Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation
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Abstract:
In this paper, we develop and analyze mixed finite element methods for the Stokes and Navier-Stokes equations. Our mixed method is based on the pseudostress-pressure-velocity formulation. The pseudostress is approximated by the Raviart-Thomas, Brezzi-Douglas-Marini, or Brezzi-Douglas-Fortin-Marini elements, the pressure and the velocity by piecewise discontinuous polynomials of appropriate degree. It is shown that these sets of finite elements are stable and yield optimal accuracy for the Stokes problem. For the pseudostress-pressure-velocity formulation of the stationary Navier-Stokes equations, the well-posedness and error estimation results are established. By eliminating the pseudostress variables in the resulting algebraic system, we obtain cell-centered finite volume schemes for the velocity and pressure variables that preserve local balance of momentum.References
- D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, DOI 10.1051/m2an/1985190100071
- Douglas N. Arnold, Jim Douglas Jr., and Chaitan P. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984), no. 1, 1–22. MR 761879, DOI 10.1007/BF01379659
- Douglas N. Arnold, Richard S. Falk, and R. Winther, Preconditioning in $H(\textrm {div})$ and applications, Math. Comp. 66 (1997), no. 219, 957–984. MR 1401938, DOI 10.1090/S0025-5718-97-00826-0
- Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $H(\textrm {div})$ and $H(\textrm {curl})$, Numer. Math. 85 (2000), no. 2, 197–217. MR 1754719, DOI 10.1007/PL00005386
- Marek A. Behr, Leopoldo P. Franca, and Tayfun E. Tezduyar, Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows, Comput. Methods Appl. Mech. Engrg. 104 (1993), no. 1, 31–48. MR 1210650, DOI 10.1016/0045-7825(93)90205-C
- F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
- Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250. MR 890035, DOI 10.1007/BF01396752
- Franco Brezzi, Jim Douglas Jr., Michel Fortin, and L. Donatella Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 581–604 (English, with French summary). MR 921828, DOI 10.1051/m2an/1987210405811
- Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- F. Brezzi, J. Rappaz, and P.-A. Raviart, Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions, Numer. Math. 36 (1980/81), no. 1, 1–25. MR 595803, DOI 10.1007/BF01395985
- Zhiqiang Cai, Jim Douglas Jr., and Moongyu Park, Development and analysis of higher order finite volume methods over rectangles for elliptic equations, Adv. Comput. Math. 19 (2003), no. 1-3, 3–33. Challenges in computational mathematics (Pohang, 2001). MR 1973457, DOI 10.1023/A:1022841012296
- Zhiqiang Cai, Barry Lee, and Ping Wang, Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems, SIAM J. Numer. Anal. 42 (2004), no. 2, 843–859. MR 2084238, DOI 10.1137/S0036142903422673
- Zhiqiang Cai and Gerhard Starke, First-order system least squares for the stress-displacement formulation: linear elasticity, SIAM J. Numer. Anal. 41 (2003), no. 2, 715–730. MR 2004196, DOI 10.1137/S003614290139696X
- Zhiqiang Cai, Charles Tong, Panayot S. Vassilevski, and Chunbo Wang, Mixed finite element methods for incompressible flow: stationary Stokes equations, Numer. Methods Partial Differential Equations 26 (2010), no. 4, 957–978. MR 2642330, DOI 10.1002/num.20467
- Zhiqiang Cai, Chunbo Wang, and Shun Zhang, Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations, SIAM J. Numer. Anal. 48 (2010), no. 1, 79–94. MR 2608359, DOI 10.1137/080718413
- Zhiqiang Cai and Yanqiu Wang, A multigrid method for the pseudostress formulation of Stokes problems, SIAM J. Sci. Comput. 29 (2007), no. 5, 2078–2095. MR 2350022, DOI 10.1137/060661429
- Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259, DOI 10.7208/chicago/9780226764320.001.0001
- M. Farhloul and M. Fortin, A new mixed finite element for the Stokes and elasticity problems, SIAM J. Numer. Anal. 30 (1993), no. 4, 971–990. MR 1231323, DOI 10.1137/0730051
- M. Farhloul and M. Fortin, Review and complements on mixed-hybrid finite element methods for fluid flows, Proceedings of the 9th International Congress on Computational and Applied Mathematics (Leuven, 2000), 2002, pp. 301–313. MR 1934446, DOI 10.1016/S0377-0427(01)00520-9
- Leonardo E. Figueroa, Gabriel N. Gatica, and Antonio Márquez, Augmented mixed finite element methods for the stationary Stokes equations, SIAM J. Sci. Comput. 31 (2008/09), no. 2, 1082–1119. MR 2466149, DOI 10.1137/080713069
- B. X. Fraeijs de Veubeke, Stress function approach , International Congress on the Finite Element Method in Structural Mechanics, Bournemouth, 1975.
- G. Gatica, M. González and S. Meddahi, A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows, Comput. Methods Appl. Mech. Eng., 193 (2004), Part I: a priori error analysis, 881-892; Part II: a posteriori error analysis, 893-911.
- Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
- Max D. Gunzburger, Finite element methods for viscous incompressible flows, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. A guide to theory, practice, and algorithms. MR 1017032
- R. Hiptmair, Multigrid method for $\mathbf H(\textrm {div})$ in three dimensions, Electron. Trans. Numer. Anal. 6 (1997), no. Dec., 133–152. Special issue on multilevel methods (Copper Mountain, CO, 1997). MR 1615161
- F. A. Milner and E.-J. Park, Mixed finite-element methods for Hamilton-Jacobi-Bellman-type equations, IMA J. Numer. Anal. 16 (1996), no. 3, 399–412. MR 1397603, DOI 10.1093/imanum/16.3.399
- J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
- P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
- Panayot S. Vassilevski and Jun Ping Wang, Multilevel iterative methods for mixed finite element discretizations of elliptic problems, Numer. Math. 63 (1992), no. 4, 503–520. MR 1189534, DOI 10.1007/BF01385872
Additional Information
- Zhiqiang Cai
- Affiliation: Department of Mathematics, Purdue University, 150 N. University Street West Lafayette, Indiana 47907-2067
- MR Author ID: 235961
- Email: zcai@math.purdue.edu
- Shun Zhang
- Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
- MR Author ID: 704861
- Email: Shun_Zhang@brown.edu
- Received by editor(s): January 15, 2010
- Received by editor(s) in revised form: April 11, 2011
- Published electronically: March 28, 2012
- Additional Notes: This work was supported in part by the National Science Foundation under grants DMS-0511430 and DMS-0810855.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 81 (2012), 1903-1927
- MSC (2010): Primary 65M15, 65M60
- DOI: https://doi.org/10.1090/S0025-5718-2012-02585-3
- MathSciNet review: 2945142