Barnett–Betcke MFS convergence-rate conjecture for Helmholtz problems

Let $\Omega$ be an exterior sound-soft Helmholtz domain with boundary $\Gamma=\partial\Omega$, and let $u$ be the scattered-field solution represented by the method of fundamental solutions (MFS), with $J$ MFS points equally distributed in conformal angle at conformal distance $R$ inside $\Gamma$. Let $\varepsilon$ be the maximum MFS error measured on $\Gamma$. Let $\rho<1$ be the conformal radius of the closest singularity of the analytic continuation of $u$, in the sense of conformal radius. Barnett–Betcke MFS convergence-rate conjecture. Then

$$\varepsilon\leq\begin{cases} K\rho^{-J/2},&\rho>R^2,\\ KR^{-J},&\rho<R^2, \end{cases}$$

where $K$ may depend on $\Gamma$, $k$, $R$, and $v$, but not on $J$. This conjecture gives the theoretical basis for the MFS point-placement strategy in the AAALS algorithm; the supplied source does not establish whether the conjecture has been resolved.