### Abstract

We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell's equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method processes the convergence efficiently, even when it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by nonlinear functions, which can be applied to estimate the denseness of the eigenvalues for the TEP.

Original language | English |
---|---|

Article number | 035009 |

Journal | Inverse Problems |

Volume | 33 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 Feb 7 |

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### Keywords

- Tellegen model
- non-equivalence deflation
- quadratic Jacobi-Davidson method
- two-dimensional transmission eigenvalue problem

### ASJC Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Mathematical Physics
- Computer Science Applications
- Applied Mathematics

### Cite this

*Inverse Problems*,

*33*(3), [035009]. https://doi.org/10.1088/1361-6420/aa5475

**An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues.** / Li, Tiexiang; Hwang, Tsung-Min; Lin, Wen Wei; Wang, Jenn Nan.

Research output: Contribution to journal › Article

*Inverse Problems*, vol. 33, no. 3, 035009. https://doi.org/10.1088/1361-6420/aa5475

}

TY - JOUR

T1 - An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues

AU - Li, Tiexiang

AU - Hwang, Tsung-Min

AU - Lin, Wen Wei

AU - Wang, Jenn Nan

PY - 2017/2/7

Y1 - 2017/2/7

N2 - We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell's equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method processes the convergence efficiently, even when it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by nonlinear functions, which can be applied to estimate the denseness of the eigenvalues for the TEP.

AB - We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell's equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method processes the convergence efficiently, even when it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by nonlinear functions, which can be applied to estimate the denseness of the eigenvalues for the TEP.

KW - Tellegen model

KW - non-equivalence deflation

KW - quadratic Jacobi-Davidson method

KW - two-dimensional transmission eigenvalue problem

UR - http://www.scopus.com/inward/record.url?scp=85014629175&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014629175&partnerID=8YFLogxK

U2 - 10.1088/1361-6420/aa5475

DO - 10.1088/1361-6420/aa5475

M3 - Article

AN - SCOPUS:85014629175

VL - 33

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 3

M1 - 035009

ER -