Time stepping procedures for the non-stationary Stokes equations

We consider first and second-order implicit time stepping procedures for the non-stationary Stokes equations in bounded domains of ?³. Using energy estimates we prove the optimal convergence properties in the Sobolev spaces H^m(G)(m = 0, 1, 2) uniformly in time, provided that the Stokes solution has a certain degree of regularity. Here in the case of the second-order scheme (method of Crank–Nicholson) the Stokes solution has to satisfy a non-local compatibility condition at the initial time t = O, which can be satisfied by a special initial construction.