On the implementation of the Dirichlet-to-Neumann radiation condition for iterative solution of the Helmholtz equation

Abstract

The Helmholtz equation posed on an unbounded domain with the Sommerfeld condition prescribed at infinity is considered. The unbounded domain is eliminated by imposing a Dirichlet-to-Neumann (DtN) map or a modified DtN map on a truncating surface and the resulting bounded domain problem is modeled using the finite element method. The resulting system of linear equations is then solved using a Krylov subspace iterative method. New, efficient algorithms to compute matrix-vector products that are based on the structure of the DtN and the modified DtN map are presented. Connections between the DtN map and the discrete Fourier transform in two dimensions and discrete spherical transform in three dimensions are established, and are utilized to develop fast implementations of matrix-vector product algorithms. Also, an SSOR-type preconditioner that is based on a local radiation condition is considered for the modified DtN formulation. An efficient implementation is proposed by extending Eisenstat's trick for the standard SSOR preconditioner. Finally, numerical examples which illustrate the efficacy of the proposed algorithms are presented.