Variational algorithms for the Helmholtz equation using time evolution and artificial boundaries

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.