# Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations II

Charles L. Epstein, Leslie Greengard, Michael O'Neil

Research output: Contribution to journalArticle

### Abstract

In this paper, we develop a new integral representation for the solution of the time harmonic Maxwell equations in media with piecewise constant dielectric permittivity and magnetic permeability in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$. This representation leads to a coupled system of Fredholm integral equations of the second kind for four scalar densities supported on the material interface. Like the classical Müller equation, it has no spurious resonances. Unlike the classical approach, however, the representation does not suffer from low-frequency breakdown. We illustrate the performance of the method with numerical examples.

Original language English (US) 753-789 37 Communications on Pure and Applied Mathematics 66 5 https://doi.org/10.1002/cpa.21420 Published - May 2013

### Fingerprint

Dielectric Constant
Fredholm Integral Equation
Maxwell equations
Permittivity
Maxwell's equations
Integral Representation
Permeability
Coupled System
Breakdown
Low Frequency
Harmonic
Numerical Solution
Scalar
Magnetic permeability
Numerical Examples
Integral equations

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

In: Communications on Pure and Applied Mathematics, Vol. 66, No. 5, 05.2013, p. 753-789.

Research output: Contribution to journalArticle

@article{28788225c6974324a05ef59117d5db31,
title = "Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations II",
abstract = "In this paper, we develop a new integral representation for the solution of the time harmonic Maxwell equations in media with piecewise constant dielectric permittivity and magnetic permeability in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$. This representation leads to a coupled system of Fredholm integral equations of the second kind for four scalar densities supported on the material interface. Like the classical M{\"u}ller equation, it has no spurious resonances. Unlike the classical approach, however, the representation does not suffer from low-frequency breakdown. We illustrate the performance of the method with numerical examples.",
author = "Epstein, {Charles L.} and Leslie Greengard and Michael O'Neil",
year = "2013",
month = "5",
doi = "10.1002/cpa.21420",
language = "English (US)",
volume = "66",
pages = "753--789",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "5",

}

TY - JOUR

T1 - Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations II

AU - Epstein, Charles L.

AU - Greengard, Leslie

AU - O'Neil, Michael

PY - 2013/5

Y1 - 2013/5

N2 - In this paper, we develop a new integral representation for the solution of the time harmonic Maxwell equations in media with piecewise constant dielectric permittivity and magnetic permeability in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$. This representation leads to a coupled system of Fredholm integral equations of the second kind for four scalar densities supported on the material interface. Like the classical Müller equation, it has no spurious resonances. Unlike the classical approach, however, the representation does not suffer from low-frequency breakdown. We illustrate the performance of the method with numerical examples.

AB - In this paper, we develop a new integral representation for the solution of the time harmonic Maxwell equations in media with piecewise constant dielectric permittivity and magnetic permeability in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$. This representation leads to a coupled system of Fredholm integral equations of the second kind for four scalar densities supported on the material interface. Like the classical Müller equation, it has no spurious resonances. Unlike the classical approach, however, the representation does not suffer from low-frequency breakdown. We illustrate the performance of the method with numerical examples.

UR - http://www.scopus.com/inward/record.url?scp=84874252475&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874252475&partnerID=8YFLogxK

U2 - 10.1002/cpa.21420

DO - 10.1002/cpa.21420

M3 - Article

VL - 66

SP - 753

EP - 789

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 5

ER -