Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation

Abstract

The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann–Dirichlet iterations by the Robin–Dirichlet ones which repairs the convergence for $k^2$ less than the first Dirichlet–Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin–Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.

Keywords:

• Helmholtz equation
• Cauchy problem
• alternating iterative method
• inverse problem
• ill-posed problem

AMS Subject Classifications:

• 35J05
• 35R25
• 35R30

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Lydie Mpinganzima was supported by the Swedish International Development Cooperation Agency (Sida) and the University of Rwanda (UR) [Sida Contribution No: 51160027–02, 51160059–02].