High Order Transparent Boundary Conditions for the Helmholtz Equation

Abstract

Abstract: We consider finite element simulations of the Helmholtz equation in unbounded domains. For computational purposes, these domains are truncated to bouded domains using transparent boundary conditions at the artificial boundaries. We present here two numerical realizations of transparent boundary conditions: the complex scaling or perfeclty matched layer method and the Hardy space infinite element method. Both methods are Galerkin methods, but their variational framework differs. Proofs of convergence of the methods are given in detail for one dimensional problems. In higher dimensions radial al well as Cartesian constructions are introduced with references to the known theory.