Boundary conditions for the Darwin model of approximation to Maxwell's equations

The Darwin model is commonly used as an approximation of Maxwell equations in electron beam simulations. It relies on the decomposition of the electric field into its transverse and longitudinal components. Such a decomposition is not unique, as long as the boundary conditions for each component are not prescribed. However, all the physically natural boundary conditions (such as the perfectly conducting wall, or the incoming wave conditions) are concerned with the total electric field, and no natural condition can be found for each component separately. Thus, one is faced with the problem of finding artificial boundary conditions which are as convenient as possible. The authors propose a different approach, leading to a different set of boundary conditions. Their approach first guarantee the uniqueness of the decomposition and second, are compatible with the perfectly conducting wall boundary condition for the overall electric field