The fictitious domain method with H1-penalty for the Stokes problem with Dirichlet boundary condition

We consider the fictitious domain method with H1-penalty for the Stokes problem with Dirichlet boundary condition. First, for the continuous penalty problem, we obtain the optimal error estimate O(ϵ) for both the velocity and pressure, where ϵ is the penalty parameter. Moreover, we investigate the Hm-regularity for the solution of the penalty problem. Then, we apply the finite element method with the P1/P1 element to the penalty problem. Since the solution to the penalty problem has a jump in the traction vector, we introduce some interpolation/projection operators, as well as an inf-sup condition with the norm depending on ϵ. With the help of these preliminaries, we derive the error estimates for the finite element approximation. The theoretical results are verified by the numerical experiments.