  A new algorithm to determine the position of the crack (discontinuity set) of
certain minimizers of Mumford-Shah functional in situations when a crack-tip
occurs is introduced. The conformal mapping $w=\sqrt{z}$ in the complex plane
is used to transform the free discontinuity problem to a new type of free
boundary problem, where the symmetry of the free boundary is an additional
constraint of a non-local nature. Instead of traditional Jacobi or Newton
iterative methods, we propose a simple iteration method which does not need the
Jacobian but is way fast than the Jacobi iteration. In each iteration, a
Laplace equation needs to be solved on an irregular domain with a Dirichlet
boundary condition on the fixed part of the boundary; and a Neumann type
boundary condition along the free boundary. The augmented immersed interface
method is employed to solve the potential problem. The numerical results agree
with the analytic analysis and provide insight into some open questions in free
discontinuity problems.
