A PENALTY METHOD FOR THE TIME-DEPENDENT STOKES PROBLEM WITH THE SLIP BOUNDARY CONDITION AND ITS FINITE ELEMENT APPROXIMATION (Numerical Analysis : New Developments for Elucidating Interdisciplinary Problems II)

We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is applied, which also facilitates the numerical implementation. For the continuous problems, the convergence of the penalty method is investigated. Then, we consider the P1/P1-stabilization or P1b/P1 finite element approximations with penalty and time-discretization. For the penalty term, we propose the reduced and non-reduced integration schemes, and obtain the error estimate for velocity and pressure. The theoretical results are verified by numerical experiments