $
$ \frac{\sqrt{2}}{32} $ 26368 3.621459-3.0381i 4.889479 35256 0.971932 0.896336-0.6230i
$ \frac{\sqrt{2}}{64} $ 108032 3.615710-3.0408i 4.875826 144248 0.959238 0.891335-0.6225i
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$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{3} $ dof $ \gamma_{1} $ $ \gamma_{3} $
$ \frac{\sqrt{2}}{32} $ 26368 3.621459-3.0381i 4.889479 35256 0.971932 0.896336-0.6230i
$ \frac{\sqrt{2}}{64} $ 108032 3.615710-3.0408i 4.875826 144248 0.959238 0.891335-0.6225i
Table 4.  The eigenvalues obtained by Algorithm 4.1 in Case 2
$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ dof $ \gamma_{3}^{A} $ dof $ \gamma_{1}^{A} $ dof $ \gamma_{3}^{A} $
3 6676 3.625462-3.0360i 6756 4.901629 8912 0.975513 9488 0.896345-0.6228i
4 6864 3.620528-3.0385i 7944 4.887311 9700 0.964711 9988 0.893712-0.6229i
5 7096 3.617741-3.0398i 8140 4.882162 10744 0.960943 10780 0.891353-0.6224i
6 7492 3.615287-3.0410i 9184 4.875004 11280 0.955730 11424 0.890078-0.6225i
7 7832 3.613902-3.0416i 9380 4.872093 12432 0.953872 12824 0.888933-0.6222i
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$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ dof $ \gamma_{3}^{A} $ dof $ \gamma_{1}^{A} $ dof $ \gamma_{3}^{A} $
3 6676 3.625462-3.0360i 6756 4.901629 8912 0.975513 9488 0.896345-0.6228i
4 6864 3.620528-3.0385i 7944 4.887311 9700 0.964711 9988 0.893712-0.6229i
5 7096 3.617741-3.0398i 8140 4.882162 10744 0.960943 10780 0.891353-0.6224i
6 7492 3.615287-3.0410i 9184 4.875004 11280 0.955730 11424 0.890078-0.6225i
7 7832 3.613902-3.0416i 9380 4.872093 12432 0.953872 12824 0.888933-0.6222i
Table 5.  The eigenvalues obtained by the Argyris element on uniform meshes in Case 3
$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{5} $ dof $ \gamma_{1} $ $ \gamma_{5} $
$ \frac{\sqrt{2}}{32} $ 26368 26.411851 40.467203 35256 5.322456 7.480636
$ \frac{\sqrt{2}}{64} $ 108032 26.332569 40.392475 144248 5.248736 7.343923
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$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{5} $ dof $ \gamma_{1} $ $ \gamma_{5} $
$ \frac{\sqrt{2}}{32} $ 26368 26.411851 40.467203 35256 5.322456 7.480636
$ \frac{\sqrt{2}}{64} $ 108032 26.332569 40.392475 144248 5.248736 7.343923
Table 6.  The eigenvalues obtained by Algorithm 4.1 in Case 3
$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $
3 9260 26.472283 31 50184 26.250754 3 12580 5.331945 31 75052 5.176916
4 10428 26.395139 32 54408 26.250749 4 13732 5.272581 32 87984 5.176912
5 10608 26.365732 33 61668 26.250747 5 14200 5.254344 33 90320 5.176911
6 10876 26.326721 34 66828 26.250745 6 15204 5.224609 34 106900 5.176909
7 11360 26.308618 35 74284 26.250744 7 16164 5.215418 35 108300 5.176908
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$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $
3 9260 26.472283 31 50184 26.250754 3 12580 5.331945 31 75052 5.176916
4 10428 26.395139 32 54408 26.250749 4 13732 5.272581 32 87984 5.176912
5 10608 26.365732 33 61668 26.250747 5 14200 5.254344 33 90320 5.176911
6 10876 26.326721 34 66828 26.250745 6 15204 5.224609 34 106900 5.176909
7 11360 26.308618 35 74284 26.250744 7 16164 5.215418 35 108300 5.176908
Table 7.  The eigenvalues obtained by the Argyris element on uniform meshes in Case 4
$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{5} $ dof $ \gamma_{1} $ $ \gamma_{5} $
$ \frac{\sqrt{2}}{32} $ 26368 26.257960 40.241343 35256 5.293323 7.445605
$ \frac{\sqrt{2}}{64} $ 108032 26.178143 40.165951 144248 5.219140 7.303772
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$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{5} $ dof $ \gamma_{1} $ $ \gamma_{5} $
$ \frac{\sqrt{2}}{32} $ 26368 26.257960 40.241343 35256 5.293323 7.445605
$ \frac{\sqrt{2}}{64} $ 108032 26.178143 40.165951 144248 5.219140 7.303772
Table 8.  The eigenvalues obtained by Algorithm 4.1 in Case 4
$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $
3 10556 26.320142 31 50004 26.095984 3 13868 5.301156 31 75572 5.147082
4 11364 26.240466 32 54228 26.095979 4 14236 5.243974 32 89092 5.147077
5 11976 26.209004 33 61596 26.095974 5 14704 5.224866 33 91104 5.147077
6 12460 26.171256 34 66936 26.095970 6 15780 5.195242 34 107800 5.147074
7 12684 26.154500 35 74392 26.095969 7 16860 5.185459 35 109236 5.147073
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$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $
3 10556 26.320142 31 50004 26.095984 3 13868 5.301156 31 75572 5.147082
4 11364 26.240466 32 54228 26.095979 4 14236 5.243974 32 89092 5.147077
5 11976 26.209004 33 61596 26.095974 5 14704 5.224866 33 91104 5.147077
6 12460 26.171256 34 66936 26.095970 6 15780 5.195242 34 107800 5.147074
7 12684 26.154500 35 74392 26.095969 7 16860 5.185459 35 109236 5.147073
Table 9.  The eigenvalues obtained by the Argyris element on uniform meshes in Case 5
$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{5} $ dof $ \gamma_{1} $ $ \gamma_{5} $
$ \frac{\sqrt{2}}{32} $ 26368 26.234587 40.205701 35256 5.288633 7.439160
$ \frac{\sqrt{2}}{64} $ 108032 26.154882 40.130586 144248 5.214645 7.298399
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$ D_{L} $ $ D_{S} $
$ h $ dof $ \gamma_{1} $ $ \gamma_{5} $ dof $ \gamma_{1} $ $ \gamma_{5} $
$ \frac{\sqrt{2}}{32} $ 26368 26.234587 40.205701 35256 5.288633 7.439160
$ \frac{\sqrt{2}}{64} $ 108032 26.154882 40.130586 144248 5.214645 7.298399
Table 10.  The eigenvalues obtained by Algorithm 4.1 in Case 5
$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $
3 8576 26.297210 24 48872 26.071283 3 10820 5.314777 12 20704 5.152103
4 9044 26.222016 25 52864 26.071256 4 11324 5.297888 31 93392 5.141918
5 9816 26.187276 26 57928 26.072189 5 12076 5.239717 32 97772 5.141916
6 10696 26.148331 27 63680 26.073274 6 12940 5.220496 33 107980 5.141892
23 45216 26.071386 28 69092 26.074004 11 19336 5.154613 34 116288 5.141958
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$ D_{L} $ $ D_{S} $
$ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $ $ l $ dof $ \gamma_{1}^{A} $
3 8576 26.297210 24 48872 26.071283 3 10820 5.314777 12 20704 5.152103
4 9044 26.222016 25 52864 26.071256 4 11324 5.297888 31 93392 5.141918
5 9816 26.187276 26 57928 26.072189 5 12076 5.239717 32 97772 5.141916
6 10696 26.148331 27 63680 26.073274 6 12940 5.220496 33 107980 5.141892
23 45216 26.071386 28 69092 26.074004 11 19336 5.154613 34 116288 5.141958
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