A numerical analysis method for melting/solidification phenomena has been developed to evaluate feasibility of the several candidate techniques in the nuclear fuel cycle. Our method is based on the extended finite element method, which has been used for moving boundary problems. The basic idea of the extended finite element method is to incorporate the signed distance function into the standard finite element interpolation to represent a discontinuous gradient of the temperature at a moving solid-liquid interface. This technique makes it possible to simulate movement of the solid-liquid interface without the use of a moving mesh. Construction of the finite element equation from the energy equation in the case of melting/solidification problems has been discussed and is reported here. The technique of quadrature and the method to solve the governing equations for the problem involving liquid flows have also been constructed in the present work. The numerical solutions of the basic problems - a one-dimensional Stefan problem, solidification in a two-dimensional square corner, and melting of pure gallium - were compared to the exact solutions or to the experimental data. Through these verifications, validity of the newly developed numerical analysis method has been demonstrated.