Fast integral equation methods for the modified Helmholtz equation

Abstract

We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, $u(\x) - α^2 Δu(\x) = 0$, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) or $O(N\log N)$ operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of the methods on several numerical examples.