A stabilized multi-level method for non-singular finite volume solutions of the stationary 3D Navier–Stokes equations Authors Jian Li Department of Mathematics Baoji University of Arts and Sciences Zhangxin Chen Department of Chemical and Petroleum Engineering, Schulich School of Engineering University of Calgary Center for Computational Geosciences, College of Mathematics and Statistics Xian Jiaotong University Yinnian He Center for Computational Geosciences, College of Mathematics and Statistics Xian Jiaotong University Article

First Online: 14 March 2012 Received: 23 December 2010 Revised: 10 October 2011 DOI :
10.1007/s00211-012-0462-z

Cite this article as: Li, J., Chen, Z. & He, Y. Numer. Math. (2012) 122: 279. doi:10.1007/s00211-012-0462-z
Abstract This paper proposes and analyzes a stabilized multi-level finite volume method (FVM) for solving the stationary 3D Navier–Stokes equations by using the lowest equal-order finite element pair without relying on any solution uniqueness condition. This multi-level stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performing one Newton correction step on each subsequent mesh, thus only solving a large linear system. An optimal convergence rate for the finite volume approximations of nonsingular solutions is first obtained with the same order as that for the usual finite element solution by using a relationship between the stabilized FVM and a stabilized finite element method. Then the multi-level finite volume approximate solution is shown to have a convergence rate of the same order as that of the stabilized finite volume solution of the stationary Navier–Stokes equations on a fine mesh with an appropriate choice of the mesh size: \({ h_{j} ~ h_{j-1}^{2}, j = 1,\ldots, J}\) . Finally, numerical results presented validate our theoretical findings.

Mathematics Subject Classification 76D05 65M08 65M12 Supported in part by NCET-11-1041, the NSF of China (No. 11071193) and (No. 10971166), Natural Science New Star of Science and Technologies Research Plan in Shaanxi Province of China (No. 2011kjxx12), Research Program of Education Department of Shaanxi Province (No. 11JK0490), the project-sponsored by SRF for ROCS, SEM, the National Basic Research Program (No. 2005CB321703), and NSERC/AERI/Foundation CMG Chair and iCORE Chair Funds in Reservoir Simulation.

References 1.

Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

MATH 2.

Bank R.E., Rose D.J.: Some error estimates for the box method. SIAM J. Numer. Anal.

24 , 777–787 (1987)

MathSciNet MATH CrossRef 3.

Bochev P., Dohrmann C.R., Gunzburger M.D.: Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J. Numer. Anal.

44 , 82–101 (2006)

MathSciNet MATH CrossRef 4.

Brezzi F., Douglas J.: Stabilized mixed methods for the Stokes problem. Numer. Math.

53 , 225–235 (1988)

MathSciNet MATH CrossRef 5.

Brezzi, F., Rappaz, J., Raviart, P.-A.: Finite element approximation of nonlinear problems. Part I: branch of nonsingular solutions. Numer. Math. 36, 1–25 (1980)

6.

Cai Z.: On the finite volume method. Numer. Math.

58 , 713–735 (1991)

MathSciNet MATH CrossRef 7.

Carstensen C., Lazarov R., Tomov S.: Explcit and averaging a posteriori error estimates for adaptive finite volume methods. SIAM J. Numer. Anal.

42 , 2496–2521 (2005)

MathSciNet MATH CrossRef 8.

Chen Z.: Finite Element Methods and Their Applications. Springer, Heidelberg (2005)

MATH 9.

Chen Z., Li R., Zhou A.: A note on the optimal

L
^{2} -estimate of finite volume element method. Adv. Comput. Math.

16 , 291–303 (2002)

MathSciNet MATH CrossRef 10.

Chou S.H.: Analysis and convergence of a covolume method for the generalized Stokes problem. Math. Comput.

66 , 85–104 (1997)

MATH CrossRef 11.

Chou S.H., Li Q.: Error estimates in

L
^{2} ,

H
^{1} and

L
^{i} in co-volume methods for elliptic and parabolic problems: a unified approach. Math. Comput.

69 , 103–120 (2000)

MathSciNet MATH 12.

Chatzipantelidis P.: A finite volume method based on the Crouzeix–Raviart element for elliptic PDEs in two dimensions. Numer. Math.

82 , 409–432 (1999)

MathSciNet MATH CrossRef 13.

Chatzipantelidis P., Lazarov R.D., Thomée V.: Error estimates for a finite volume method for parabolic equations in convex polygonal domains. Numer. Methods PDEs

20 , 650–674 (2004)

MATH CrossRef 14.

Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

MATH 15.

Ewing R.E., Lazarov R.D., Lin Y.: Finite volume element approximations of nonlocal reactive flows in porous media. Numer. Methods Partial Differ. Equ.

16 , 285–311 (2000)

MathSciNet MATH CrossRef 16.

Ewing R.E., Lin T., Lin Y.: On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal.

39 , 1865–1888 (2002)

MathSciNet MATH CrossRef 17.

Girault V., Raviart P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)

MATH CrossRef 18.

He Y.N., Li J.: A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equation. Appl. Numer. Math.

58 , 1503–1514 (2008)

MathSciNet MATH CrossRef 19.

He Y.N.: Two-level method based on finite element and Crank–Nicolson extrapolation for the time-dependent Navier–Stokes equations. SIAM J. Numer. Anal.

41 , 1263–1285 (2003)

MathSciNet MATH CrossRef 20.

He Y.N., Liu K., Sun W.: Multi-level spectral galerkin method for the Navier–Stokes problem I: spatial discretization. Numer. Math.

111 , 501–522 (2005)

MathSciNet CrossRef 21.

Heywood J.G., Rannacher R.: Finite-element approximations of the nonstationary Navier–Stokes problem. Part I: Regularity of solutions and second-order spatial discretization. SIAM J. Numer. Anal.

19 , 275–311 (1982)

MathSciNet MATH CrossRef 22.

Layton W.: A two level discretization method for the Navier–Stokes equations. Comput. Math. Appl.

26 , 33–38 (1993)

MathSciNet MATH CrossRef 23.

Layton W., Lenferink W.: A multilevel mesh independence principle for the Navier–Stokes equations. SIAM J. Numer. Anal.

33 , 17–30 (1996)

MathSciNet MATH CrossRef 24.

Layton W., Lee H.K., Peterson J.: Numerical solution of the stationary Navier–Stokes equations using a multilevel finite element method. SIAM J. Sci. Comput.

20 , 1–12 (1998)

MathSciNet CrossRef 25.

Li J., Chen Z.: A New Stabilized Finite Volume Method for the Stationary Stokes Equations. Adv. Comput. Math.

30 , 141–152 (2009)

MathSciNet MATH CrossRef 26.

Li, J., Chen, Z.: Stability and convergence of a stabilized finite volume method for the transient Navier–Stokes equations. Adv. Comp. Math. (2012, in press)

27.

Li, J., Chen, Z.: Optimal and maximum-norm analysis of a finite volume method for the stationary Navier–Stokes equations with large data (2012, in press)

28.

Li J., He Y.N.: A stabilized finite element method based on two local Gauss integrations for the stokes equations. J. Comput. Appl. Math.

214 , 58–65 (2008)

MathSciNet MATH CrossRef 29.

Li R.: Generalized difference methods for a nonlinear Dirichlet problem. SIAM J. Numer. Anal.

24 , 77–88 (1987)

MathSciNet MATH CrossRef 30.

Li R., Chen Z., Wu W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000)

MATH 31.

Li J., He Y.N., Xu H.: A multi-level stabilized finite element method for the stationary Navier–Stoke equations. Comput. Methods Appl. Mech. Eng.

196 , 2852–2862 (2007)

MathSciNet MATH CrossRef 32.

Li J., Shen L., Chen Z.: Convergence and stability of a stabilized finite volume method for the stationary Navier–Stokes equations. BIT Numer. Math.

50 , 823–842 (2010)

MathSciNet MATH CrossRef 33.

Li R., Zhu P.: Generalized difference methods for second order elliptic partial differential equations (I)-triangle grids. Numer. Math. J. Chin. Univ. 2 , 140–152 (1982)

34.

Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1984)

35.

Wang J., Wang Y., Ye X.: A new finite volume method for the stokes problems. Int. J. Numer. Anal. Model.

7 , 281–302 (2009)

MathSciNet 36.

Xu J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput.

15 , 231–237 (1994)

MathSciNet MATH CrossRef 37.

Xu J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal.

33 , 1759–1778 (1996)

MathSciNet MATH CrossRef 38.

Xu J., Zou Q.: Analysis of linear and quadratic simplicial finite volume methods for elliptic equations. Numer. Math.

111 , 469–492 (2009)

MathSciNet MATH CrossRef 39.

Xu, J., Zhu, Y., Zou, Q.: New adaptive finite volume methods and convergence analysis (2012, in press)

40.

Ye X.: On the relationship between finite volume and finite element methods applied to the Stokes equations. Numer. Mathods Partial Differ. Equ.

5 , 440–453 (2001)

CrossRef