Numerical instabilities of the integral approach to the interior boundary‐value problem for the two‐dimensional Helmholtz equation

Abstract

Abstract The integral equations arising from the Green's formula, applied to the two‐dimensional Helmholtz equation defined in a limited domain, are considered and the presence of instabilities in their numerical solution, when a real Green's function is adopted, is pointed out. A complete study has been carried out for a circular domain and the conditions under which such instabilities can occur in a domain of arbitrary geometry have been investigated. In particular, it has been shown that in every case the use of a complex Green's function is able to avoid their presence.