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

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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.

Classification : 65N30, 35Q30
Keywords: Nonconforming FEM, Stokes equations, slip boundary condition, domain perturbation, discrete H^1/2-norm

[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. | MR | Zbl | Numdam | DOI

[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. | MR | Zbl | Numdam | DOI

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

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

[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. | MR | Zbl | DOI

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

[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. | MR | Zbl | DOI

[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. | MR | Zbl | Numdam

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

[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. | MR | Zbl | DOI

[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. | MR | Zbl | DOI

[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. | Zbl | MR | DOI

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

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

[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. | MR | Zbl | DOI

[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. | MR | Zbl | DOI