In this paper, finite difference schemes based on asymptotic analysis and the augmented immersed interface method are proposed for potential problems with an inclusion whose
characteristic width is much smaller than the characteristic length in one and two dimensions.
We call such a problem as a crack problem for simplicity. In the proposed methods, we use asymptotic
analysis to approximate the problem with a single sharp interface. The jump conditions
for the interface problem are derived. For one-dimensional problem, or two-dimensional problems
in which the center line of the crack is parallel to one of axis, we can simply modify the finite
difference scheme with added correction terms at irregular grid points. The coefficient matrix
of the finite difference equations is still an M-matrix. For problems with a general thin crack,
an augmented variable along the center line of the crack is introduced so that we can apply the
immersed interface method to get the discretization. The augmented equation is the asymptotic
jump condition. Numerical experiments including the case with large jump discontinuity in the
coefficient are presented.