Affiliations: Université de Paris VII, Mathématiques, 3ème cycle, Tour 45–55, 5e etage, porte 16, 2, place Jussieu, 75251 Paris Cedex 05, France | Mathematics Department, University of Michigan, Ann Arbor, USA
Note: [] Correspondence to: C. Bardos, Université de Paris VII, Mathématiques, 3ème cycle, Tour 45-55, 5e etage, porte 16, 2, place Jussieu, 75251 Paris Cedex 05, France.
Note: [] Partially supported by the Direction Générale de l'Armement under contract DRET ERS92/1441/A300/DRET/DS/SR.
Note: [] Partially supported by the National Science Foundation and the Office of Naval Research under grants DMS 90 03256 and N 014 92 J 1245, respectively.
Abstract: This paper is devoted to the mathematical analysis of some algorithms for the computation of the outgoing solution of the Helmholtz equation in an exterior domain. In a first approximation an artificial boundary with absorbing boundary condition is inserted. One then computes the periodic response in this bounded dissipative setting to a periodic forcing term. The response is characterized as the unique minimum of a convex functional. The functional is computed from solution of the time dependent problem in the artificially bounded domain. One such algorithm is due to Glowinski who proposed the functional J2 described below, it has been implemented by Bristeau et al. [5]. We propose a different functional J1 which is unconditionally coercive while the coerciveness of J2 depends in a subtle way on the geometry of the domain. It is coercive for non-trapping obstacles. The coerciveness property is essential for the convergence of the numerical method.
