Grid computing accomplishes high throughput computing by using a very large number of unexploited computing resources. We present a parallel method GMRES to solve large sparse linear systems by the use of a lightweight GRID XtremWeb. This global computing platform, just as many popular GRID systems, is devoted to multi-parameters generic applications. We have implemented an important algorithm GMRES which is one of the key methods to resolve large, nonsymmetric, linear problems. We discuss the performances of this algorithm deployed on two XtremWeb networks: a local network with 128 non-dedicated PCs in Polytech-Lille of University of Lille I in France, a remote network with 3 clusters of SCGN Grid that includes 91 CPUs totally in the High Performance Center for Computer Science of University of Tsukuba in Japan. We compare these performances with those of a MPI implementation of GMRES on the same platform.We present the advantages and shortcomings of our implementation on this lightweight GRID XtremWeb.