Stabilized finite element method for Navier–Stokes equations with physical boundary conditions

Amara, M.; Capatina-Papaghiuc, D.; Trujillo, D.

doi:10.1090/S0025-5718-07-01929-1

Stabilized finite element method for Navier–Stokes equations with physical boundary conditions
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by M. Amara, D. Capatina-Papaghiuc and D. Trujillo;

Math. Comp. 76 (2007), 1195-1217

DOI: https://doi.org/10.1090/S0025-5718-07-01929-1

Published electronically: March 15, 2007

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Abstract:

This paper deals with the numerical approximation of the 2D and 3D Navier-Stokes equations, satisfying nonstandard boundary conditions. This lays on the finite element discretisation of the corresponding Stokes problem, which is achieved through a three-fields stabilized mixed formulation. A priori and a posteriori error bounds are established for the nonlinear problem, ascertaining the convergence of the method. Finally, numerical tests are presented, including mesh refinement via error indicators.

References

Mohamed Amara and Christine Bernardi, Convergence of a finite element discretization of the Navier-Stokes equations in vorticity and stream function formulation, M2AN Math. Model. Numer. Anal. 33 (1999), no. 5, 1033–1056 (English, with English and French summaries). MR 1726723, DOI 10.1051/m2an:1999133
Amara, M., Capatina-Papaghiuc, D., Trujillo, D.: Stabilized method for the Navier-Stokes equations with nonstandard boundary conditions, Preprint LMA UPPA, n. 0325, p. 1- 29 (2003) (http://lma.univ-pau.fr/publis/publis_pre.php)
Amara, M., Capatina-Papaghiuc, D., Trujillo, D.: A 3D numerical model for the vorticity-velocity-pressure formulation of the Navier-Stokes problem, Preprint LMA UPPA, n. 0510, p.1-11 (2005) (http://lma.univ-pau.fr/publis/publis_pre.php)
M. Amara, E. Chacón Vera, and D. Trujillo, Vorticity-velocity-pressure formulation for Stokes problem, Math. Comp. 73 (2004), no. 248, 1673–1697. MR 2059731, DOI 10.1090/S0025-5718-03-01615-6
C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal. 35 (1998), no. 5, 1893–1916. MR 1639966, DOI 10.1137/S0036142995293766
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
Gabriel Caloz and Jacques Rappaz, Numerical analysis for nonlinear and bifurcation problems, Handbook of numerical analysis, Vol. V, Handb. Numer. Anal., V, North-Holland, Amsterdam, 1997, pp. 487–637. MR 1470227, DOI 10.1016/S1570-8659(97)80004-X
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
C. Conca, C. Parés, O. Pironneau, and M. Thiriet, Navier-Stokes equations with imposed pressure and velocity fluxes, Internat. J. Numer. Methods Fluids 20 (1995), no. 4, 267–287. MR 1316046, DOI 10.1002/fld.1650200402
Martin Costabel, A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains, Math. Methods Appl. Sci. 12 (1990), no. 4, 365–368. MR 1048563, DOI 10.1002/mma.1670120406
François Dubois, Michel Salaün, and Stéphanie Salmon, Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem, J. Math. Pures Appl. (9) 82 (2003), no. 11, 1395–1451 (English, with English and French summaries). MR 2020806, DOI 10.1016/j.matpur.2003.09.002
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
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
J. Pousin and J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems, Numer. Math. 69 (1994), no. 2, 213–231. MR 1310318, DOI 10.1007/s002110050088