Implementation of exact non-reflecting boundary conditions in the finite element method for the time-dependent wave equation

When solving the wave equation in infinite regions using finite element methods, the domain must be truncated. We investigate the accuracy of time-dependent non-reflecting boundary conditions (NRBC) derived in Grote, Keller (1995), when implemented in the finite element method. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary. High-order temporal derivatives are formulated as a system of first-order ordinary differential equations. Several versions of implicit and explicit multi-step, time-integration schemes are presented for solution of the finite element equations concurrently with the first-order system appearing in the NRBC. An alternative scaling of the boundary variables is introduced which leads to a well-conditioned coefficient matrix. Although the boundary conditions are global over the boundary, when implemented in the finite element method, they only require inner products of spherical harmonics within the force vector, and as a result, they are easy to implement and do not disturb the banded/sparse structure of the matrix equations. Several numerical examples are presented which demonstrate the improvement in accuracy over standard finite element methods.