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

TL;DR

This paper develops and analyzes a Robin-Dirichlet alternating iterative method for solving the Helmholtz equation's Cauchy problem in unbounded domains, demonstrating convergence through numerical experiments with finite difference methods.

Contribution

It extends the Robin-Dirichlet alternating iterative method to unbounded domains for Helmholtz problems and establishes convergence conditions with numerical validation.

Findings

The iterative method converges with suitable domain truncation and Robin parameters.

Numerical experiments confirm the effectiveness of the proposed approach.

The method provides a practical solution for Helmholtz problems in unbounded domains.

Abstract

We consider the Cauchy problem for the Helmholtz equation with a domain in R^d, d>2 with N cylindrical outlets to infinity with bounded inclusions in 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 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 Mpinganzima et al. 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 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, the Robin-Dirichlet alternating iterative procedure is convergent.