# American Institute of Mathematical Sciences

October  2018, 23(8): 3195-3212. doi: 10.3934/dcdsb.2018281

## Nonconforming elements of class $L^2$ for Helmholtz transmission eigenvalue problems

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

* Corresponding author: hanjiayu126@126.com

Received  June 2017 Revised  November 2017 Published  October 2018 Early access  August 2018

Fund Project: Supported by National Natural Science Foundation of China (grant Nos. 11561014 and 11761022).

For solving the Helmholtz transmission eigenvalue problem, we use the mixed formulation of Cakoni et al. to construct a new nonconforming element discretization. Based on the discretization, this paper first discuss the nonconforming element methods of class $L^2$, and prove the error estimates of the discrete eigenvalues obtained by the cubic tetrahedron element, incomplete cubic tetrahedral element and Morley element et al. We report some numerical examples using the nonconforming elements mixed with linear Lagrange element to show that our discretization can obtain the transmission eigenvalues of higher accuracy in 3D domains than the nonconforming element discretization in the existing literature.

Citation: Jiayu Han. Nonconforming elements of class $L^2$ for Helmholtz transmission eigenvalue problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3195-3212. doi: 10.3934/dcdsb.2018281
##### References:
 [1] J. An and J. Shen, A spectral-element method for transmission eigenvalue problems, J. Sci. Comput., 57 (2013), 670-688.  doi: 10.1007/s10915-013-9720-1. [2] I. Babuska, J. E. Osborn, Eigenvalue Problems, in: P. G. Ciarlet, J. L. Lions, (Ed. ), Finite Element Methods (Part 1), Handbook of Numerical Analysis, Elsevier Science Publishers, North-Holand, 2 (1991), 641-787. [3] H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math Method Appl Sci, 2 (1980), 556-581.  doi: 10.1002/mma.1670020416. [4] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Springer-Verlag, New york, 2002. doi: 10.1007/978-1-4757-3658-8. [5] F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.  doi: 10.1080/00036810802713966. [6] F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scttering problem for anisotropic media, Inverse Problems, 26 (2010), 074004, 14pp. doi: 10.1088/0266-5611/26/7/074004. [7] F. Cakoni, P. Monk and J. Sun, Error analysis for the finite element approximation of transmission eigenvalues, Comput. Meth. Appl. Math., 14 (2014), 419-427.  doi: 10.1515/cmam-2014-0021. [8] L. Chen, iFEM: An integrated finite element method package in MATLAB., Technical Report, University of California at Irvine, 2009. [9] P. G. Ciarlet, Basic error estimates for elliptic proplems, in: P. G. Ciarlet, J. L. Lions, (Ed. ), Finite Element Methods (Part1), Handbook of Numerical Analysis, vol. 2, Elsevier Science Publishers, North-Holand, 1991, 21-343. [10] D. Colton, P. Monk and J. Sun, Analytical and computational methods for transmission eigenvalues, Inverse Problems, 26 (2010), 045011, 16pp. doi: 10.1088/0266-5611/26/4/045011. [11] D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Problem Imaging, 1 (2007), 13-28.  doi: 10.3934/ipi.2007.1.13. [12] H. Geng, X. Ji, J. Sun and L. Xu, $C^0IP$ methods for the transmission eigenvalue problem, J. Sci. Comput., 68 (2016), 326-338.  doi: 10.1007/s10915-015-0140-2. [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. [14] J. Han and Y. Yang, An adaptive finite element method for the transmission eigenvalue problem, J. Sci. Comput., 69 (2016), 1-22.  doi: 10.1007/s10915-016-0234-5. [15] X. Ji, J. Sun and T. Turner, Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues, ACM Transaction on Math. Soft., 38 (2012), Art. 29, 8 pp. doi: 10.1145/2331130.2331137. [16] X. Ji, J. Sun and H. Xie, A multigrid method for Helmholtz transmission eigenvalue problems, J. Sci. Comput., 60 (2014), 276-294.  doi: 10.1007/s10915-013-9794-9. [17] A. Kleefeld, A numerical method to compute interior transmission eigenvalues, Inverse Problems, 29 (2013), 104012, 20pp. doi: 10.1088/0266-5611/29/10/104012. [18] J. T. Oden and J. N. Reddy, An Introduction to the Mathematical Theory of Finite Elements, New York-London-Sydney, 1976. [19] B. P. Rynne and B. D. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal., 22 (1991), 1755-1762.  doi: 10.1137/0522109. [20] Z. Shi and M. Wang, Finite Element Methods, Beijing, Scientific Publishers, 2013. [21] J. Sun, Estimation of transmission eigenvalues and the index of refraction from Cauchy data, Inverse Probl, 27 (2011), 015009, 24pp. doi: 10.1088/0266-5611/27/1/015009. [22] J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860-1874.  doi: 10.1137/100785478. [23] M. Wang, Z. Shi and J. Xu, A new class of Zienkiewicz-type nonconforming element in any dimensions, Numer. Math., 106 (2007), 335-347.  doi: 10.1007/s00211-007-0063-4. [24] M. Wang and J. Xu, Nonconforming tetrahedral finite elements for fourth order elliptic equations, Math. Comp., 76 (2007), 1-18.  doi: 10.1090/S0025-5718-06-01889-8. [25] Y. Yang, J. Han, H. Bi, Error estimates and a two grid scheme for approximating transmission eigenvalues, arXiv: 1506.06486 V2 [math. NA] 2 Mar 2016. [26] Y. Yang, H. Bi, H. Li and J. Han, Mixed method for the helmholtz transmission eigenvalues, SIAM J. Sci. Comput., 38 (2016), A1383-A1403.  doi: 10.1137/15M1050756. [27] Y. Yang, J. Han and H. Bi, Non-conforming finite element methods for transmission eigenvalue problem, Comput. Methods Appl. Mech. Engrg., 307 (2016), 144-163.  doi: 10.1016/j.cma.2016.04.021. [28] F. Zeng, J. Sun and L. Xu, A spectral projection method for transmission eigenvalues, Sci China Math, 59 (2016), 1613-1622.  doi: 10.1007/s11425-016-0289-8.

show all references

##### References:
 [1] J. An and J. Shen, A spectral-element method for transmission eigenvalue problems, J. Sci. Comput., 57 (2013), 670-688.  doi: 10.1007/s10915-013-9720-1. [2] I. Babuska, J. E. Osborn, Eigenvalue Problems, in: P. G. Ciarlet, J. L. Lions, (Ed. ), Finite Element Methods (Part 1), Handbook of Numerical Analysis, Elsevier Science Publishers, North-Holand, 2 (1991), 641-787. [3] H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math Method Appl Sci, 2 (1980), 556-581.  doi: 10.1002/mma.1670020416. [4] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Springer-Verlag, New york, 2002. doi: 10.1007/978-1-4757-3658-8. [5] F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.  doi: 10.1080/00036810802713966. [6] F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scttering problem for anisotropic media, Inverse Problems, 26 (2010), 074004, 14pp. doi: 10.1088/0266-5611/26/7/074004. [7] F. Cakoni, P. Monk and J. Sun, Error analysis for the finite element approximation of transmission eigenvalues, Comput. Meth. Appl. Math., 14 (2014), 419-427.  doi: 10.1515/cmam-2014-0021. [8] L. Chen, iFEM: An integrated finite element method package in MATLAB., Technical Report, University of California at Irvine, 2009. [9] P. G. Ciarlet, Basic error estimates for elliptic proplems, in: P. G. Ciarlet, J. L. Lions, (Ed. ), Finite Element Methods (Part1), Handbook of Numerical Analysis, vol. 2, Elsevier Science Publishers, North-Holand, 1991, 21-343. [10] D. Colton, P. Monk and J. Sun, Analytical and computational methods for transmission eigenvalues, Inverse Problems, 26 (2010), 045011, 16pp. doi: 10.1088/0266-5611/26/4/045011. [11] D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Problem Imaging, 1 (2007), 13-28.  doi: 10.3934/ipi.2007.1.13. [12] H. Geng, X. Ji, J. Sun and L. Xu, $C^0IP$ methods for the transmission eigenvalue problem, J. Sci. Comput., 68 (2016), 326-338.  doi: 10.1007/s10915-015-0140-2. [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. [14] J. Han and Y. Yang, An adaptive finite element method for the transmission eigenvalue problem, J. Sci. Comput., 69 (2016), 1-22.  doi: 10.1007/s10915-016-0234-5. [15] X. Ji, J. Sun and T. Turner, Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues, ACM Transaction on Math. Soft., 38 (2012), Art. 29, 8 pp. doi: 10.1145/2331130.2331137. [16] X. Ji, J. Sun and H. Xie, A multigrid method for Helmholtz transmission eigenvalue problems, J. Sci. Comput., 60 (2014), 276-294.  doi: 10.1007/s10915-013-9794-9. [17] A. Kleefeld, A numerical method to compute interior transmission eigenvalues, Inverse Problems, 29 (2013), 104012, 20pp. doi: 10.1088/0266-5611/29/10/104012. [18] J. T. Oden and J. N. Reddy, An Introduction to the Mathematical Theory of Finite Elements, New York-London-Sydney, 1976. [19] B. P. Rynne and B. D. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal., 22 (1991), 1755-1762.  doi: 10.1137/0522109. [20] Z. Shi and M. Wang, Finite Element Methods, Beijing, Scientific Publishers, 2013. [21] J. Sun, Estimation of transmission eigenvalues and the index of refraction from Cauchy data, Inverse Probl, 27 (2011), 015009, 24pp. doi: 10.1088/0266-5611/27/1/015009. [22] J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860-1874.  doi: 10.1137/100785478. [23] M. Wang, Z. Shi and J. Xu, A new class of Zienkiewicz-type nonconforming element in any dimensions, Numer. Math., 106 (2007), 335-347.  doi: 10.1007/s00211-007-0063-4. [24] M. Wang and J. Xu, Nonconforming tetrahedral finite elements for fourth order elliptic equations, Math. Comp., 76 (2007), 1-18.  doi: 10.1090/S0025-5718-06-01889-8. [25] Y. Yang, J. Han, H. Bi, Error estimates and a two grid scheme for approximating transmission eigenvalues, arXiv: 1506.06486 V2 [math. NA] 2 Mar 2016. [26] Y. Yang, H. Bi, H. Li and J. Han, Mixed method for the helmholtz transmission eigenvalues, SIAM J. Sci. Comput., 38 (2016), A1383-A1403.  doi: 10.1137/15M1050756. [27] Y. Yang, J. Han and H. Bi, Non-conforming finite element methods for transmission eigenvalue problem, Comput. Methods Appl. Mech. Engrg., 307 (2016), 144-163.  doi: 10.1016/j.cma.2016.04.021. [28] F. Zeng, J. Sun and L. Xu, A spectral projection method for transmission eigenvalues, Sci China Math, 59 (2016), 1613-1622.  doi: 10.1007/s11425-016-0289-8.
Error curves computed by MZ element mixed with linear element for $n = 16$ on the unit square (left top) and on the L-shaped (right top), for $n = 8+x_1-x_2$ on the unit square (left bottom) and on the L-shaped (right bottom)
The eigenvalues obtained by MZ element mixed with linear element in 2D domains
 $n=16$ $n=8+x_1-x_2$ $h$ $j$ $k^S_{j, h}$ $k^L_{j, h}$ $j$ $k^S_{j, h}$ $k^L_{j, h}$ $\frac{\sqrt2}{32}$ 1 1.879083492 1.4773077 1 2.819407434 2.302531586 $\frac{\sqrt2}{64}$ 1 1.879447918 1.4767038 1 2.821437358 2.302554268 $\frac{\sqrt2}{128}$ 1 1.879554183 1.4763944 1 2.821996985 2.302391353 $\frac{\sqrt2}{32}$ 2 2.443168293 1.5694361 2 3.534452084 2.394060079 $\frac{\sqrt2}{64}$ 2 2.443929469 1.5696511 2 3.537524631 2.395270328 $\frac{\sqrt2}{128}$ 2 2.444156467 1.5697071 2 3.538394853 2.395585637 $\frac{\sqrt2}{32}$ 3 2.443168293 1.7049725 5 4.498450-0.871213i 2.928086-0.563389i $\frac{\sqrt2}{64}$ 3 2.443929469 1.7051045 5 4.496942-0.871413i 2.925438-0.564575i $\frac{\sqrt2}{128}$ 3 2.444156467 1.7050978 5 4.496644-0.871466i 2.924644-0.564720i $\frac{\sqrt2}{32}$ 4 2.865221584 1.7827065 6 4.498450+0.871213i 2.928086+0.563389i $\frac{\sqrt2}{64}$ 4 2.866032777 1.7830026 6 4.496942+0.871413i 2.925438+0.564575i $\frac{\sqrt2}{128}$ 4 2.866329734 1.7830869 6 4.496644+0.871466i 2.924644+0.564720i
 $n=16$ $n=8+x_1-x_2$ $h$ $j$ $k^S_{j, h}$ $k^L_{j, h}$ $j$ $k^S_{j, h}$ $k^L_{j, h}$ $\frac{\sqrt2}{32}$ 1 1.879083492 1.4773077 1 2.819407434 2.302531586 $\frac{\sqrt2}{64}$ 1 1.879447918 1.4767038 1 2.821437358 2.302554268 $\frac{\sqrt2}{128}$ 1 1.879554183 1.4763944 1 2.821996985 2.302391353 $\frac{\sqrt2}{32}$ 2 2.443168293 1.5694361 2 3.534452084 2.394060079 $\frac{\sqrt2}{64}$ 2 2.443929469 1.5696511 2 3.537524631 2.395270328 $\frac{\sqrt2}{128}$ 2 2.444156467 1.5697071 2 3.538394853 2.395585637 $\frac{\sqrt2}{32}$ 3 2.443168293 1.7049725 5 4.498450-0.871213i 2.928086-0.563389i $\frac{\sqrt2}{64}$ 3 2.443929469 1.7051045 5 4.496942-0.871413i 2.925438-0.564575i $\frac{\sqrt2}{128}$ 3 2.444156467 1.7050978 5 4.496644-0.871466i 2.924644-0.564720i $\frac{\sqrt2}{32}$ 4 2.865221584 1.7827065 6 4.498450+0.871213i 2.928086+0.563389i $\frac{\sqrt2}{64}$ 4 2.866032777 1.7830026 6 4.496942+0.871413i 2.925438+0.564575i $\frac{\sqrt2}{128}$ 4 2.866329734 1.7830869 6 4.496644+0.871466i 2.924644+0.564720i
The eigenvalues obtained by CT element mixed with linear element on the cube
 $Dof$ $j$ $k^C_{j, h}$($n=16$) $Dof$ $j$ $k^C_{j, h}$($n=8+x_1-x_2$) $1895$ 1 2.0449 1895 1 2.9579 $16179$ 1 2.0604 16179 1 3.0051 $55903$ 1 2.0641 55903 1 3.0162 $1895$ 2, 3, 4 2.5461 1895 2 3.6098 $16179$ 2, 3, 4 2.5714 16179 2 3.6853 $55903$ 2, 3, 4 2.5785 55903 2 3.7049 $1895$ 5, 6, 7 2.9402 1895 3, 4 3.6119, 3.6121 $16179$ 5, 6, 7 2.9683 16179 3, 4 3.6878, 3.6879 $55903$ 5, 6, 7 2.9781 55903 3, 4 3.7074, 3.7075
 $Dof$ $j$ $k^C_{j, h}$($n=16$) $Dof$ $j$ $k^C_{j, h}$($n=8+x_1-x_2$) $1895$ 1 2.0449 1895 1 2.9579 $16179$ 1 2.0604 16179 1 3.0051 $55903$ 1 2.0641 55903 1 3.0162 $1895$ 2, 3, 4 2.5461 1895 2 3.6098 $16179$ 2, 3, 4 2.5714 16179 2 3.6853 $55903$ 2, 3, 4 2.5785 55903 2 3.7049 $1895$ 5, 6, 7 2.9402 1895 3, 4 3.6119, 3.6121 $16179$ 5, 6, 7 2.9683 16179 3, 4 3.6878, 3.6879 $55903$ 5, 6, 7 2.9781 55903 3, 4 3.7074, 3.7075
The eigenvalues obtained by MZ element on the cube
 $Dof$ $j$ $k^C_{j, h}$($n=16$) $Dof$ $j$ $k^C_{j, h}$($n=8+x_1-x_2$) $3608$ 1 2.1651 3608 1 3.2181 $30648$ 1 2.0935 30648 1 3.0747 $105688$ 1 2.0791 105688 1 3.0474 $3608$ 2, 3, 4 2.7794, 2.7803, 2.7807 3608 2 4.0750 $30648$ 2, 3, 4 2.6350, 2.6350, 2.6351 30648 2 3.8098 $105688$ 2, 3, 4 2.6067 105688 2 3.7596 $3608$ 5, 6, 7 3.2927, 3.2959, 3.2973 3608 3, 4 4.0790, 4.0800 $30648$ 5, 6, 7 3.0680, 3.0682, 3.0684 30648 3, 4 3.8126, 3.8126 $105688$ 5, 6, 7 3.0234 105688 3, 4 3.7622, 3.7623
 $Dof$ $j$ $k^C_{j, h}$($n=16$) $Dof$ $j$ $k^C_{j, h}$($n=8+x_1-x_2$) $3608$ 1 2.1651 3608 1 3.2181 $30648$ 1 2.0935 30648 1 3.0747 $105688$ 1 2.0791 105688 1 3.0474 $3608$ 2, 3, 4 2.7794, 2.7803, 2.7807 3608 2 4.0750 $30648$ 2, 3, 4 2.6350, 2.6350, 2.6351 30648 2 3.8098 $105688$ 2, 3, 4 2.6067 105688 2 3.7596 $3608$ 5, 6, 7 3.2927, 3.2959, 3.2973 3608 3, 4 4.0790, 4.0800 $30648$ 5, 6, 7 3.0680, 3.0682, 3.0684 30648 3, 4 3.8126, 3.8126 $105688$ 5, 6, 7 3.0234 105688 3, 4 3.7622, 3.7623
The eigenvalues obtained by CT element mixed with linear element on the tetrahedron, $n = 16$
 $Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$ $1071$ 2.7560 3.3119, 3.3119, 3.3617 3.9131, 3.9623, 3.9738 $9955$ 2.7650 3.3220, 3.3239, 3.3239 3.9144, 3.9151, 3.9151 $85963$ 2.7758 3.3371, 3.3396, 3.3396 3.9320, 3.9320, 3.9342
 $Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$ $1071$ 2.7560 3.3119, 3.3119, 3.3617 3.9131, 3.9623, 3.9738 $9955$ 2.7650 3.3220, 3.3239, 3.3239 3.9144, 3.9151, 3.9151 $85963$ 2.7758 3.3371, 3.3396, 3.3396 3.9320, 3.9320, 3.9342
The eigenvalues obtained by MZ element on the tetrahedron, $n = 16$
 $Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$ $2072$ 3.2333 4.0873, 4.0873, 4.1929 5.1997, 5.1997, 5.2041 $19000$ 2.8870 3.5128, 3.5128, 3.5666 4.2089, 4.2528, 4.2557 $162936$ 2.8051 3.3852, 3.3852, 3.3980 4.0036, 4.0185, 4.0185
 $Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$ $2072$ 3.2333 4.0873, 4.0873, 4.1929 5.1997, 5.1997, 5.2041 $19000$ 2.8870 3.5128, 3.5128, 3.5666 4.2089, 4.2528, 4.2557 $162936$ 2.8051 3.3852, 3.3852, 3.3980 4.0036, 4.0185, 4.0185
The eigenvalues obtained by CT element mixed with linear element on the thick L-shaped
 $Dof$ $j$ $k^{TL}_{j, h}$($n=16$) $j$ $k^{TL}_{j, h}$($n=8+x_1-x_2$) $257$ 1 2.4603 1 3.3865 $2575$ 1 1.9756 1 2.8223 $23171$ 1 1.8209 1 2.6248 $257$ 2 2.5244 2 3.4304 $2575$ 2 2.0359 2 2.8372 $23171$ 2 1.8801 2 2.6415 $257$ 3 2.5311 3 3.5598 $2575$ 3 2.0974 3 3.0510 $23171$ 3 1.9431 3 2.8327
 $Dof$ $j$ $k^{TL}_{j, h}$($n=16$) $j$ $k^{TL}_{j, h}$($n=8+x_1-x_2$) $257$ 1 2.4603 1 3.3865 $2575$ 1 1.9756 1 2.8223 $23171$ 1 1.8209 1 2.6248 $257$ 2 2.5244 2 3.4304 $2575$ 2 2.0359 2 2.8372 $23171$ 2 1.8801 2 2.6415 $257$ 3 2.5311 3 3.5598 $2575$ 3 2.0974 3 3.0510 $23171$ 3 1.9431 3 2.8327
The eigenvalues obtained by MZ element on the thick L-shaped
 $Dof$ $j$ $k^{TL}_{j, h}$($n=16$) $j$ $k^{TL}_{j, h}$($n=8+x_1-x_2$) $504$ 1 3.6272 1 4.5809 - 2.0747i $4952$ 1 2.5041 1 3.9589 $44088$ 1 1.9305 1 2.8511 $504$ 2 3.9842 2 4.5809 + 2.0747i $4952$ 2 2.7611 2 4.5904 $44088$ 2 1.9943 2 2.8939 $504$ 3 4.6844 3 5.2114 - 2.2381i $4952$ 3 2.8373 3 4.9593 $44088$ 3 2.0851 3 3.1157
 $Dof$ $j$ $k^{TL}_{j, h}$($n=16$) $j$ $k^{TL}_{j, h}$($n=8+x_1-x_2$) $504$ 1 3.6272 1 4.5809 - 2.0747i $4952$ 1 2.5041 1 3.9589 $44088$ 1 1.9305 1 2.8511 $504$ 2 3.9842 2 4.5809 + 2.0747i $4952$ 2 2.7611 2 4.5904 $44088$ 2 1.9943 2 2.8939 $504$ 3 4.6844 3 5.2114 - 2.2381i $4952$ 3 2.8373 3 4.9593 $44088$ 3 2.0851 3 3.1157
The eigenvalues obtained by CT element mixed with linear element on the sphere, $n = 16$
 $Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$ 1608 2.3431 2.9533, 2.9550, 2.9564 3.5575, 3.5597, 3.5600 16643 2.3075 2.9250, 2.9252, 2.9254 3.5257, 3.5260, 3.5261 51277 2.3035 2.9229, 2.9229, 2.9230 3.5261, 3.5262, 3.5263
 $Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$ 1608 2.3431 2.9533, 2.9550, 2.9564 3.5575, 3.5597, 3.5600 16643 2.3075 2.9250, 2.9252, 2.9254 3.5257, 3.5260, 3.5261 51277 2.3035 2.9229, 2.9229, 2.9230 3.5261, 3.5262, 3.5263
The eigenvalues obtained by MZ element on the sphere, $n = 16$
 $Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$ 3076 2.4860 3.3110, 3.3259, 3.3441 4.2252, 4.2297, 4.2574 31568 2.3306 2.9875, 2.9886, 2.9905 3.6451, 3.6471, 3.6520 97048 2.3147 2.9520, 2.9521, 2.9529 3.5817, 3.5825, 3.5848
 $Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$ 3076 2.4860 3.3110, 3.3259, 3.3441 4.2252, 4.2297, 4.2574 31568 2.3306 2.9875, 2.9886, 2.9905 3.6451, 3.6471, 3.6520 97048 2.3147 2.9520, 2.9521, 2.9529 3.5817, 3.5825, 3.5848
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