A Jump Condition Capturing Finite Difference Scheme for Elliptic Interface Problems

We propose a simple finite difference scheme for the elliptic interface problem with a discontinuous diffusion coefficient using a body-fitted curvilinear coordinate system. The resulting matrix is symmetric and positive definite. Standard techniques of acceleration such as PCG and multigrid can be used to invert the matrix. The main advantage of the scheme is its simplicity: the entries of the matrix are simply the centered difference second order approximation of the metric tensor $g^{\alpha\beta}$. In addition, the interface jump conditions are naturally built into the finite difference discretization. No interpolation/extrapolation process is involved in the derivation of the scheme. Both the solution and the flux are observed to have second order accuracy.