Summary: The Stokes problem is approximated by a mixed finite element method using a new finite element, which has properties analogous to the finite volume methods, namely, the local conservation of the momentum and the mass. Estimates of optimal order are derived for the errors in the velocity, the pressure, and the gradient of the velocity. This new finite element also works for the elasticity problem, and all estimates are valid uniformly with respect to the compressibility. Finally, some numerical results for the incompressible Navier-Stokes equations are presented.