Convergence of the ghost fluid method for elliptic equations with interfaces

Abstract

This paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps. A weak formulation of the problem is first presented, which then yields the ex- istence and uniqueness of a solution to the problem by classical methods. 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 con- vergence of finite difference methods derived from a weak problem, and as a consequence, the ghost fluid method is proved to be convergent.