A vector absorbing boundary condition for vector potential satisfying the Lorentz gauge

A vector absorbing boundary condition is derived for a vector potential which satisfies the Lorentz gauge, based on a form of the Representation theorem for field quantities having non-zero divergence. The potential is assumed to be the linear superposition of those of arbitrarily arranged Hertzian dipoles, which potentials are individually particular solutions of the Helmholtz equation. Compared with ABCs derived for fields a penalty is introduced by the non-zero divergence and for a first-order ABC the error is shown to be 0(r^-2). A second-order condition is derived, and using a Galerkin formulation both conditions can yield symmetric matrices