American Institute of Mathematical Sciences

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

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

Jiayu Han ,

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.

Keywords: Transmission eigenvalue, nonconforming finite elements, error estimates.
Mathematics Subject Classification: 65N25, 65N30.
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.

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

$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
Table Options

Download as excel

$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
Table Options

Download as excel

$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
Table Options

Download as excel

$Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$
$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
Table Options

Download as excel

$Dof$ $k^T_{1, h}$ $k^T_{2, h}, k^T_{3, h}, k^T_{4, h}$ $k^T_{5, h}, k^T_{6, h}, k^T_{7, h}$
$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
Table Options

Download as excel

$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
Table Options

Download as excel

$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
Table Options

Download as excel

$Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$
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
Table Options

Download as excel

$Dof$ $k^{Sp}_{1, h}$ $k^{Sp}_{2, h}, k^{Sp}_{3, h}, k^{Sp}_{4, h}$ $k^{Sp}_{5, h}, k^{Sp}_{6, h}, k^{Sp}_{7, h}$
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
[1]

Zhangxin Chen, Qiaoyuan Jiang, Yanli Cui. Locking-free nonconforming finite elements for planar linear elasticity. Conference Publications, 2005, 2005 (Special) : 181-189. doi: 10.3934/proc.2005.2005.181
[2]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems and Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795
[3]

Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051
[4]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[5]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663
[6]

Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic and Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040
[7]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control and Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041
[8]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089
[9]

Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062
[10]

Peter Monk, Jiguang Sun. Inverse scattering using finite elements and gap reciprocity. Inverse Problems and Imaging, 2007, 1 (4) : 643-660. doi: 10.3934/ipi.2007.1.643
[11]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure and Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297
[12]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control and Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014
[13]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial and Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
[14]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems and Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
[15]

Natalia P. Bondarenko, Vjacheslav A. Yurko. A new approach to the inverse discrete transmission eigenvalue problem. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2021073
[16]

Eric Dubach, Robert Luce, Jean-Marie Thomas. Pseudo-Conform Polynomial Lagrange Finite Elements on Quadrilaterals and Hexahedra. Communications on Pure and Applied Analysis, 2009, 8 (1) : 237-254. doi: 10.3934/cpaa.2009.8.237
[17]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[18]

Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055
[19]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[20]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (211)
  • HTML views (128)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy