Analysis and Numerical Methods for Some Crack Problems

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.