In a generic multi-scale computational homogenization (CH) procedure, the crucial point is the definition and the solution of the Unit Cell (UC) Boundary Value Problem (BVP). The main aspects to be chosen for the formulation of the UC BVP are: (i) geometry; (ii) boundary conditions (BCs); (iii) material models; (iv) numerical approximation techniques. All these components play a key-role in the efficiency of the multi-scale procedure. In the present study, the UC BVP is formulated for running bond masonry according to a displacement based variational formulation, where the material of the blocks is considered indefinitely elastic and the mortar joints are simulated by zero-thickness elasto-plastic interfaces. The choice of adopting an elasto-plastic response of mortar represents a good compromise between ease of applicability and effective representation of the decohesion process occurring at the joint level. Linear BCs are used to apply the macroscopic strain tensor to the UC. The numerical discretization is original with respect to the more common FE mesoscopic discretization, it is in fact formulated in the framework of meshless methods. It will be showed that the meshless discretization allows to obtain a considerable computational gain with respect to a standard FE discretization. Numerical simulations focus on the FEM-Meshless comparison for the pure modes of failu. © 2017 Centro Servizi d'Ateneo S.r.l. All rights reserved.