The fictitious domain method with L 2 ‐penalty for the Stokes problem with the Dirichlet boundary condition

We consider the fictitious domain method with L 2 ‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order O ( ϵ 1 4) in H 1 ‐norm for the velocity and in L 2 ‐norm for the pressure, where ϵ is the penalty parameter. The L 2 ‐norm error estimate for the velocity is upgraded to O ( ϵ ) . Moreover, we derive the a priori estimates depending on ϵ for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate O ( h + ϵ 1 4) in H 1 ‐norm for the velocity and in L 2 ‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L 2 ‐norm, the convergence rate is improved to O ( h + ϵ 1 2) . The theoretical results are verified by the numerical experiments.