Boundary Decomposition for Helmholtz and Maxwell equations 1: disjoint sub‐scatterers

Article type: Research Article

Authors: Balabane, Mikhael

Affiliations: University of Paris 13, Inst. Galilee, Math. Dpt, av. J.B. Clement, 93430 Villetaneuse, France E‐mail: balabane@math.univ‐paris13.fr

Abstract: In order to compute solutions of the Helmholtz or the Maxwell equations, in the complementary set in $\mathbb{R}$3 of a scatterer, one use usually a Domain Decomposition technique when the scatterer is huge. In this paper, we give an alternative to this technique, that we call Boundary Decomposition. We decompose the scatterer itself, into sub‐scatterers, and give the solution as a sum of contributions of each sub‐scatterer. From this point of view, it is an homological type decomposition. It is given here for disjoint sub‐scatterers, and the equations are solved in the frequency domain. Each contribution of a sub‐scaterer is computed as a sum of a series: an algorithm of Jacobi type is given and prove to converge. The theoretical results needed are set and the proofs for convergence of the algorithm are given. Note that the algorithm does not rely on properties of any specific numerical method for solving the equations. It only relies on properties of the equation itself. So at each step, one can use a specific solver adapted to the geometry of the sub‐scatterer dealt with at that step. This algorithm fits distributed computations. It is now in use in industrial CAD, namely at Aerospatiale‐EADS France.1

Journal: Asymptotic Analysis, vol. 38, no. 1, pp. 1-10, 2004