Investigation of adaptive absorbing boundary condition for finite element solution of three-dimensional scattering

Abstract

When solving open-region scattering problems using the finite element method, the infinite region exterior to the scatterer must be truncated with an artificial boundary to limit the number of unknowns. Consequently, a boundary condition must be introduced at this artificial boundary for a unique finite element solution. Two classes of such boundary conditions have been developed and employed extensively in the finite element solution of scattering problems. The first class of boundary conditions is derived from the boundary integral equations involving Green's functions. The second class of boundary conditions is derived from the differential wave equations. These boundary conditions relate the field at one point of the boundary only to those at its neighboring points and, thus, are called local boundary conditions. As a result, the corresponding numerical system is a sparse banded matrix, which is similar to the FEM matrix. Such matrices can be stored and solved efficiently. However, these boundary conditions do not lead to exact solutions because they do not possess zero reflection for all incidence angles, and, to minimize the solution error, the artificial boundary must be placed at some distance away from the scatterer, resulting in a large discretization domain. Recently, Li and Cendes (1994) proposed a method to improve the accuracy of the finite element solution of scattering using ABCs. In that method, an adaptive scheme is introduced to update the ABCs using the boundary integral equation in the process of the finite element solution. Li and Cendes applied this method to 2D scattering problems. In this paper, we apply this method to more complicated 3D scattering problems and investigate its advantages, shortcomings, and possible solutions.