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CONTINUOUS Q1–Q1 STOKES ELEMENTS STABILIZED WITH NON-CONFORMING NULL EDGE AVERAGE VELOCITY FUNCTIONS

Department of Mathematics, University of Colorado at Denver, P. O. Box 173364, Denver, CO 80217-3364, USA

Laboratório de Geofísica de Exploração de Petróleo, CPGG-UFBA, Campus Universitário da Federação, Salvador-BA, 40210-340, Brazil

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MARCUS SARKIS

Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro-RJ, 22460-320, Brazil

Worcester Polytechnic Institute, Mathematical Sciences Department, 100 Institute Road, Worcester, MA 01609, USA

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https://doi.org/10.1142/S021820250700198XCited by:3 (Source: Crossref)

Abstract

We present a stabilized finite element method for Stokes equations with piecewise continuous bilinear approximations for both velocity and pressure variables. The velocity field is enriched with piecewise polynomial bubble functions with null average at element edges. These functions are statically condensed at the element level and therefore they can be viewed as a continuous Q1–Q1 stabilized finite element method. The enriched velocity-pressure pair satisfies optimal inf–sup conditions and approximation properties. Numerical experiments show that the proposed discretization outperforms the Galerkin least-squares method.

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References

D. N. Arnold, D. Boffi and R. S. Falk, Math. Comput. 71, 909 (2002), DOI: 10.1090/S0025-5718-02-01439-4. Crossref, Web of Science, Google Scholar
D. N. Arnold, F. Brezzi and M. Fortin, Calcolo 21, 337 (1984), DOI: 10.1007/BF02576171. Crossref, Google Scholar
I. Babuška, Numer. Math. 20, 179 (1973), DOI: 10.1007/BF01436561. Crossref, Web of Science, Google Scholar
W. Bai, Comput. Methods Appl. Mech. Engrg. 143, 41 (1997), DOI: 10.1016/S0045-7825(96)01146-2. Crossref, Web of Science, Google Scholar
S. C. Brenner and L. R. Scott , The Mathematical Theory of Finite Element Methods ( Springer-Verlag , 1994 ) . Crossref, Google Scholar
F. Brezzi, Rev. Franc. Automat. Inform. Rech. Oper. 8, 129 (1974). Google Scholar
F. Brezzi and M. Fortin , Mixed and Hybrid Finite Element Methods ( Springer-Verlag , 1991 ) . Crossref, Google Scholar
F. Brezzi and J. Pitkäranta, Efficient Solutions of Elliptic Systems (Kiel, 1984) (Vieweg, 1984) pp. 11–19. Crossref, Google Scholar
P. G. Ciarlet, Handbook of Numerical Analysis 2, eds. P. G. Ciarlet and J. L. Lions (North-Holland, 1991) pp. 17–351. Google Scholar
B. Cockburnet al., SIAM J. Numer. Anal. 40, 319 (2002), DOI: 10.1137/S0036142900380121. Crossref, Web of Science, Google Scholar
L. P. Franca and S. Frey, Comput. Meth. Appl. Mech. Engrg. 99, 209 (1992), DOI: 10.1016/0045-7825(92)90041-H. Crossref, Web of Science, Google Scholar
L. P. Franca and T. J. R. Hughes, Comput. Meth. Appl. Mech. Engrg. 69, 89 (1988), DOI: 10.1016/0045-7825(88)90168-5. Crossref, Web of Science, Google Scholar
L. P. Franca and T. J. R. Hughes, Comput. Meth. Appl. Mech. Engrg. 105, 285 (1993), DOI: 10.1016/0045-7825(93)90126-I. Crossref, Web of Science, Google Scholar
L. P. Franca , T. J. R. Hughes and R. Stenberg , Incompressible Computational Fluid Dynamics — Trends and Advances , eds. M. Gunzburger and R. Nicolaides ( Cambridge Univ. Press , 1993 ) . Google Scholar
U. Ghia, K. N. Ghia and C. T. Shin, J. Comput. Phys. 48, 387 (1982), DOI: 10.1016/0021-9991(82)90058-4. Crossref, Web of Science, Google Scholar
V. Girault and P.-A. Raviart , Finite Element Methods for Navier–Stokes Equations — Theory and Algorithms ( Springer–Verlag , 1986 ) . Crossref, Google Scholar
P. Goldfeld, L. F. Pavarino and O. Widlund, Numer. Math. 95, 283 (2003), DOI: 10.1007/s00211-002-0450-9. Crossref, Web of Science, Google Scholar
T. J. R. Hughes, L. P. Franca and M. Balestra, Comput. Meth. Appl. Mech. Engrg. 59, 85 (1986), DOI: 10.1016/0045-7825(86)90025-3. Crossref, Web of Science, Google Scholar
B. Khoromskij and G. Wittum , Numerical Solution of Elliptic Differential Equations by Reduction to the Interface 36 ( Springer-Verlag , 2004 ) . Crossref, Google Scholar
P. Mons and G. Rogé, Math. Mod. Numer. Anal. 26, 507 (1992). Crossref, Web of Science, Google Scholar
L. F. Pavarino and O. Widlund, Comm. Pure Appl. Math. 55, 302 (2002), DOI: 10.1002/cpa.10020. Crossref, Web of Science, Google Scholar
R. Pierre, SIAM J. Numer. Anal. 32, 1210 (1995), DOI: 10.1137/0732056. Crossref, Web of Science, Google Scholar
R. Stenberg, Math. Comput. 42, 9 (1984), DOI: 10.2307/2007557. Web of Science, Google Scholar