In this work, we present a new computational method based on a nonconforming domain decomposition technique for modeling of phase transitions. Phase transitions are the result of thermal or mechanical loading in ferromagnetic materials or shape memory materials. Modeling of phase transitions is important because it can help to predict or control the behavior of these materials. This thesis will focus on phase transitions characterized by two directions of magnetization in the case of ferromagnets and two variants of Martensite in the case of shape memory materials. In both types of materials, branching occurs near an internal surface which is characterized by complex microstructures. These microstructures occur at a minimum energy state. The new computational method simulates the branching behavior of these microstructures near an internal surface. We approximate the microstructures via energy minimization. We minimize the total stored energy stored in vicinity of internal surface with the minimizing function representing the microstructures. We compare the numerical results obtained by the new technique with those obtained by a more standard technique, one not incorporating nonconforming domain decomposition. Furthermore, we verify the various energy scaling laws used to predict the total stored energy near an internal surface. Among these laws, we verify the local-in-y scaling property which has been conjectured but not proven.