Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator

In this paper, we describe a new class of fast solvers for separable elliptic partial differential equations in cylindrical coordinates $(r,θ,z)$ with free-space radiation conditions. By combining integral equation methods in the radial variable $r$ with Fourier methods in $θ$ and $z$, we show that high-order accuracy can be achieved in both the governing potential and its derivatives. A weak singularity arises in the Fourier transform with respect to $z$ that is handled with special purpose quadratures. We show how these solvers can be applied to the evaluation of the Coulomb collision operator in kinetic models of ionized gases.