A well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition

Abstract

Abstract. The construction of a well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition is proposed. A suitable parametrix is obtained by using a new unknown and an approximation of the transparency condition. We prove the well-posedness of the equation for any wavenumber. Finally, some numerical comparisons with well-tried method prove the efficiency of the new formulation.Mathematics Subject Classification. 65R20, 15A12, 65N38, 65F10, 65Z05.

Approximate local magnetic-to-electric surface operators for time-harmonic Maxwell's equations

“…where u is given by (54). The nice point with (58) is that there is no coupling between the auxiliary vector fields φ j , j = 1, ..., N p .…”

“…When an approximation of the MtE surface operator is known, it can be suitably injected into a standard integral equation for preconditioning or can be used to build new well-conditioned and stable integral equations with respect to k. These directions have now been widely studied in the case of three-dimensional acoustics [10,9,[11][12][13]22,23,2,19,42], where the MtE operator is then the Dirichlet-to-Neumann map (DtN(u, ∂ n u) = 0 on Γ ) [59,37,36,14,7], but much less deeply for electromagnetic scattering [28,22,23,2,19,54,41]. In [28], Darbas presented a construction of a square-root OSRC generalizing to electromagnetism the construction proposed in [14] for acoustics.…”

“…In [28], Darbas presented a construction of a square-root OSRC generalizing to electromagnetism the construction proposed in [14] for acoustics. This operator was discretized and extended by Pernet [54] in an integral equation framework for the Leontovich boundary-value problem. The finite element discretization was based on the Helmholtz/Hodge decomposition, which is clearly different from the approach presented here since no integral framework is required.…”

A Simple Preconditioned Domain Decomposition Method for Electromagnetic Scattering Problems

“…It has been shown that the underlying systems arising from integral equations are usually badly conditioned and there is a need to develop preconditioning strategies in order to accelerate the convergence of the iterative solver [10], [22], [30], [31]. For instance, the so-called GCSIE methodology has been developed which turns out to be particularly efficient in the case where the object has no cavities and no singularities, by building intrinsically well conditioned integral equations [1], [6], [14], [23], [27].…”

Well‐conditioned boundary integral formulations for high‐frequency elastic scattering problems in three dimensions

“…Such integral formulations can be interpreted as generalizations of the well-known Brakhage-Werner integral equation (BW) [13] and combined field integral equation (CFIE) [14]. Several well-conditioned integral equations based on this formalism have already been proposed in acoustic and electromagnetic scattering (e.g., [15][16][17][18][19][20][21]). A pseudo inverse of the principal classical symbol of the single layer boundary integral operator -or equivalently the principal classical symbol of the Neumann trace of the double layer boundary integral operator -is used to approach the DtN map [17,19,22] in the framework of the on-surface radiation condition (OSRC) methods (e.g., [23][24][25]).…”