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.

Published in Computers & Mathematics with Applications

Cited by in corpus (9)

• An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion
• Hybrid asymptotic-numerical approach for estimating first passage time densities of the two-dimensional narrow capture problem
• Partition of Unity Extension of Functions on Complex Domains
• Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution
• A Kernel-Based Explicit Unconditionally Stable Scheme for Hamilton-Jacobi Equations on Nonuniform Meshes
• Vortices and backflow in hydrodynamic heat transport
• An integral equation method for the advection-diffusion equation on time-dependent domains in the plane
• Spectrally accurate Ewald summation for the Yukawa potential in two dimensions
• Integral equation methods for the Yukawa-Beltrami equation on the sphere