Second Order Semi-Discretization for Unsteady Stokes Flow

Abstract

We consider a second order implicit time stepping procedure for the unsteady Stokes equations in bounded domains of R3. Using energy estimates we prove optimal convergence properties in a series of Sobolev spaces Hm(G) (m = 0, 1, 2) uniformely in time, provided that the Stokes solution has a high degree of regularity uniformly in time. Here in the case of our second order method the solution of the Stokes equations has to satisfy a certain non-local compatibility condition at the parabolic boundary being virtually uncheckable for given data, which can be satisfied by a special initial construction.