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An adaptive finite element method for the elastic transmission eigenvalue problem
1. | School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China |
2. | School of Biology & Engineering, Guizhou Medical University, Guiyang 550025, China |
We discuss the a posteriori error estimates of the H$ ^{2} $-conforming finite element for the elastic transmission eigenvalue problem, which is of fourth order and non-selfadjoint, with vector-valued eigenfunctions. We first introduce the error indicators for primal eigenfunction, dual eigenfunction and eigenvalue, respectively. Then the reliability and efficiency of the indicators for both eigenfunctions are proved and an adaptive algorithm of residual type is designed. Finally, we implement numerical experiments to validate our theoretical analysis and show the robustness of the algorithm.
References:
[1] |
M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience, New York, 2000.
doi: 10.1002/9781118032824. |
[2] |
I. Babuska and W. C. Rheinboldt,
Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), 736-754.
doi: 10.1137/0715049. |
[3] |
G. Bao, G. Hu, J. Sun and T. Yin,
Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301.
doi: 10.1016/j.matpur.2018.01.007. |
[4] |
C. Bellis, F. Cakoni and B. B. Guzina,
Nature of the transmission eigenvalue spectrum for elastic bodies, IMA J. Appl. Math., 78 (2013), 895-923.
doi: 10.1093/imamat/hxr070. |
[5] |
Ju. M. Berezanski$\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. |






dof | dof | |||||
26368 | 13.614085 | 24.446563 | 35256 | 3.059038 | 4.867139 | |
108032 | 13.582410 | 24.426026 | 144248 | 3.028473 | 4.857275 |
dof | dof | |||||
26368 | 13.614085 | 24.446563 | 35256 | 3.059038 | 4.867139 | |
108032 | 13.582410 | 24.426026 | 144248 | 3.028473 | 4.857275 |
dof | dof | dof | dof | |||||
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 |
dof | dof | dof | dof | |||||
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 |
dof | dof | |||||
26368 | 3.621459-3.0381i | 4.889479 | 35256 | 0.971932 | 0.896336-0.6230i | |
108032 | 3.615710-3.0408i | 4.875826 | 144248 | 0.959238 | 0.891335-0.6225i |
dof | dof | |||||
26368 | 3.621459-3.0381i | 4.889479 | 35256 | 0.971932 | 0.896336-0.6230i | |
108032 | 3.615710-3.0408i | 4.875826 | 144248 | 0.959238 | 0.891335-0.6225i |
dof | dof | dof | dof | |||||
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 |
dof | dof | dof | dof | |||||
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 |
dof | dof | |||||
26368 | 26.411851 | 40.467203 | 35256 | 5.322456 | 7.480636 | |
108032 | 26.332569 | 40.392475 | 144248 | 5.248736 | 7.343923 |
dof | dof | |||||
26368 | 26.411851 | 40.467203 | 35256 | 5.322456 | 7.480636 | |
108032 | 26.332569 | 40.392475 | 144248 | 5.248736 | 7.343923 |
dof | dof | dof | dof | ||||||||
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 |
dof | dof | dof | dof | ||||||||
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 |
dof | dof | |||||
26368 | 26.257960 | 40.241343 | 35256 | 5.293323 | 7.445605 | |
108032 | 26.178143 | 40.165951 | 144248 | 5.219140 | 7.303772 |
dof | dof | |||||
26368 | 26.257960 | 40.241343 | 35256 | 5.293323 | 7.445605 | |
108032 | 26.178143 | 40.165951 | 144248 | 5.219140 | 7.303772 |
dof | dof | dof | dof | ||||||||
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 |
dof | dof | dof | dof | ||||||||
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 |
dof | dof | |||||
26368 | 26.234587 | 40.205701 | 35256 | 5.288633 | 7.439160 | |
108032 | 26.154882 | 40.130586 | 144248 | 5.214645 | 7.298399 |
dof | dof | |||||
26368 | 26.234587 | 40.205701 | 35256 | 5.288633 | 7.439160 | |
108032 | 26.154882 | 40.130586 | 144248 | 5.214645 | 7.298399 |
dof | dof | dof | dof | ||||||||
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 |
dof | dof | dof | dof | ||||||||
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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