In this paper we discuss lowest order stabilizations of Stokes finite elements. We study the behavior of the constants in front of the error estimates in terms of the stabilization parameters and confirm with numerical tests that the bounds are sharp. Moreover, we investigate the local mass conservation properties of the considered schemes and analyze new schemes with enhanced pressure approximation, which guarantee a better local discretization of the divergence free constraint.
@article{BUMI_2012_9_5_3_543_0,
author = {Daniele Boffi and Nicola Cavallini and Francesca Gardini and Lucia Gastaldi},
title = {Stabilized Stokes Elements and Local Mass Conservation},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {5},
year = {2012},
pages = {543-573},
zbl = {1291.76197},
mrnumber = {3051736},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2012_9_5_3_543_0}
}
Boffi, Daniele; Cavallini, Nicola; Gardini, Francesca; Gastaldi, Lucia. Stabilized Stokes Elements and Local Mass Conservation. Bollettino dell'Unione Matematica Italiana, Tome 5 (2012) pp. 543-573. http://gdmltest.u-ga.fr/item/BUMI_2012_9_5_3_543_0/
[1] - , Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math., 33 (2) (1979), 211-224. | MR 549450 | Zbl 0423.65058
[2] - - , Finite elements for the Stokes problem. In and , editors, Mixed finite elements, compatibility conditions, and applications, volume 1939 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 2008), 45-100. | MR 2459075 | Zbl 1182.76895
[3] - - - , Immersed boundary method: performance analysis of popular finite element spaces. In , and , editors, COUPLED PROBLEMS 2011. Computational Methods for Coupled Problems in Science and Engineering IV (Cimne, 2011).
[4] - - - , Local Mass Conservation of Stokes Finite Elements, J. Sci. Comput., to appear. | MR 2948699 | Zbl 1264.74259
[5] - - , Finite element approach to immersed boundary method with different fluid and solid densities., Math. Models Methods Appl. Sci., 21 (12) (2011), 2523-2550. | MR 2864640 | Zbl 1242.76190
[6] - - - , On the hyper-elastic formulation of the immersed boundary method, Comput. Methods Appl. Mech. Engrg., 197 (25-28) (2008), 2210-2231. | MR 2412821
[7] - , Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics (Springer-Verlag, New York, 1991). | MR 1115205 | Zbl 0788.73002
[8] - , On the stabilization of finite element approximations of the Stokes equations. In Efficient solutions of elliptic systems (Kiel, 1984), volume 10 of Notes Numer. Fluid Mech. (Vieweg, Braunschweig, 1984), 11-19. | MR 804083
[9] , The finite element method for elliptic problems, North-Holland Publishing Co. (Amsterdam, 1978). | MR 520174 | Zbl 0383.65058
[10] , Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (R-2) (1975), 77-84. | MR 400739 | Zbl 0368.65008
[11] - - , A Lagrangian finite element approach for the analysis of fluid-structure interaction problems, Internat. J. Numer. Methods Engrg., 84 (5) (2010), 610-630. | MR 2761810 | Zbl 1202.74164
[12] - - , Stabilized finite element methods. In Incompressible computational fluid dynamics: trends and advances (Cambridge Univ. Press, Cambridge, 2008), 87-107. | MR 2504357 | Zbl 1189.76339
[13] - , Error analysis of Galerkin least squares methods for the elasticity equations, SIAM J. Numer. Anal., 28 (6) (1991), 1680-1697. | MR 1135761 | Zbl 0759.73055
[14] - , A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg., 65 (1) (1987), 85-96. | MR 914609 | Zbl 0635.76067
[15] - - , Errata: ``A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations'', Comput. Methods Appl. Mech. Engrg., 62 (1) (1987), 111. | MR 889303 | Zbl 0622.76077
[16] - , The challenge of mass conservation in the solution of free-surface flows with the fractional-step method: problems and solutions, Int. J. Numer. Methods Biomed. Eng., 26 (10) (2010), 1313-1330. | MR 2731638 | Zbl 1274.76191
[17] - , The stabilisation of low order mixed finite element methods for incompressible flow. In Proceedings of the Fifth International Symposium on Numerical Methods in Engineering, Vol. 1, 2 (Lausanne, 1989) (Southampton, 1989), 111-116. Comput. Mech. | MR 1052962
[18] - , Analysis of locally stabilized mixed finite element methods for the Stokes problem, Math. Comp., 58 (197) (1992), 1-10. | MR 1106973 | Zbl 0738.76040
[19] - , Fluid structure interaction with large structural displacements, Comput. Methods Appl. Mech. Engrg., 190 (2001), 3039-3067. | Zbl 1001.74040
[20] - , Mass conservation of finite element methods for coupled flow-transport problems, Int. J. Comput. Sci. Math., 1 (2-4) (2007), 293-307. | MR 2396384 | Zbl 1185.65214
[21] - - - , The relevance of conservation for stability and accuracy of numerical methods for fluid-structure interaction, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4195-4215. | MR 1986021 | Zbl 1181.74156
[22] - - - , Improving mass conservation in simulation of incompressible flows, Int. J. Numer. Meth. Engng. (2012). | MR 2931179 | Zbl 1246.76059
[23] - , Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem, Comput. Methods Appl. Mech. Engrg., 79 (1) (1990), 71-86. | MR 1044204 | Zbl 0706.76075