The authors prove the convergence of the ghost fluid finite difference method for a second-order elliptic equation with discontinuous coefficients and given interfacial jumps. The ghost fluid method (GFM) is simple, efficient and robust. One of the novelties, and advantages, of the method is the arm-by-arm splitting technique, which makes the method in many dimensions as simple as in one dimension. Another advantage is that the resulting linear system of the method is the same as the linear system obtained from the simplest standard five point stencil finite difference method for the equation without discontinuous coefficients or given interfacial jumps. Therefore the resulting linear system is symmetric and positive definite, and can be efficiently solved. This finite difference method captures the sharp solution profile at the interfaces without smearing.
The convergence proof starts with the formulation of the problem, in terms of a uniformly elliptic bilinear form. Existence and uniqueness of a weak solution \(v\) follow immediately by elementary functional analysis. The solution space is the same as \(H_0^1 \), except for a different, but equivalent, inner product induced by the bilinear form. The source term includes in effect a delta distribution along the interface to account for the jump in the normal derivative.
The solution of the interface problem is then recovered by the addition of a background function to correct for the jump in the solution across the interface as well as the boundary conditions. Discretizing this weak formulation in a natural way results in a discrete weak problem, which is equivalent to the finite difference method by X.-D. Liu, R. Fedkiw and M. Kang [J. Comput. Phys. 160, 151-178 (2000; Zbl 0958.65105)]. As expected, the discrete weak formulation inherits the structural conditions (uniform boundedness, extension, consistency) in the discrete sense; hence existence, uniqueness, and uniform boundedness of the family \(v^h \) of discrete solutions hold. An abstract framework for the convergence proof is then provided.
In addition to the uniform structural conditions for the weak problem and its discretizations, the existence of a uniformly bounded family of extension operators \(T^h \) from the discrete spaces to the solution space, which satisfy a strong approximation property is postulated. Finally, the weak consistency of the discrete problem with the original problem is imposed. Under these assumptions, the sequence of extended weak solutions \(T^h \left( {v^h } \right)\) converges weakly to the solution v in \(H_0^1 \).
There is a similarity between the standard convergence proof of finite element methods and suggested approach. Both of them use structural conditions. For finite element methods, the structural conditions and weak consistency are inherited directly from the weak problem for the partial differential equation, because the discrete bilinear forms are obtained by restriction to finite dimensional subspaces. Cea’s lemma then says that the extended discrete solution is the closest function in the finite dimensional subspace to the true solution. This reduces convergence and error estimation to a problem in approximation theory.
For finite difference methods, further approximations are made so that the discrete problem cannot be obtained by restriction to a finite dimensional subspace. In the case, the difference between the finite difference scheme and the finite element method is that point values of the coefficients are used rather than cell averages. Because the coefficients in considered problem are discontinuous, the solution lies only in \(H_0^1 \) and not \(H^2 \cap H_0^1 \), and therefore, the authors obtain convergence, but without a rate.