A numerical algorithm to reduce ill-conditioning in meshless methods for the Helmholtz equation

Some meshless methods have been applied to the numerical solution of boundary value problems involving the Helmholtz equation. In this work, we focus on the method of fundamental solutions and the plane waves method. It is well known that these methods can be highly accurate assuming smoothness of the domains and the boundary data. However, the matrices involved are often ill-conditioned and the effect of this ill-conditioning may drastically reduce the accuracy. In this work, we propose a numerical algorithm to reduce the ill-conditioning in both methods. The idea is to perform a suitable change of basis. This allows to obtain new basis functions that span exactly the same space as the original meshless method, but are much better conditioned. In the case of circular domains, this technique allows to obtain errors close to machine precision, with condition numbers of order O(1), independently of the number of basis functions in the expansion.