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

• * Corresponding author: Jun Zhang

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.

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

 Citation: • • 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

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

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

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

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

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

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

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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