Integral equations via saddle point problems for time-harmonic Maxwell's equations
type: Item
instance of: scholarly article
title: Integral equations via saddle point problems for time-harmonic Maxwell's equations
zbMATH Open document ID: 1016.65110
DOI: 10.1016/S0377-0427(02)00658-1
author: Francis Collino
author: Bruno Després
published in: Journal of Computational and Applied Mathematics
publication date: 2003-03-12T00:00:00Z
review text: The authors propose and study a new system of integral equations for solving obstacle scattering by time-harmonic electromagnetic waves, with particular emphasis on impedance boundary conditions. The space \(\mathcal W\) of incoming and outgoing solutions of Maxwell's equations is introduced, on which a suitable quadratic functional depending on a boundary data is defined. It is proved that the minimum of this functional is reached at a point that coincides with the outgoing solution of Maxwell's equation satisfying a boundary condition linked to the data boundary. Using a parametrization of \(\mathcal W\) by means of currents, it is shown that the system has an interpretation as a minimization problem with constraints. Introducing a Lagrange multiplier and the associated Lagrangian, the optimality conditions allow to recover the system previously obtained. One interesting feature of the new system of equations, comparing to the classical ones, is that the space of solutions is \(L^2\). A penalization procedure may be used to get more coercivity on the multiplier, even if the inf-sup condition is already true for the nonpenalized formulation. This coercive framework might appear as unusual when compared with the standard theory of integral operators for electromagnetism. However, the proof of well-posedness of the weak formulation is based on some well known fundamental properties of the exterior Calderón projectors for Maxwell's equations. An iterative algorithm to solve the system is proposed, and its convergence is proved. Other numerical issues are also discussed. In order to give a better understanding of the new system, the spectrum of the integral operator in the special case of a spherical scatterer is analytically determined. The method presented in this work has some advantages from the computational point of view when compared to more classical integral equations like electric field integral equation or magnetic or combined field integral equation.
reviewed by: Nikolay Yakovlevich Tikhonenko
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 78A45
zbMATH DE Number: 1880342
zbMATH Keywords: Maxwell equations
zbMATH Keywords: obstacle scattering
zbMATH Keywords: time-harmonic electromagnetic waves
zbMATH Keywords: system of integral equation
zbMATH Keywords: Green function
zbMATH Keywords: incoming and outgoing electromagnetic fields
zbMATH Keywords: asymptotic behavior of electromagnetic pairs
zbMATH Keywords: quadratic functional
zbMATH Keywords: minimization problem with constraints
zbMATH Keywords: Lagrange multiplier
zbMATH Keywords: penalized system
zbMATH Keywords: well-posedness
zbMATH Keywords: saddle point problems
zbMATH Keywords: impedance boundary conditions
zbMATH Keywords: convergence
zbMATH Keywords: spectrum
zbMATH Keywords: spherical scatterer
zbMATH Keywords: exterior Calderon projectors
zbMATH Keywords: iterative algorithm
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1880342