Finite Element Formulation of Exact Dirichlet-to-Neumann Radiation Conditions on Elliptic and Spheroidal Boundaries

Abstract

Exact Dirichlet-to-Neumann (DtN) radiation boundary conditions are derived in elliptic and spheroidal coordinates and formulated in a finite element method for the Helmholtz equation in unbounded domains. The DtN map matches the first N wave harmonics exactly at the artificial boundary. The use of elliptic and spheroidal boundaries enables the efficient solution of scattering from elongated objects in two- and three-dimensions respectively. Modified DtN conditions based on first and second order local boundary operators are also derived in elliptic and spheroidal coordinates, in a form suitable for finite element implementation. The modified DtN conditions are more accurate than the DtN boundary condition, yet require no extra memory and little extra cost. Direct implementation involves non-local spatial integrals leading to a dense, fully populated submatrix. A matrix-free interpretation of the non-local DtN map for elliptic and spheroidal boundaries, suitable for iterative solution of the resulting complex-symmetric system is described. For both the DtN and modified DtN conditions, we describe efficient and effective SSOR preconditioners with Eisenstat’s trick based on the matrix partition associated with the interior mesh and local boundary operator. Numerical examples of scattering from elliptic and spheroidal boundaries are computed to demonstrate the efficiency and accuracy of the boundary treatments for elongated structures.