Fast integral equation methods for the modified Helmholtz equation
Provides efficient numerical solvers for a specific PDE in computational science and engineering, but the approach is incremental (combining known techniques).
The paper presents fast integral equation methods for solving the modified Helmholtz equation in 2D, achieving O(N) or O(N log N) complexity via fast multipole methods and high-order discretization.
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.