It is well known that the solution of the Laplace equation in a non convex polygonal domain of $R^2$ has a singular behaviour near non convex corners. Consequently we investigate three refined finite volume methods (cell-center, conforming finite volume-element and non conforming finite volume-element) to approximate the solution of such a problem and restore optimal orders of convergence as for smooth solutions. Numerical tests are presented and confirm the theoretical rates of convergence.