Stabilized Stokes Elements and Local Mass Conservation
Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 3, pp. 543-573.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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_a4,
     author = {Boffi, Daniele and Cavallini, Nicola and Gardini, Francesca and Gastaldi, Lucia},
     title = {Stabilized {Stokes} {Elements} and {Local} {Mass} {Conservation}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {543--573},
     publisher = {mathdoc},
     volume = {Ser. 9, 5},
     number = {3},
     year = {2012},
     zbl = {1291.76197},
     mrnumber = {3051736},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2012_9_5_3_a4/}
}
TY  - JOUR
AU  - Boffi, Daniele
AU  - Cavallini, Nicola
AU  - Gardini, Francesca
AU  - Gastaldi, Lucia
TI  - Stabilized Stokes Elements and Local Mass Conservation
JO  - Bollettino della Unione matematica italiana
PY  - 2012
SP  - 543
EP  - 573
VL  - 5
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2012_9_5_3_a4/
LA  - en
ID  - BUMI_2012_9_5_3_a4
ER  - 
%0 Journal Article
%A Boffi, Daniele
%A Cavallini, Nicola
%A Gardini, Francesca
%A Gastaldi, Lucia
%T Stabilized Stokes Elements and Local Mass Conservation
%J Bollettino della Unione matematica italiana
%D 2012
%P 543-573
%V 5
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2012_9_5_3_a4/
%G en
%F BUMI_2012_9_5_3_a4
Boffi, Daniele; Cavallini, Nicola; Gardini, Francesca; Gastaldi, Lucia. Stabilized Stokes Elements and Local Mass Conservation. Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 3, pp. 543-573. http://geodesic.mathdoc.fr/item/BUMI_2012_9_5_3_a4/

[1] M. Bercovier - O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math., 33 (2) (1979), 211-224. | fulltext EuDML | DOI | MR | Zbl

[2] D. Boffi - F. Brezzi - M. Fortin, Finite elements for the Stokes problem. In D. Boffi and L. Gastaldi, editors, Mixed finite elements, compatibility conditions, and applications, volume 1939 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 2008), 45-100. | DOI | MR | Zbl

[3] D. Boffi - N. Cavallini - F. Gardini - L. Gastaldi, Immersed boundary method: performance analysis of popular finite element spaces. In M. Papadrakakis, E. Onate and B. Schrefler, editors, COUPLED PROBLEMS 2011. Computational Methods for Coupled Problems in Science and Engineering IV (Cimne, 2011).

[4] D. Boffi - N. Cavallini - F. Gardini - L. Gastaldi, Local Mass Conservation of Stokes Finite Elements, J. Sci. Comput., to appear. | DOI | MR | Zbl

[5] D. Boffi - N. Cavallini - L. Gastaldi, Finite element approach to immersed boundary method with different fluid and solid densities., Math. Models Methods Appl. Sci., 21 (12) (2011), 2523-2550. | DOI | MR | Zbl

[6] D. Boffi - L. Gastaldi - L. Heltai - C. S. Peskin, On the hyper-elastic formulation of the immersed boundary method, Comput. Methods Appl. Mech. Engrg., 197 (25-28) (2008), 2210-2231. | DOI | MR

[7] F. Brezzi - M. Fortin, Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics (Springer-Verlag, New York, 1991). | DOI | MR | Zbl

[8] F. Brezzi - J. Pitkäranta, 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

[9] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co. (Amsterdam, 1978). | MR | Zbl

[10] P. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (R-2) (1975), 77-84. | fulltext EuDML | MR | Zbl

[11] M. Cremonesi - A. Frangi - U. Perego, A Lagrangian finite element approach for the analysis of fluid-structure interaction problems, Internat. J. Numer. Methods Engrg., 84 (5) (2010), 610-630. | DOI | MR | Zbl

[12] L. P. Franca - T. J. R. Hughes - R. Stenberg, Stabilized finite element methods. In Incompressible computational fluid dynamics: trends and advances (Cambridge Univ. Press, Cambridge, 2008), 87-107. | MR | Zbl

[13] L. P. Franca - R. Stenberg, Error analysis of Galerkin least squares methods for the elasticity equations, SIAM J. Numer. Anal., 28 (6) (1991), 1680-1697. | DOI | MR | Zbl

[14] T. J. R. Hughes - L. P. Franca, 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. | DOI | MR | Zbl

[15] T. J. R. Hughes - L. P. Franca - M. Balestra, 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. | DOI | MR | Zbl

[16] S. R. Idelsohn - E. Oñate, 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. | DOI | MR | Zbl

[17] N. Kechkar - D. J. Silvester, 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

[18] N. Kechkar - D. J. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem, Math. Comp., 58 (197) (1992), 1-10. | DOI | MR | Zbl

[19] P. Le Tallec - J. Mouro, Fluid structure interaction with large structural displacements, Comput. Methods Appl. Mech. Engrg., 190 (2001), 3039-3067. | Zbl

[20] G. Matthies - L. Tobiska, Mass conservation of finite element methods for coupled flow-transport problems, Int. J. Comput. Sci. Math., 1 (2-4) (2007), 293-307. | DOI | MR | Zbl

[21] C. Michler - E. H. Van Brummelen - S. J. Hulshoff - R. De Borst, The relevance of conservation for stability and accuracy of numerical methods for fluid-structure interaction, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4195-4215. | DOI | MR | Zbl

[22] P. Ryzhakov - E. Oñate - R. Rossi - S. Idelsohn, Improving mass conservation in simulation of incompressible flows, Int. J. Numer. Meth. Engng. (2012). | DOI | MR | Zbl

[23] D. J. Silvester - N. Kechkar, 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. | DOI | MR | Zbl