Additive manufacturing (AM) is a new production method based on material addition. AM brings new possibilities in geometries and shapes, as well as reduction on material and energy wastage, that place it in a position towards sustainable industry. AM is based on a heat source melting metal in order to form the desired geometry. Direct energy deposition (DED) methods are a type of AM procedures, based on continuous addition of material as the heat source melts it. DED methods do not need a closed environment to work so the size of products is only limited by the range of the robotic tools. Numerical simulations are required by the AM industry in order to reduced failures in fabrications. Both, previous and in real time, simulations can save time and money in the design and fabrication of a piece by AM processes. The main challenge that AM DED processes pose is the computational cost, due to the intrinsic multi-scale of the processes and the continuously growing physical domain. The state of the art numerical procedures to deal with the material addition is the use of the quiet element method or the inactive element method. Both methods use a unique mesh for the piece and activate elements as the heat source moves above them. The main disadvantage of these methods is the mesh size. As the critical micro scale behaviors go over the whole domain, the mesh must be fine enough to capture them. Away from the heat source, there is no need for a fine mesh so it could be coarser, what will lead to remeshing and its subsequent problems. To reduce the computational costs, an Arlequin based method is presented. To deal with the multi-scale of the problem, the Arlequin method uses two distinct meshes: A coarse mesh of the whole domain and a fine mesh that moves along with the heating source to capture the high thermal gradients near the melt pool. In addition to the Arlequin method, a change of variable is introduced to transform the moving fine mesh in a fixed mesh so the calculations on each time step are simplified. The Arlequin method uses two solutions, one for each mesh, and couples them over a subdomain of the intersection between meshes. To achieve this coupling, a mesh intersection method is used. Different coupling strategies can be chosen and a comparison between them is analyzed. As representative example, a 2D thermal model of a wall was chosen to validate the results.