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

Kashiwabara, Takahito; Oikawa, Issei; Zhou, Guanyu

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 869-891.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain $\Omega \subset {ℝ}^N(N=2,3)$. We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix–Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition $u·n_{\partial \Omega }=g$on $\partial \Omega$. Because the original domain $\Omega$ must be approximated by a polygonal (or polyhedral) domain ${\Omega }_h$ before applying the finite element method, we need to take into account the errors owing to the discrepancy $\Omega \ne {\Omega }_h$, that is, the issues of domain perturbation. In particular, the approximation of $n_{\partial \Omega }$ by $n_{\partial {\Omega }_h}$ makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator $H^1(\Omega {)}^N\to H^{1/2}\left(\partial \Omega \right)$; $u\mapsto u·n_{\partial \Omega }$. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates $O(h^{\alpha }+ϵ)$ and $O(h^{2\alpha }+ϵ)$ for the velocity in the $H^1$- and $L^2$-norms respectively, where $\alpha =1$ if $N=2\mathrm{and}\alpha =1/2\mathrm{if}N=3$. This improve the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016) 705–740] obtained for he conforming approximation in the sense that there appears no reciprocal of the penalty parameter $\in$ in the estimates.

[1] E. Bänsch and K. Deckelnick, Optimal error estimates for the Stokes and Navier–Stokes equations with slip-boundary condition. ESAIM: M2AN 33 (1999) 923–938. | DOI | Numdam | MR | Zbl

[2] H. Beirão Da Veiga, Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions. Adv. Differ. Equ. 9 (2004) 1079–1114. | MR | Zbl

[3] C. Bernardi, F. Hecht and O. Pironneau, Coupling Darcy and Stokes equations for porus media with cracks. ESAIM: M2AN 39 (2005) 7–35. | DOI | Numdam | MR | Zbl

[4] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications. Springer (2013). | DOI | MR | Zbl

[5] S.C. Brenner, Korn’s inequalities for piecewise H¹ vecor fields. Math. Comput. 73 (2003) 1067–1087. | DOI | MR | Zbl

[6] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edition. Springer (2007). | MR | Zbl

[7] S.C. Brenner, L.-Y. Sung and Y. Zhang, A Quadratic C⁰ Interior Penalty Method for an Elliptic Optimal Control Problem with State Constraints, edited by X. Feng, et al. In: Recent developments in discontinuous galerkin finite element methods for partial differential equations (2014) 97–132. | DOI | MR | Zbl

[8] E. Burman and P. Hansbo, Stabilized Crouzeix–Raviart element for the Darcy–Stokes problem. Numer. Methods. Partial Differ. Equ. 21 (2005) 986–997. | DOI | MR | Zbl

[9] A. Çağlar and A. Liakos, Weak imposition of boundary conditions for the Navier–Stokes equations by a penalty method. Int. J. Numer. Methods. Fluids 61 (2009) 411–431. | DOI | MR | Zbl

[10] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR | Zbl

[11] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO: Numer. Anal. 7 (1973) 33–76. | Numdam | MR | Zbl

[12] I. Dione and J. Urquiza, Penalty: finite element approximation of Stokes equations with slip boundary conditions. Numer. Math. 129 (2015) 587–610. | DOI | MR | Zbl

[13] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer (1998). | MR | Zbl

[14] M. Juntunen and R. Stenberg, Analysis of finite element methods for the Brinkman problem. Calcolo 47 (2010) 129–147. | DOI | MR | Zbl

[15] T. Kashiwabara and T. Kemmochi, L^∞- and W^1,∞-error estimates of linear finite element method for Neumann boundary value problem in a smooth domainPreprint arXiv:1804.00390 (2018) .

[16] T. Kashiwabara, I. Oikawa and G. Zhou, Penalty method with P1/P1 finite element approximation for the stokes equations under the slip boundary condition. Numer. Math. 134 (2016) 705–740. | DOI | MR | Zbl

[17] P. Knobloch, Variational crimes in a finite element discretization of 3D Stokes equations with nonstandard boundary conditions. East-West J. Numer. Math. 7 (1999) 133–158. | MR | Zbl

[18] W. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal. 40 (2003) 2195–2218. | DOI | MR | Zbl

[19] A. Logg, K.-A. Mardal, G.N. Wells, et al., Automated Solution of Differential Equations by the Finite Element Method. Springer (2012). | DOI | MR | Zbl

[20] R. Verfürth, Finite element approximation of incompressible Navier–Stokes equations with slip boundary condition. Numer. Math. 50 (1987) 697–721. | DOI | MR | Zbl

[21] G. Zhou, T. Kashiwabara and I. Oikawa, Penalty method for the stationary Navier–Stokes problems under the slip boundary condition. J. Sci. Comput. 68 (2016) 339–374. | DOI | MR | Zbl

[22] G. Zhou, T. Kashiwabara, I. Oikawa, A penalty method for the time-dependent Stokes problem with the slip boundary condition and its finite element approximation, Appl. Math. 62 (2017) 377–403. | DOI | MR | Zbl