Finite element formulation of exact non-reflecting boundary conditions for the time-dependent wave equation

A modi ed version of an exact Non-re ecting Boundary Condition (NRBC) rst derived by Grote and Keller is implemented in a nite element formulation for the scalar wave equation. The NRBC annihilate the rst N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second- order local boundary condition derived by Bayliss and Turkel. Two alternative nite element formulations are given. In the rst, the boundary operator is implemented directly as a `naturalʹ boundary condition in the weak form of the initial{boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time- integration schemes are presented for solution of the nite element semidiscrete equations concurrently with the rst-order di erential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the nite element method. The results demonstrate that the nite element formulation of the (modi ed) NRBC is remarkably robust, and highly accurate.