Abstract: A semi-discrete method is proposed for finding the numerical solution of the boundary value problem(BVP) of Navier's equations on the polygon with a single edge-crack. After a suitable transformation of the coordinates, the BVP is reduced to a discontinuous coefficients problem on a semi-infinite strip, and the semi-discrete approximation of the problem is obtained, equivalent to a BVP of a system of O.D.E's with constant coefficients. Furthermore, the semi-discrete approximation of the BVP can be acquired by a direct method. It is worthwhile to note that, the semi-discrete approximation in the form of separable variables naturally possesses the singularity of the given problem. Finally numerical examples show the effectiveness of the method to calculate the approximation of the stress intensity factors.