This project deals with the numerical approximation of the Schwarz alternat ing method for a linear elliptic boundary value problem in the context of two overlapping subdomains with non-matching grids. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independent triangulation. Three types of non-matching grids discretizations are presented and studied, and error estimates in the maximum norm, on each sub domain, are established, by two different approaches: the maximum principle and the Banach's fixed point principle. Moreover, numerical experiments are conducted to support the theory.