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Integral equations via saddle point problems for time-harmonic Maxwell’s equations. (English) Zbl 1016.65110
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.

MSC:
65R20 Numerical methods for integral equations
65F05 Direct numerical methods for linear systems and matrix inversion
78A45 Diffraction, scattering
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