On the comparison of the boundary integral equations method and the method of fundamental solutions for a 3D boundary value problem for the Helmholtz equation

This article presents a comparative study of the Boundary Integral Equations Method (BIEM) and the Method of Fundamental Solutions (MFS) for the numerical solution of the interior Dirichlet problem for the 3D Helmholtz equation with complex wave number. The BIEM approach is based on the representation of the solution via a double-layer potential and involves solving a Fredholm integral equation of the second kind using quadrature rules on surfaces diffeomorphic to the sphere. The MFS approximates the solution using a linear combination of fundamental solutions with source points placed outside the domain and determines the unknown coefficients by collocation on the boundary. Numerical experiments are conducted on geometries with analytic parametrizations, using exact solutions for error validation. The results demonstrate that while MFS often achieves higher accuracy, it suffers from severe ill-conditioning, unlike the more stable BIEM system. The study highlights the trade-offs in accuracy, stability, and computational complexity inherent to both methods.