### Abstract

In this paper, we develop a new representation for outgoing solutions to the time harmonic Maxwell equations in unbounded domains in R{double struck}^{3}. This representation leads to a Fredholm integral equation of the second kind for solving the problem of scattering from a perfect conductor, which does not suffer from spurious resonances or low-frequency breakdown, although it requires the inversion of the scalar surface Laplacian on the domain boundary. In the course of our analysis, we give a new proof of the existence of nontrivial families of time-harmonic solutions with vanishing normal components that arise when the boundary of the domain is not simply connected. We refer to these as k-Neumann fields, since they generalize, to nonzero wave numbers, the classical harmonic Neumann fields. The existence of k-Neumann fields was established earlier by Kress.

Original language | English (US) |
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Pages (from-to) | 413-463 |

Number of pages | 51 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 63 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2010 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Debye sources and the numerical solution of the time harmonic maxwell equations.** / Epstein, Charles L.; Greengard, Leslie.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 63, no. 4, pp. 413-463. https://doi.org/10.1002/cpa.20313

}

TY - JOUR

T1 - Debye sources and the numerical solution of the time harmonic maxwell equations

AU - Epstein, Charles L.

AU - Greengard, Leslie

PY - 2010/4

Y1 - 2010/4

N2 - In this paper, we develop a new representation for outgoing solutions to the time harmonic Maxwell equations in unbounded domains in R{double struck}3. This representation leads to a Fredholm integral equation of the second kind for solving the problem of scattering from a perfect conductor, which does not suffer from spurious resonances or low-frequency breakdown, although it requires the inversion of the scalar surface Laplacian on the domain boundary. In the course of our analysis, we give a new proof of the existence of nontrivial families of time-harmonic solutions with vanishing normal components that arise when the boundary of the domain is not simply connected. We refer to these as k-Neumann fields, since they generalize, to nonzero wave numbers, the classical harmonic Neumann fields. The existence of k-Neumann fields was established earlier by Kress.

AB - In this paper, we develop a new representation for outgoing solutions to the time harmonic Maxwell equations in unbounded domains in R{double struck}3. This representation leads to a Fredholm integral equation of the second kind for solving the problem of scattering from a perfect conductor, which does not suffer from spurious resonances or low-frequency breakdown, although it requires the inversion of the scalar surface Laplacian on the domain boundary. In the course of our analysis, we give a new proof of the existence of nontrivial families of time-harmonic solutions with vanishing normal components that arise when the boundary of the domain is not simply connected. We refer to these as k-Neumann fields, since they generalize, to nonzero wave numbers, the classical harmonic Neumann fields. The existence of k-Neumann fields was established earlier by Kress.

UR - http://www.scopus.com/inward/record.url?scp=77952823014&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952823014&partnerID=8YFLogxK

U2 - 10.1002/cpa.20313

DO - 10.1002/cpa.20313

M3 - Article

VL - 63

SP - 413

EP - 463

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 4

ER -