When using the equations in their original form with the divergence condition as part of the system, there is a necessary restriction on the boundary data. We formulate the boundary conditions in a new way, such that this problem is eliminated, and the boundary value problem becomes non-singular and well conditioned. By using a difference approximation on a staggered grid, we are able to derive a non-singular approximation in a direct way, and such that we get the same type of estimates as for the continuous case.
Numerical experiments confirm the theoretical results.
We will present a brief survey of this work as well.