  In this paper, a weak formulation of the discontinuous variable coefficient
Poisson equation with interfacial jumps is studied. The existence, uniqueness
and regularity of solutions of this problem are obtained. It is shown that the
application of the Ghost Fluid Method by Fedkiw, Kang, and Liu to this problem
can be obtained in a natural way through discretization of the weak
formulation. An abstract framework is given for proving the convergence of
finite difference methods derived from a weak problem, and as a consequence,
the Ghost Fluid Method is proven to be convergent.
