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

  • * Corresponding author: Jun Zhang

    * Corresponding author: Jun Zhang 

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

Abstract / Introduction Full Text(HTML) Figure(6) / Table(7) Related Papers Cited by
  • 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:

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  • 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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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