September  2019, 24(9): 4799-4813. doi: 10.3934/dcdsb.2019031

An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries

1. 

Computational mathematics research center/School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, China

2. 

School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, China

* Corresponding author: Jun Zhang

Received  January 2018 Revised  August 2018 Published  September 2019 Early access  February 2019

Fund Project: The first author is supported by Guizhou University of Finance and Economics(No. 2017XZD01).

In this work, we develop an efficient spectral method to solve the Helmholtz transmission eigenvalue problem in polar geometries. An essential difficulty is that the polar coordinate transformation introduces the polar singularities. In order to overcome this difficulty, we introduce some pole conditions and the corresponding weighted Sobolev space. The polar coordinate transformation and variable separation techniques are presented to transform the original problem into a series of equivalent one-dimensional eigenvalue problem, and error estimate for the approximate eigenvalues and corresponding eigenfunctions are obtained. Finally, numerical simulations are performed to confirm the validity of the numerical method.

Citation: Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031
References:
[1]

J. An and J. Shen, A Spectral-Element method for transmission eigenvalue problems, Journal of Scientific Computing, 57 (2013), 670-688.  doi: 10.1007/s10915-013-9720-1.

[2]

J. An and J. Shen, Spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem, Computers and Mathematics with Applications, 69 (2015), 1132-1143.  doi: 10.1016/j.camwa.2015.03.002.

[3]

J. An, A Legendre-Galerkin spectral approximation and estimation of the index of refraction for transmission eigenvalues, Applied Numerical Mathematics, 108 (2016), 171-184.  doi: 10.1016/j.apnum.2015.11.007.

[4]

J. An and J. Shen, Efficient spectral methods for transmission eigenvalues and estimation of the index of refraction, J. Math. Study, 47 (2014), 1-20. 

[5]

J. An, H. Li and Z. Zhang, Spectral-galerkin approximation and optimal error estimate for stokes eigenvalue problems in polar geometries, arXiv: 1610.08647.

[6]

I. Babu$\breve{s}$ka and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, 2 (1991), 641-787. 

[7]

F. Cakoni, M. Cayoren and D. Colton, Transmission eigenvalues and the nondestructive testing of dielectrics, Inverse Problems, 24 (2008), 065016, 15pp. doi: 10.1088/0266-5611/24/6/065016.

[8]

F. CakoniD. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM Journal on Mathematical Analysis, 42 (2010), 237-255.  doi: 10.1137/090769338.

[9]

F. CakoniD. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, Comptes Rendus Mathematique, 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.

[10]

F. CakoniD. Colton and P. Monk, On the use of transmission eigenvalues to estimate the index of refraction from far field data, Inverse Problems, 23 (2007), 507-522.  doi: 10.1088/0266-5611/23/2/004.

[11]

F. Cakoni, D. Colton, P. Monk and J. G. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inverse Problems, 26 (2010), 074004, 14pp. doi: 10.1088/0266-5611/26/7/074004.

[12]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Applicable Analysis, 88 (2009), 475-493.  doi: 10.1080/00036810802713966.

[13]

F. CakoniD. Colton and H. Haddar, On the determination of dirichlet or transmission eigenvalues from far field data, Comptes Rendus Mathematique, 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.

[14]

F. CakoniP. Monk and J. Sun, Error analysis for the finite element approximation of transmission eigenvalues, Computational Methods in Applied Mathematics, 14 (2014), 419-427.  doi: 10.1515/cmam-2014-0021.

[15]

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

[16]

D. Colton, P. Monk and J. G. Sun, Analytical and computational methods for transmission eigenvalues, Inverse Problems, 26 (2010), 045011, 16pp. doi: 10.1088/0266-5611/26/4/045011.

[17]

X. JiY. Xi and H. Xie, Nonconforming finite element method for the transmission eigenvalue problem, Advances in Applied Mathematics and Mechanics, 9 (2017), 92-103.  doi: 10.4208/aamm.2015.m1295.

[18]

X. Ji, J. G. Sun and T. Turner, Algorithm 922: A mixed finite element method for Helmholtz Transmission eigenvalues, ACM Transactions on Mathematical Software (TOMS), 38 (2012), Art. 29, 8 pp. doi: 10.1145/2331130.2331137.

[19]

X. JiJ. Sun and H. Xie, A multigrid method for Helmholtz transmission eigenvalue problems, Journal of Scientific Computing, 60 (2014), 276-294.  doi: 10.1007/s10915-013-9794-9.

[20]

A. Kirsch, On the existence of transmission eigenvalues, Inverse Problems and Imaging, 3 (2009), 155-172.  doi: 10.3934/ipi.2009.3.155.

[21]

L. Paivarinta and J. Sylvester, Transmission eigenvalues, SIAM Journal on Mathematical Analysis, 40 (2008), 738-753.  doi: 10.1137/070697525.

[22]

B. P. Rynne and B. D. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous media, SIAM Journal on Mathematical Analysis, 22 (1991), 1755-1762.  doi: 10.1137/0522109.

[23]

J. Shen, Efficient spectral-Galerkin methods Ⅲ: Polar and cylindrical geometries, SIAM J. Sci. Comput., 18 (1997), 1583-1604.  doi: 10.1137/S1064827595295301.

[24]

J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, 2006.

[25]

J. Sun, Iterative methods for transmission eigenvalues, SIAM Journal on Numerical Analysis, 49 (2011), 1860-1874.  doi: 10.1137/100785478.

[26]

Y. Yang, H. Bi, H. Li, et al, Mixed methods for the helmholtz transmission eigenvalues, SIAM Journal on Scientific Computing, 38 (2016), A1383–A1403. doi: 10.1137/15M1050756.

show all references

References:
[1]

J. An and J. Shen, A Spectral-Element method for transmission eigenvalue problems, Journal of Scientific Computing, 57 (2013), 670-688.  doi: 10.1007/s10915-013-9720-1.

[2]

J. An and J. Shen, Spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem, Computers and Mathematics with Applications, 69 (2015), 1132-1143.  doi: 10.1016/j.camwa.2015.03.002.

[3]

J. An, A Legendre-Galerkin spectral approximation and estimation of the index of refraction for transmission eigenvalues, Applied Numerical Mathematics, 108 (2016), 171-184.  doi: 10.1016/j.apnum.2015.11.007.

[4]

J. An and J. Shen, Efficient spectral methods for transmission eigenvalues and estimation of the index of refraction, J. Math. Study, 47 (2014), 1-20. 

[5]

J. An, H. Li and Z. Zhang, Spectral-galerkin approximation and optimal error estimate for stokes eigenvalue problems in polar geometries, arXiv: 1610.08647.

[6]

I. Babu$\breve{s}$ka and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, 2 (1991), 641-787. 

[7]

F. Cakoni, M. Cayoren and D. Colton, Transmission eigenvalues and the nondestructive testing of dielectrics, Inverse Problems, 24 (2008), 065016, 15pp. doi: 10.1088/0266-5611/24/6/065016.

[8]

F. CakoniD. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM Journal on Mathematical Analysis, 42 (2010), 237-255.  doi: 10.1137/090769338.

[9]

F. CakoniD. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, Comptes Rendus Mathematique, 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.

[10]

F. CakoniD. Colton and P. Monk, On the use of transmission eigenvalues to estimate the index of refraction from far field data, Inverse Problems, 23 (2007), 507-522.  doi: 10.1088/0266-5611/23/2/004.

[11]

F. Cakoni, D. Colton, P. Monk and J. G. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inverse Problems, 26 (2010), 074004, 14pp. doi: 10.1088/0266-5611/26/7/074004.

[12]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Applicable Analysis, 88 (2009), 475-493.  doi: 10.1080/00036810802713966.

[13]

F. CakoniD. Colton and H. Haddar, On the determination of dirichlet or transmission eigenvalues from far field data, Comptes Rendus Mathematique, 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.

[14]

F. CakoniP. Monk and J. Sun, Error analysis for the finite element approximation of transmission eigenvalues, Computational Methods in Applied Mathematics, 14 (2014), 419-427.  doi: 10.1515/cmam-2014-0021.

[15]

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

[16]

D. Colton, P. Monk and J. G. Sun, Analytical and computational methods for transmission eigenvalues, Inverse Problems, 26 (2010), 045011, 16pp. doi: 10.1088/0266-5611/26/4/045011.

[17]

X. JiY. Xi and H. Xie, Nonconforming finite element method for the transmission eigenvalue problem, Advances in Applied Mathematics and Mechanics, 9 (2017), 92-103.  doi: 10.4208/aamm.2015.m1295.

[18]

X. Ji, J. G. Sun and T. Turner, Algorithm 922: A mixed finite element method for Helmholtz Transmission eigenvalues, ACM Transactions on Mathematical Software (TOMS), 38 (2012), Art. 29, 8 pp. doi: 10.1145/2331130.2331137.

[19]

X. JiJ. Sun and H. Xie, A multigrid method for Helmholtz transmission eigenvalue problems, Journal of Scientific Computing, 60 (2014), 276-294.  doi: 10.1007/s10915-013-9794-9.

[20]

A. Kirsch, On the existence of transmission eigenvalues, Inverse Problems and Imaging, 3 (2009), 155-172.  doi: 10.3934/ipi.2009.3.155.

[21]

L. Paivarinta and J. Sylvester, Transmission eigenvalues, SIAM Journal on Mathematical Analysis, 40 (2008), 738-753.  doi: 10.1137/070697525.

[22]

B. P. Rynne and B. D. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous media, SIAM Journal on Mathematical Analysis, 22 (1991), 1755-1762.  doi: 10.1137/0522109.

[23]

J. Shen, Efficient spectral-Galerkin methods Ⅲ: Polar and cylindrical geometries, SIAM J. Sci. Comput., 18 (1997), 1583-1604.  doi: 10.1137/S1064827595295301.

[24]

J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, 2006.

[25]

J. Sun, Iterative methods for transmission eigenvalues, SIAM Journal on Numerical Analysis, 49 (2011), 1860-1874.  doi: 10.1137/100785478.

[26]

Y. Yang, H. Bi, H. Li, et al, Mixed methods for the helmholtz transmission eigenvalues, SIAM Journal on Scientific Computing, 38 (2016), A1383–A1403. doi: 10.1137/15M1050756.

Figure 1.  Numerical errors of example 1 for $ m = 0 $.
Figure 2.  Numerical errors of example 1 for $ m = 2 $.
Figure 3.  Numerical errors of example 2 for $ m = 1 $.
Figure 4.  Numerical errors of example 2 for m = 0.
Figure 5.  Numerical errors of example 2 for m = 1.
Figure 6.  Numerical errors of example 3 for m = 0.
Table 1.  The first four eigenvalues for m = 0
$N \setminus k_{0N}^i $ $k_{0N}^1$ $k_{0N}^2$ $k_{0N}^3$ $k_{0N}^4$
$N = 10$ 1.9879951237731613.7409250902496916.5822782981638128.386644958117056
$N = 15$ 1.9879951237713753.7409249351004446.5810305728250588.273172861972871
$N = 20$ 1.9879951237713763.7409249351004466.5810305497202578.273170634133454
$N = 25$ 1.9879951237713783.7409249351004436.5810305497202558.273170634125105
$N = 30$ 1.9879951237713783.7409249351004436.5810305497202558.273170634125105
$N \setminus k_{0N}^i $ $k_{0N}^1$ $k_{0N}^2$ $k_{0N}^3$ $k_{0N}^4$
$N = 10$ 1.9879951237731613.7409250902496916.5822782981638128.386644958117056
$N = 15$ 1.9879951237713753.7409249351004446.5810305728250588.273172861972871
$N = 20$ 1.9879951237713763.7409249351004466.5810305497202578.273170634133454
$N = 25$ 1.9879951237713783.7409249351004436.5810305497202558.273170634125105
$N = 30$ 1.9879951237713783.7409249351004436.5810305497202558.273170634125105
Table 2.  The first four eigenvalues for $m = 1$
$N \setminus k_{0N}^i $ $k_{1N}^1$ $k_{1N}^2$ $k_{0N}^3$ $k_{1N}^4$
$N = 10$ 2.6129299642085494.2958107711784455.9897750231532219.091957821822609
$N = 15$ 2.6129299639027934.2958099366585835.9885009103153738.841298655023424
$N = 20$ 2.6129299639027894.2958099366584515.9885009101069528.841280530061555
$N = 25$ 2.6129299639027924.2958099366584595.9885009101069548.841280529974766
$N = 30$ 2.6129299639027944.2958099366584485.9885009101069548.841280529974791
$N \setminus k_{0N}^i $ $k_{1N}^1$ $k_{1N}^2$ $k_{0N}^3$ $k_{1N}^4$
$N = 10$ 2.6129299642085494.2958107711784455.9897750231532219.091957821822609
$N = 15$ 2.6129299639027934.2958099366585835.9885009103153738.841298655023424
$N = 20$ 2.6129299639027894.2958099366584515.9885009101069528.841280530061555
$N = 25$ 2.6129299639027924.2958099366584595.9885009101069548.841280529974766
$N = 30$ 2.6129299639027944.2958099366584485.9885009101069548.841280529974791
Table 3.  The first four eigenvalues for $m = 2$
$N \setminus k_{2N}^i $ $k_{2N}^1$ $k_{2N}^2$ $k_{2N}^3$ $k_{2N}^4$
$N = 10$ 3.2266479487189864.9418893327206796.6051581796652958.397840671676448
$N = 15$ 3.2266479478902504.9418345576933486.6038344917423358.279736371734421
$N = 20$ 3.2266479478902504.9418345576926686.6038344679396338.279734012410303
$N = 25$ 3.2266479478902514.9418345576926666.6038344679396398.279734012401546
$N = 30$ 3.2266479478902514.9418345576926636.6038344679396368.279734012401549
$N \setminus k_{2N}^i $ $k_{2N}^1$ $k_{2N}^2$ $k_{2N}^3$ $k_{2N}^4$
$N = 10$ 3.2266479487189864.9418893327206796.6051581796652958.397840671676448
$N = 15$ 3.2266479478902504.9418345576933486.6038344917423358.279736371734421
$N = 20$ 3.2266479478902504.9418345576926686.6038344679396338.279734012410303
$N = 25$ 3.2266479478902514.9418345576926666.6038344679396398.279734012401546
$N = 30$ 3.2266479478902514.9418345576926636.6038344679396368.279734012401549
Table 4.  The first four eigenvalues for $m = 0$
$N \setminus k_{0N}^i $ $k_{0N}^1$ $k_{0N}^2$ $k_{0N}^3$ $k_{0N}^4$
$N = 10$ 2.7594351391661496.6676595732850059.04916438713020313.35822517979881
$N = 15$ 2.7594351391589746.6675470345741039.04704049505956212.95759442163584
$N = 20$ 2.7594351391589786.6675470345536979.04704040735104412.95695038075036
$N = 25$ 2.7594351391589786.6675470345536979.04704040735104412.95695038075036
$N = 30$ 2.7594351391589816.6675470345536989.04704040735104812.95695038075038
$N \setminus k_{0N}^i $ $k_{0N}^1$ $k_{0N}^2$ $k_{0N}^3$ $k_{0N}^4$
$N = 10$ 2.7594351391661496.6676595732850059.04916438713020313.35822517979881
$N = 15$ 2.7594351391589746.6675470345741039.04704049505956212.95759442163584
$N = 20$ 2.7594351391589786.6675470345536979.04704040735104412.95695038075036
$N = 25$ 2.7594351391589786.6675470345536979.04704040735104412.95695038075036
$N = 30$ 2.7594351391589816.6675470345536989.04704040735104812.95695038075038
Table 5.  The first four eigenvalues for $m = 1$
$N \setminus k_{1N}^i $ $k_{1N}^1$ $k_{1N}^2$ $k_{1N}^3$ $k_{1N}^4$
$N = 10$ 3.5272761555210525.8952946424691029.83750280906308912.78302633393358
$N = 15$ 3.5272761554993295.8952856699749789.81049195339294812.18568470580060
$N = 20$ 3.5272761554993295.8952856699749599.81049186060655712.18556255236016
$N = 25$ 3.5272761554993275.8952856699749579.81049186060632612.18556255231528
$N = 30$ 3.5272761554993315.8952856699749589.81049186060631912.18556255231531
$N \setminus k_{1N}^i $ $k_{1N}^1$ $k_{1N}^2$ $k_{1N}^3$ $k_{1N}^4$
$N = 10$ 3.5272761555210525.8952946424691029.83750280906308912.78302633393358
$N = 15$ 3.5272761554993295.8952856699749789.81049195339294812.18568470580060
$N = 20$ 3.5272761554993295.8952856699749599.81049186060655712.18556255236016
$N = 25$ 3.5272761554993275.8952856699749579.81049186060632612.18556255231528
$N = 30$ 3.5272761554993315.8952856699749589.81049186060631912.18556255231531
Table 6.  The first four eigenvalues for $m = 2$
$N \setminus k_{2N}^i $ $k_{2N}^1$ $k_{2N}^2$ $k_{2N}^3$ $k_{2N}^4$
$N = 10$ 4.3079726597709616.6485931699587279.08685403542764513.34452507236263
$N = 15$ 4.3079726359714196.6484866952933519.08368918127876612.94522396389963
$N = 20$ 4.3079726359714156.6484866952765069.08368905330218212.94460023197706
$N = 25$ 4.3079726359714176.6484866952765089.08368905330217112.94460022855666
$N = 30$ 4.3079726359714186.6484866952765089.08368905330217512.94460022855666
$N \setminus k_{2N}^i $ $k_{2N}^1$ $k_{2N}^2$ $k_{2N}^3$ $k_{2N}^4$
$N = 10$ 4.3079726597709616.6485931699587279.08685403542764513.34452507236263
$N = 15$ 4.3079726359714196.6484866952933519.08368918127876612.94522396389963
$N = 20$ 4.3079726359714156.6484866952765069.08368905330218212.94460023197706
$N = 25$ 4.3079726359714176.6484866952765089.08368905330217112.94460022855666
$N = 30$ 4.3079726359714186.6484866952765089.08368905330217512.94460022855666
Table 7.  The first four eigenvalues for $m = 0$
$N \setminus k_{1N}^i $ $k_{1N}^1$ $k_{1N}^2$ $k_{1N}^3$ $k_{1N}^4$
$N = 10$ 3.4335642722321994.6006854510154958.1135119669037339.405045505293291
$N = 15$ 3.4335642067521734.6006418869644747.9663111665781049.146084137724058
$N = 20$ 3.4335642067521724.6006418869634697.9663093604662839.145821838198342
$N = 25$ 3.4335642067521724.6006418869634747.9663093604586889.145821837979225
$N = 30$ 3.4335642067521754.6006418869634727.9663093604586919.145821837979231
$N \setminus k_{1N}^i $ $k_{1N}^1$ $k_{1N}^2$ $k_{1N}^3$ $k_{1N}^4$
$N = 10$ 3.4335642722321994.6006854510154958.1135119669037339.405045505293291
$N = 15$ 3.4335642067521734.6006418869644747.9663111665781049.146084137724058
$N = 20$ 3.4335642067521724.6006418869634697.9663093604662839.145821838198342
$N = 25$ 3.4335642067521724.6006418869634747.9663093604586889.145821837979225
$N = 30$ 3.4335642067521754.6006418869634727.9663093604586919.145821837979231
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