Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition

Abstract

We consider the penalty method for the stationary Navier-Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate O( ) in H k -norm, where is the penalty parameter. We are concerned with the finite element approximation with the P1b/P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate O(h + √ + h/ √ ) for the non-reduced-integration scheme with d = 2, 3, and the reduced-integration scheme with d = 3, where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with d = 2, we prove the convergence order O(h + √ + h 2 / √ ). The theoretical results are verified by numerical experiments.

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

“…, we obtain (with a similar argument to [23], Proof of Lemma 4.1], [24], Appendix] or [25], §2.2.2]):…”

Penalty method with Crouzeix–Raviart approximation for the Stokes equations under slip boundary condition

“…In particular, the optimal rate of convergence O(h) was achieved by choosing ǫ = O(h 2 ) in the two-dimensional case. This strategy was then extended to the stationary Navier-Stokes equations in [20] and to the non-stationary Stokes equations in [21].…”

A penalty method for the time-dependent Stokes problem with the slip boundary condition and its finite element approximation

“…To prove the optimal error estimate, the inf-sup conditions of pressure and Lagrange multiplier have been used (cf. [4], [6], [28]). However, these arguments are not applicable to the non-stationary problem.…”

“…Now we turn our attention to the finite element approximation for the penalty problem. For the stationary Stokes/Navier-Stokes problem with the slip boundary condition, the FEM without penalty has been studied by Verfürth [25], [26], [27], Knobloch [14] and Bäncsh and Deckelnick [1], and the case of the penalty method has been investigated by Dione and Urquiza [6] and [12], [28]. The error estimates of all the above works become sub-optimal if the difference between n and n h is carefully taken into account (see Introduction of [12] for a comprehensive description of these works).…”

“…The error estimates of all the above works become sub-optimal if the difference between n and n h is carefully taken into account (see Introduction of [12] for a comprehensive description of these works). We mention that the error can be upgraded to optimal in the two-dimensional case by introducing a reduced integration for the penalty term (see [12], [28]).…”