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

Jun Zhang; Xinyue Fan

doi:10.3934/dcdsb.2019031

Article Contents

2019, Volume 24, Issue 9: 4799-4813. Doi: 10.3934/dcdsb.2019031

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

Jun Zhang^1, , and
Xinyue Fan^2,

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
^* Corresponding author: Jun Zhang

Received: January 2018

Revised: August 2018

Early access: February 2019

Published: July 2019

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35J05, 35P15; Secondary: 76M22.

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Figure 1. Numerical errors of example 1 for $ m = 0 $.

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Figure 2. Numerical errors of example 1 for $ m = 2 $.

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Figure 3. Numerical errors of example 2 for $ m = 1 $.

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Figure 4. Numerical errors of example 2 for m = 0.

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Figure 5. Numerical errors of example 2 for m = 1.

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Figure 6. Numerical errors of example 3 for m = 0.

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Table 1. The first four eigenvalues for m = 0

$N \setminus k_{0N}^i $	$k_{0N}^1$	$k_{0N}^2$	$k_{0N}^3$	$k_{0N}^4$
$N = 10$	1.987995123773161	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

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Table 2. The first four eigenvalues for $m = 1$

$N \setminus k_{0N}^i $	$k_{1N}^1$	$k_{1N}^2$	$k_{0N}^3$	$k_{1N}^4$
$N = 10$	2.612929964208549	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

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Table 3. The first four eigenvalues for $m = 2$

$N \setminus k_{2N}^i $	$k_{2N}^1$	$k_{2N}^2$	$k_{2N}^3$	$k_{2N}^4$
$N = 10$	3.226647948718986	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

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Table 4. The first four eigenvalues for $m = 0$

$N \setminus k_{0N}^i $	$k_{0N}^1$	$k_{0N}^2$	$k_{0N}^3$	$k_{0N}^4$
$N = 10$	2.759435139166149	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

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Table 5. The first four eigenvalues for $m = 1$

$N \setminus k_{1N}^i $	$k_{1N}^1$	$k_{1N}^2$	$k_{1N}^3$	$k_{1N}^4$
$N = 10$	3.527276155521052	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

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Table 6. The first four eigenvalues for $m = 2$

$N \setminus k_{2N}^i $	$k_{2N}^1$	$k_{2N}^2$	$k_{2N}^3$	$k_{2N}^4$
$N = 10$	4.307972659770961	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

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Table 7. The first four eigenvalues for $m = 0$

$N \setminus k_{1N}^i $	$k_{1N}^1$	$k_{1N}^2$	$k_{1N}^3$	$k_{1N}^4$
$N = 10$	3.433564272232199	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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