Robin–Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain

Abstract

Abstract We consider the Cauchy problem for the Helmholtz equation with a domain in ℝ d {\mathbb{R}^{d}} , d ≥ 2 {d\geq 2} with N cylindrical outlets to infinity with bounded inclusions in ℝ d - 1 . {\mathbb{R}^{d-1}.} Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz’ya [14] proposed an alternating iterative method for solving Cauchy problems associated with elliptic, self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) [4] for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in ℝ 2 {\mathbb{R}^{2}} that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters μ 0 {\mu_{0}} and μ 1 {\mu_{1}} , the Robin–Dirichlet alternating iterative procedure is convergent.