# American Institute of Mathematical Sciences

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  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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [6] I. Babu$\breve{s}$ka and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, 2 (1991), 641-787.   Google Scholar [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.  Google Scholar [8] F. Cakoni, D. 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.  Google Scholar [9] F. Cakoni, D. 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.  Google Scholar [10] F. Cakoni, D. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] F. Cakoni, D. 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.  Google Scholar [14] F. Cakoni, P. 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.  Google Scholar [15] D. Colton, L. Paivarinta and J. Sylvester, The interior transmission problem, Inverse Problems and Imaging, 1 (2007), 13-28.  doi: 10.3934/ipi.2007.1.13.  Google Scholar [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.  Google Scholar [17] X. Ji, Y. 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.  Google Scholar [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.  Google Scholar [19] X. Ji, J. 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.  Google Scholar [20] A. Kirsch, On the existence of transmission eigenvalues, Inverse Problems and Imaging, 3 (2009), 155-172.  doi: 10.3934/ipi.2009.3.155.  Google Scholar [21] L. Paivarinta and J. Sylvester, Transmission eigenvalues, SIAM Journal on Mathematical Analysis, 40 (2008), 738-753.  doi: 10.1137/070697525.  Google Scholar [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.  Google Scholar [23] J. Shen, Efficient spectral-Galerkin methods Ⅲ: Polar and cylindrical geometries, SIAM J. Sci. Comput., 18 (1997), 1583-1604.  doi: 10.1137/S1064827595295301.  Google Scholar [24] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, 2006.  Google Scholar [25] J. Sun, Iterative methods for transmission eigenvalues, SIAM Journal on Numerical Analysis, 49 (2011), 1860-1874.  doi: 10.1137/100785478.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [6] I. Babu$\breve{s}$ka and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, 2 (1991), 641-787.   Google Scholar [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.  Google Scholar [8] F. Cakoni, D. 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.  Google Scholar [9] F. Cakoni, D. 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.  Google Scholar [10] F. Cakoni, D. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] F. Cakoni, D. 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.  Google Scholar [14] F. Cakoni, P. 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.  Google Scholar [15] D. Colton, L. Paivarinta and J. Sylvester, The interior transmission problem, Inverse Problems and Imaging, 1 (2007), 13-28.  doi: 10.3934/ipi.2007.1.13.  Google Scholar [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.  Google Scholar [17] X. Ji, Y. 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.  Google Scholar [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.  Google Scholar [19] X. Ji, J. 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.  Google Scholar [20] A. Kirsch, On the existence of transmission eigenvalues, Inverse Problems and Imaging, 3 (2009), 155-172.  doi: 10.3934/ipi.2009.3.155.  Google Scholar [21] L. Paivarinta and J. Sylvester, Transmission eigenvalues, SIAM Journal on Mathematical Analysis, 40 (2008), 738-753.  doi: 10.1137/070697525.  Google Scholar [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.  Google Scholar [23] J. Shen, Efficient spectral-Galerkin methods Ⅲ: Polar and cylindrical geometries, SIAM J. Sci. Comput., 18 (1997), 1583-1604.  doi: 10.1137/S1064827595295301.  Google Scholar [24] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, 2006.  Google Scholar [25] J. Sun, Iterative methods for transmission eigenvalues, SIAM Journal on Numerical Analysis, 49 (2011), 1860-1874.  doi: 10.1137/100785478.  Google Scholar [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.  Google Scholar
Numerical errors of example 1 for $m = 0$.
Numerical errors of example 1 for $m = 2$.
Numerical errors of example 2 for $m = 1$.
Numerical errors of example 2 for m = 0.
Numerical errors of example 2 for m = 1.
Numerical errors of example 3 for m = 0.
The first four eigenvalues for m = 0
 $N \setminus k_{0N}^i$ $k_{0N}^1$ $k_{0N}^2$ $k_{0N}^3$ $k_{0N}^4$ $N = 10$ 1.987995123773161 3.740925090249691 6.582278298163812 8.386644958117056 $N = 15$ 1.987995123771375 3.740924935100444 6.581030572825058 8.273172861972871 $N = 20$ 1.987995123771376 3.740924935100446 6.581030549720257 8.273170634133454 $N = 25$ 1.987995123771378 3.740924935100443 6.581030549720255 8.273170634125105 $N = 30$ 1.987995123771378 3.740924935100443 6.581030549720255 8.273170634125105
 $N \setminus k_{0N}^i$ $k_{0N}^1$ $k_{0N}^2$ $k_{0N}^3$ $k_{0N}^4$ $N = 10$ 1.987995123773161 3.740925090249691 6.582278298163812 8.386644958117056 $N = 15$ 1.987995123771375 3.740924935100444 6.581030572825058 8.273172861972871 $N = 20$ 1.987995123771376 3.740924935100446 6.581030549720257 8.273170634133454 $N = 25$ 1.987995123771378 3.740924935100443 6.581030549720255 8.273170634125105 $N = 30$ 1.987995123771378 3.740924935100443 6.581030549720255 8.273170634125105
The first four eigenvalues for $m = 1$
 $N \setminus k_{0N}^i$ $k_{1N}^1$ $k_{1N}^2$ $k_{0N}^3$ $k_{1N}^4$ $N = 10$ 2.612929964208549 4.295810771178445 5.989775023153221 9.091957821822609 $N = 15$ 2.612929963902793 4.295809936658583 5.988500910315373 8.841298655023424 $N = 20$ 2.612929963902789 4.295809936658451 5.988500910106952 8.841280530061555 $N = 25$ 2.612929963902792 4.295809936658459 5.988500910106954 8.841280529974766 $N = 30$ 2.612929963902794 4.295809936658448 5.988500910106954 8.841280529974791
 $N \setminus k_{0N}^i$ $k_{1N}^1$ $k_{1N}^2$ $k_{0N}^3$ $k_{1N}^4$ $N = 10$ 2.612929964208549 4.295810771178445 5.989775023153221 9.091957821822609 $N = 15$ 2.612929963902793 4.295809936658583 5.988500910315373 8.841298655023424 $N = 20$ 2.612929963902789 4.295809936658451 5.988500910106952 8.841280530061555 $N = 25$ 2.612929963902792 4.295809936658459 5.988500910106954 8.841280529974766 $N = 30$ 2.612929963902794 4.295809936658448 5.988500910106954 8.841280529974791
The first four eigenvalues for $m = 2$
 $N \setminus k_{2N}^i$ $k_{2N}^1$ $k_{2N}^2$ $k_{2N}^3$ $k_{2N}^4$ $N = 10$ 3.226647948718986 4.941889332720679 6.605158179665295 8.397840671676448 $N = 15$ 3.226647947890250 4.941834557693348 6.603834491742335 8.279736371734421 $N = 20$ 3.226647947890250 4.941834557692668 6.603834467939633 8.279734012410303 $N = 25$ 3.226647947890251 4.941834557692666 6.603834467939639 8.279734012401546 $N = 30$ 3.226647947890251 4.941834557692663 6.603834467939636 8.279734012401549
 $N \setminus k_{2N}^i$ $k_{2N}^1$ $k_{2N}^2$ $k_{2N}^3$ $k_{2N}^4$ $N = 10$ 3.226647948718986 4.941889332720679 6.605158179665295 8.397840671676448 $N = 15$ 3.226647947890250 4.941834557693348 6.603834491742335 8.279736371734421 $N = 20$ 3.226647947890250 4.941834557692668 6.603834467939633 8.279734012410303 $N = 25$ 3.226647947890251 4.941834557692666 6.603834467939639 8.279734012401546 $N = 30$ 3.226647947890251 4.941834557692663 6.603834467939636 8.279734012401549
The first four eigenvalues for $m = 0$
 $N \setminus k_{0N}^i$ $k_{0N}^1$ $k_{0N}^2$ $k_{0N}^3$ $k_{0N}^4$ $N = 10$ 2.759435139166149 6.667659573285005 9.049164387130203 13.35822517979881 $N = 15$ 2.759435139158974 6.667547034574103 9.047040495059562 12.95759442163584 $N = 20$ 2.759435139158978 6.667547034553697 9.047040407351044 12.95695038075036 $N = 25$ 2.759435139158978 6.667547034553697 9.047040407351044 12.95695038075036 $N = 30$ 2.759435139158981 6.667547034553698 9.047040407351048 12.95695038075038
 $N \setminus k_{0N}^i$ $k_{0N}^1$ $k_{0N}^2$ $k_{0N}^3$ $k_{0N}^4$ $N = 10$ 2.759435139166149 6.667659573285005 9.049164387130203 13.35822517979881 $N = 15$ 2.759435139158974 6.667547034574103 9.047040495059562 12.95759442163584 $N = 20$ 2.759435139158978 6.667547034553697 9.047040407351044 12.95695038075036 $N = 25$ 2.759435139158978 6.667547034553697 9.047040407351044 12.95695038075036 $N = 30$ 2.759435139158981 6.667547034553698 9.047040407351048 12.95695038075038
The first four eigenvalues for $m = 1$
 $N \setminus k_{1N}^i$ $k_{1N}^1$ $k_{1N}^2$ $k_{1N}^3$ $k_{1N}^4$ $N = 10$ 3.527276155521052 5.895294642469102 9.837502809063089 12.78302633393358 $N = 15$ 3.527276155499329 5.895285669974978 9.810491953392948 12.18568470580060 $N = 20$ 3.527276155499329 5.895285669974959 9.810491860606557 12.18556255236016 $N = 25$ 3.527276155499327 5.895285669974957 9.810491860606326 12.18556255231528 $N = 30$ 3.527276155499331 5.895285669974958 9.810491860606319 12.18556255231531
 $N \setminus k_{1N}^i$ $k_{1N}^1$ $k_{1N}^2$ $k_{1N}^3$ $k_{1N}^4$ $N = 10$ 3.527276155521052 5.895294642469102 9.837502809063089 12.78302633393358 $N = 15$ 3.527276155499329 5.895285669974978 9.810491953392948 12.18568470580060 $N = 20$ 3.527276155499329 5.895285669974959 9.810491860606557 12.18556255236016 $N = 25$ 3.527276155499327 5.895285669974957 9.810491860606326 12.18556255231528 $N = 30$ 3.527276155499331 5.895285669974958 9.810491860606319 12.18556255231531
The first four eigenvalues for $m = 2$
 $N \setminus k_{2N}^i$ $k_{2N}^1$ $k_{2N}^2$ $k_{2N}^3$ $k_{2N}^4$ $N = 10$ 4.307972659770961 6.648593169958727 9.086854035427645 13.34452507236263 $N = 15$ 4.307972635971419 6.648486695293351 9.083689181278766 12.94522396389963 $N = 20$ 4.307972635971415 6.648486695276506 9.083689053302182 12.94460023197706 $N = 25$ 4.307972635971417 6.648486695276508 9.083689053302171 12.94460022855666 $N = 30$ 4.307972635971418 6.648486695276508 9.083689053302175 12.94460022855666
 $N \setminus k_{2N}^i$ $k_{2N}^1$ $k_{2N}^2$ $k_{2N}^3$ $k_{2N}^4$ $N = 10$ 4.307972659770961 6.648593169958727 9.086854035427645 13.34452507236263 $N = 15$ 4.307972635971419 6.648486695293351 9.083689181278766 12.94522396389963 $N = 20$ 4.307972635971415 6.648486695276506 9.083689053302182 12.94460023197706 $N = 25$ 4.307972635971417 6.648486695276508 9.083689053302171 12.94460022855666 $N = 30$ 4.307972635971418 6.648486695276508 9.083689053302175 12.94460022855666
The first four eigenvalues for $m = 0$
 $N \setminus k_{1N}^i$ $k_{1N}^1$ $k_{1N}^2$ $k_{1N}^3$ $k_{1N}^4$ $N = 10$ 3.433564272232199 4.600685451015495 8.113511966903733 9.405045505293291 $N = 15$ 3.433564206752173 4.600641886964474 7.966311166578104 9.146084137724058 $N = 20$ 3.433564206752172 4.600641886963469 7.966309360466283 9.145821838198342 $N = 25$ 3.433564206752172 4.600641886963474 7.966309360458688 9.145821837979225 $N = 30$ 3.433564206752175 4.600641886963472 7.966309360458691 9.145821837979231
 $N \setminus k_{1N}^i$ $k_{1N}^1$ $k_{1N}^2$ $k_{1N}^3$ $k_{1N}^4$ $N = 10$ 3.433564272232199 4.600685451015495 8.113511966903733 9.405045505293291 $N = 15$ 3.433564206752173 4.600641886964474 7.966311166578104 9.146084137724058 $N = 20$ 3.433564206752172 4.600641886963469 7.966309360466283 9.145821838198342 $N = 25$ 3.433564206752172 4.600641886963474 7.966309360458688 9.145821837979225 $N = 30$ 3.433564206752175 4.600641886963472 7.966309360458691 9.145821837979231
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