Fast integral equation methods for the modified Helmholtz equation

TL;DR

This paper introduces fast, high-order integral equation methods for efficiently solving the two-dimensional modified Helmholtz equation in complex domains, utilizing well-conditioned formulations and fast multipole techniques.

Contribution

It develops well-conditioned second-kind integral equations and a fast multipole-based iterative solver for the modified Helmholtz equation in complex geometries.

Findings

The methods achieve O(N) or O(N log N) computational complexity.

Numerical examples demonstrate high accuracy and efficiency.

The approach handles both Dirichlet and Neumann boundary conditions.

Abstract

We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, $u(\x) - \alpha^2 \Delta u(\x) = 0$, 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.