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  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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

F. CakoniP. 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. Google Scholar

[8]

L. Chen, iFEM: An integrated finite element method package in MATLAB., Technical Report, University of California at Irvine, 2009.Google Scholar

[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.Google Scholar

[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. Google Scholar

[11]

D. ColtonL. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Problem Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13. Google Scholar

[12]

H. GengX. JiJ. 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. Google Scholar

[13]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

X. JiJ. 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. Google Scholar

[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. Google Scholar

[18]

J. T. Oden and J. N. Reddy, An Introduction to the Mathematical Theory of Finite Elements, New York-London-Sydney, 1976. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[22]

J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860-1874. doi: 10.1137/100785478. Google Scholar

[23]

M. WangZ. 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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[26]

Y. YangH. BiH. Li and J. Han, Mixed method for the helmholtz transmission eigenvalues, SIAM J. Sci. Comput., 38 (2016), A1383-A1403. doi: 10.1137/15M1050756. Google Scholar

[27]

Y. YangJ. 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. Google Scholar

[28]

F. ZengJ. 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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

F. CakoniP. 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. Google Scholar

[8]

L. Chen, iFEM: An integrated finite element method package in MATLAB., Technical Report, University of California at Irvine, 2009.Google Scholar

[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.Google Scholar

[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. Google Scholar

[11]

D. ColtonL. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Problem Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13. Google Scholar

[12]

H. GengX. JiJ. 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. Google Scholar

[13]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

X. JiJ. 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. Google Scholar

[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. Google Scholar

[18]

J. T. Oden and J. N. Reddy, An Introduction to the Mathematical Theory of Finite Elements, New York-London-Sydney, 1976. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[22]

J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860-1874. doi: 10.1137/100785478. Google Scholar

[23]

M. WangZ. 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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[26]

Y. YangH. BiH. Li and J. Han, Mixed method for the helmholtz transmission eigenvalues, SIAM J. Sci. Comput., 38 (2016), A1383-A1403. doi: 10.1137/15M1050756. Google Scholar

[27]

Y. YangJ. 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. Google Scholar

[28]

F. ZengJ. 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. Google Scholar

Figure 1.  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)
Table 1.  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}$11.8790834921.477307712.8194074342.302531586
$\frac{\sqrt2}{64}$11.8794479181.476703812.8214373582.302554268
$\frac{\sqrt2}{128}$11.8795541831.476394412.8219969852.302391353
$\frac{\sqrt2}{32}$22.4431682931.569436123.5344520842.394060079
$\frac{\sqrt2}{64}$22.4439294691.569651123.5375246312.395270328
$\frac{\sqrt2}{128}$22.4441564671.569707123.5383948532.395585637
$\frac{\sqrt2}{32}$32.4431682931.704972554.498450-0.871213i2.928086-0.563389i
$\frac{\sqrt2}{64}$32.4439294691.705104554.496942-0.871413i2.925438-0.564575i
$\frac{\sqrt2}{128}$32.4441564671.705097854.496644-0.871466i2.924644-0.564720i
$\frac{\sqrt2}{32}$42.8652215841.782706564.498450+0.871213i2.928086+0.563389i
$\frac{\sqrt2}{64}$42.8660327771.783002664.496942+0.871413i2.925438+0.564575i
$\frac{\sqrt2}{128}$42.8663297341.783086964.496644+0.871466i2.924644+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}$11.8790834921.477307712.8194074342.302531586
$\frac{\sqrt2}{64}$11.8794479181.476703812.8214373582.302554268
$\frac{\sqrt2}{128}$11.8795541831.476394412.8219969852.302391353
$\frac{\sqrt2}{32}$22.4431682931.569436123.5344520842.394060079
$\frac{\sqrt2}{64}$22.4439294691.569651123.5375246312.395270328
$\frac{\sqrt2}{128}$22.4441564671.569707123.5383948532.395585637
$\frac{\sqrt2}{32}$32.4431682931.704972554.498450-0.871213i2.928086-0.563389i
$\frac{\sqrt2}{64}$32.4439294691.705104554.496942-0.871413i2.925438-0.564575i
$\frac{\sqrt2}{128}$32.4441564671.705097854.496644-0.871466i2.924644-0.564720i
$\frac{\sqrt2}{32}$42.8652215841.782706564.498450+0.871213i2.928086+0.563389i
$\frac{\sqrt2}{64}$42.8660327771.783002664.496942+0.871413i2.925438+0.564575i
$\frac{\sqrt2}{128}$42.8663297341.783086964.496644+0.871466i2.924644+0.564720i
Table 2.  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$12.0449189512.9579
$16179$12.06041617913.0051
$55903$12.06415590313.0162
$1895$2, 3, 42.5461189523.6098
$16179$2, 3, 42.57141617923.6853
$55903$2, 3, 42.57855590323.7049
$1895$5, 6, 72.940218953, 43.6119, 3.6121
$16179$5, 6, 72.9683161793, 43.6878, 3.6879
$55903$5, 6, 72.9781559033, 43.7074, 3.7075
$Dof$$j$ $k^C_{j, h}$($n=16$)$Dof$ $j$$k^C_{j, h}$($n=8+x_1-x_2$)
$1895$12.0449189512.9579
$16179$12.06041617913.0051
$55903$12.06415590313.0162
$1895$2, 3, 42.5461189523.6098
$16179$2, 3, 42.57141617923.6853
$55903$2, 3, 42.57855590323.7049
$1895$5, 6, 72.940218953, 43.6119, 3.6121
$16179$5, 6, 72.9683161793, 43.6878, 3.6879
$55903$5, 6, 72.9781559033, 43.7074, 3.7075
Table 3.  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$12.1651360813.2181
$30648$12.09353064813.0747
$105688$12.079110568813.0474
$3608$2, 3, 42.7794, 2.7803, 2.7807360824.0750
$30648$2, 3, 42.6350, 2.6350, 2.63513064823.8098
$105688$2, 3, 42.606710568823.7596
$3608$5, 6, 73.2927, 3.2959, 3.297336083, 44.0790, 4.0800
$30648$5, 6, 73.0680, 3.0682, 3.0684306483, 43.8126, 3.8126
$105688$5, 6, 73.02341056883, 43.7622, 3.7623
$Dof$$j$ $ k^C_{j, h}$($n=16$)$Dof$ $j$$ k^C_{j, h}$($n=8+x_1-x_2$)
$3608$12.1651360813.2181
$30648$12.09353064813.0747
$105688$12.079110568813.0474
$3608$2, 3, 42.7794, 2.7803, 2.7807360824.0750
$30648$2, 3, 42.6350, 2.6350, 2.63513064823.8098
$105688$2, 3, 42.606710568823.7596
$3608$5, 6, 73.2927, 3.2959, 3.297336083, 44.0790, 4.0800
$30648$5, 6, 73.0680, 3.0682, 3.0684306483, 43.8126, 3.8126
$105688$5, 6, 73.02341056883, 43.7622, 3.7623
Table 4.  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$2.75603.3119, 3.3119, 3.36173.9131, 3.9623, 3.9738
$9955$2.76503.3220, 3.3239, 3.32393.9144, 3.9151, 3.9151
$85963$2.77583.3371, 3.3396, 3.33963.9320, 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$2.75603.3119, 3.3119, 3.36173.9131, 3.9623, 3.9738
$9955$2.76503.3220, 3.3239, 3.32393.9144, 3.9151, 3.9151
$85963$2.77583.3371, 3.3396, 3.33963.9320, 3.9320, 3.9342
Table 5.  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$3.23334.0873, 4.0873, 4.19295.1997, 5.1997, 5.2041
$19000$2.88703.5128, 3.5128, 3.56664.2089, 4.2528, 4.2557
$162936$2.80513.3852, 3.3852, 3.39804.0036, 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$3.23334.0873, 4.0873, 4.19295.1997, 5.1997, 5.2041
$19000$2.88703.5128, 3.5128, 3.56664.2089, 4.2528, 4.2557
$162936$2.80513.3852, 3.3852, 3.39804.0036, 4.0185, 4.0185
Table 6.  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$12.460313.3865
$2575$11.975612.8223
$23171$11.820912.6248
$257$22.524423.4304
$2575$22.035922.8372
$23171$21.880122.6415
$257$32.531133.5598
$2575$32.097433.0510
$23171$31.943132.8327
$Dof$$j$ $k^{TL}_{j, h}$($n=16$) $j$$k^{TL}_{j, h}$($n=8+x_1-x_2$)
$257$12.460313.3865
$2575$11.975612.8223
$23171$11.820912.6248
$257$22.524423.4304
$2575$22.035922.8372
$23171$21.880122.6415
$257$32.531133.5598
$2575$32.097433.0510
$23171$31.943132.8327
Table 7.  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$13.627214.5809 - 2.0747i
$4952$12.504113.9589
$44088$11.930512.8511
$504$23.984224.5809 + 2.0747i
$4952$22.761124.5904
$44088$21.994322.8939
$504$34.684435.2114 - 2.2381i
$4952$32.837334.9593
$44088$32.085133.1157
$Dof$$j$ $k^{TL}_{j, h}$($n=16$) $j$$k^{TL}_{j, h}$($n=8+x_1-x_2$)
$504$13.627214.5809 - 2.0747i
$4952$12.504113.9589
$44088$11.930512.8511
$504$23.984224.5809 + 2.0747i
$4952$22.761124.5904
$44088$21.994322.8939
$504$34.684435.2114 - 2.2381i
$4952$32.837334.9593
$44088$32.085133.1157
Table 8.  The eigenvalues obtained by CT element mixed with linear element on the sphere, $n = 16$
$Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$
16082.34312.9533, 2.9550, 2.95643.5575, 3.5597, 3.5600
166432.30752.9250, 2.9252, 2.92543.5257, 3.5260, 3.5261
512772.30352.9229, 2.9229, 2.92303.5261, 3.5262, 3.5263
$Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$
16082.34312.9533, 2.9550, 2.95643.5575, 3.5597, 3.5600
166432.30752.9250, 2.9252, 2.92543.5257, 3.5260, 3.5261
512772.30352.9229, 2.9229, 2.92303.5261, 3.5262, 3.5263
Table 9.  The eigenvalues obtained by MZ element on the sphere, $n = 16$
$Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$
30762.48603.3110, 3.3259, 3.34414.2252, 4.2297, 4.2574
315682.33062.9875, 2.9886, 2.99053.6451, 3.6471, 3.6520
970482.31472.9520, 2.9521, 2.95293.5817, 3.5825, 3.5848
$Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$
30762.48603.3110, 3.3259, 3.34414.2252, 4.2297, 4.2574
315682.33062.9875, 2.9886, 2.99053.6451, 3.6471, 3.6520
970482.31472.9520, 2.9521, 2.95293.5817, 3.5825, 3.5848
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