The fictitious domain method with L²‐penalty for the Stokes problem with the Dirichlet boundary condition

Abstract

We consider the fictitious domain method with L2‐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 H1‐norm for the velocity and in L2‐norm for the pressure, where ϵ is the penalty parameter. The L2‐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 H1‐norm for the velocity and in L2‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L2‐norm, the convergence rate is improved to O ( h + ϵ 1 2 ) . The theoretical results are verified by the numerical experiments.