The main goal of the current study is developing an advanced and robust numerical tool for accurate capturing heat front propagation. In some applications such as impermeable medium, Heat transfer in the surrounding domain of fracture acts just as a conduction process but the heat transfer through the fractures appears as a convection process. From a mathematical point of view, a parabolic partial differential equation (PDE) should be solved in the surrounding domain whereas a hyperbolic PDE should be solved in the domain of fractures. In fact, they have completely different treatments and this is one of the complicated problems in this area. In this paper, the presence of fractures and discontinuities are considered with the aim of eXtended Finite Element Method (X-FEM). In the proposed numerical approach, the domain is decomposed into local and global scales. Global and local domains are solved by the X-FEM and Least Square Method (LSM) techniques, respectively. As a final result, it is determined that the treatment of coupling term between two scales is one of the most important factors for system performance. Increasing its effect can significantly improve the efficiency of the whole system.

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