An iterative finite element algorithm is proposed for numerically solving two-phase generalized Stefan interface problems with discontinuous solutions, co-normal derivatives, and coefficients. This algorithm iteratively solves small subregion problems for each single phase with good accuracy and exchange information at the interface to advance the iteration until convergence, following the idea of Schwarz Alternating Methods. The finite element grids on different phases do not have to match each other at the interface. Numerical experiments are performed to show the accuracy of the algorithm for capturing discontinuities in the solutions and coefficients.