Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations (original) (raw)
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Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations II
Communications on Pure and Applied Mathematics, 2012
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 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.
On a transmission problem for the time–harmonic Maxwell equations
Riassunto: In questo lavoro si considera il problema di trasmissione per le equazioni di Maxwell armoniche nel tempo, per un diffusore a infiniti strati omogenei, costituiti da materiali diversi. Si dimostra l'esistenza e l'unicità della soluzione. Inoltre si costruisce una rappresentazione integrale del campo esterno totale e si esamina il comportamento asintotico dell'onda diffusa nella regione di radiazione.
The Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scattering
Communications on Pure and Applied Mathematics, 2015
We present a new formulation for the problem of electromagnetic scattering from perfect electric conductors. While our representation for the electric and magnetic fields is based on the standard vector and scalar potentials A; in the Lorenz gauge, we establish boundary conditions on the potentials themselves rather than on the field quantities. This permits the development of a wellconditioned second-kind Fredholm integral equation that has no spurious resonances, avoids low-frequency breakdown, and is insensitive to the genus of the scatterer. The equations for the vector and scalar potentials are decoupled. That is, the unknown scalar potential defining the scattered field, scat , is determined entirely by the incident scalar potential inc. Likewise, the unknown vector potential defining the scattered field, A scat , is determined entirely by the incident vector potential A inc. This decoupled formulation is valid not only in the static limit but for arbitrary ! 0.
Existence and Uniqueness of Scattering Solutions in Non-smooth Domains
Journal of Mathematical Analysis and Applications, 1996
A short and self-contained proof of the existence of the scattering solution in exterior domains is presented for some class of second order elliptic equations. The method does not use the integral equation; it is based on Fredholm theory and the limiting absorption principle for solutions in the whole space. It covers domains with Lipschitz boundaries, domains satisfying a cone condition, and those with the so-called local compactness property.
A boundary integral equation for the transmission eigenvalue problem for Maxwell equation
Mathematical Methods in the Applied Sciences, 2018
We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two-by-two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric-to-magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation. KEYWORDS boundary integral equations, inhomogeneous media, inverse scattering, transmission eigenvalues Dedicated to Erich Martensen on the occasion of his 90th birthday
Mathematical Models and Methods in Applied Sciences, 2000
We analyze the solution of the time-harmonic Maxwell equations with vanishing electric permittivity in bounded domains and subject to absorbing boundary conditions. The problem arises naturally in magnetotellurics when considering the propagation of electromagnetic waves within the earth's interior. Existence and uniqueness are shown under the assumption that the source functions are square integrable. In this case, the electric and magnetic fields belong to H(curl; Ω). If, in addition, the divergences of the source functions are square integrable and the coefficients are Lipschitz-continuous, a stronger regularity result is obtained. A decomposition of the space of square integrable vector functions and a new compact imbedding result are exploited. *
The Conductive Boundary Condition for Maxwell’s Equations
SIAM Journal on Applied Mathematics, 1992
The transmission boundary value problem for a perturbed Dirac operator on arbitrary bounded Lipschitz domains in R 3 is formulated and solved in terms of layer potentials of Clifford-Cauchy type. As a byproduct of this analysis, an elliptization procedure for the Maxwell system is devised which allows us to show that the Maxwell and Helmholtz transmission boundary value problems are well-posed as a corollary of the unique solvability of this more general Dirac transmission problem.
On the solution of Maxwell's equations in axisymmetric domains with edges
ZAMM, 2005
In this paper we present the basic mathematical tools for treating boundary value problems for the Maxwell equations in three-dimensional axisymmetric domains with reentrant edges using the Fourier-finite-element method. We consider both the classical and the regularized time-harmonic Maxwell equations subject to perfect conductor boundary conditions. The partial Fourier decomposition reduces the three-dimensional boundary value problem into an infinite sequence of twodimensional boundary value problems in the plane meridian domain of the axsiymmetric domain. Here, suitable weighted Sobolev spaces that characterize the solutions of the reduced problems are given, and their trace properties on the rotational axis are proved. In these spaces, it is proved that the reduced problems are well posed, and the asymptotic behavior of the solutions near reentrant corners of the meridian domain is explicitly described by suitable singularity functions. Finally, a finite number of the two-dimensional problems is considered and treated using H 1-conforming finite elements. An approximation of the solution of the three-dimensional problem is obtained by Fourier synthesis. For domains with reentrant edges, the singular field method is employed to compensate the singular behavior of the solutions of the reduced problems. Emphases are given to convergence analysis of the combined approximations in H 1 under different regularity assumptions on the solution.
Approximate transmission conditions for time-harmonic Maxwell equations in a domain with thin layer
We study the behavior of the electromagnetic field in a biological cell modelized by a medium surrounded by a thin layer and embedded in an ambient medium. We derive approximate transmission conditions in order to replace the membrane by these conditions on the boundary of the interior domain. Our approach is essentially geometric and based on a suitable change of variables in the thin layer. Few notions of differential calculus are given in order to obtain our asymptotic conditions in a simple way. This paper extends to time-harmonic Maxwell equations the previous works presented in . Asymptotic transmission conditions at any order are given in Appendix 1. Conditions de transmission approchées pour leś equations de Maxwell en régime harmonique dans un milieuà couche mince Résumé : Nousétudions le comportement asymptotique du champélectromagnétique dans une cellule biologique plongée dans un milieu ambiant. La cellule est composée d'un cytoplasme entouré d'une fine membrane. Nous obtenons des conditions de transmission sur le bord du cytoplasmeéquivalentesà la couche mince. Notre approche est essentiellement géométrique et basée sur un changement de variables adéquat dans la couche mince. Quelques notions de calcul différentiel sont rappelées afin d'obtenir directement notre développement asymptotique. Par ailleurs des estimations d'erreur sont démontrées. En appendice, nous présentons le développement asymptotiqueà tout ordre. 2 Using the notations of the electrical engineeering community, q = ω 2`ǫ − i σ ω´, where ω is the frequency, ǫ the permittivity and σ the conductivity of the domain [3]. INRIA inria-00347971, version 3 -14 Sep 2009 Péron& Poignard where