Abstract : The development of numerical methods to simulate the Navier-Stokes' equations with rigid boundaries is a very active topic. While literature abounds with convergence and stability results, fractional step methods are known to introduce numerical boundary layers errors in the pressure. This problem already presents at conforming boundaries and becomes particularly critical near boundaries with non-conforming mesh, a situation which occurs when we want to be independent from any form of remeshing with moving objects, for example. The purpose of this thesis is to solve the incompressible Navier-Stokes equations with accurate velocity and pressure fields by a finite difference method. This question led us to reformulate the non-stationary Stokes problem in two dimensions using the streamfunction psyand a function phi conjugated to pressure. We have implemented a first method for solving the Stokes problem in this representation. We have also developed an original scheme to solve the Poisson Neumann problem in a irregular domain, and proposed a second method in order to solve the incompressible Stokes Problem in a more conventional formulation pressure-velocity-vorticity. Both types of algorithms gives us a convergence of order 2 in space for all fields of the problem.