Vorticity-velocity-pressure formulation for Stokes problem

Amara, M.; Chacón Vera, E.; Trujillo, D.

doi:10.1090/S0025-5718-03-01615-6

Vorticity-velocity-pressure formulation for Stokes problem
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by M. Amara, E. Chacón Vera and D. Trujillo;

Math. Comp. 73 (2004), 1673-1697

DOI: https://doi.org/10.1090/S0025-5718-03-01615-6

Published electronically: October 27, 2003

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Abstract:

We propose a three-field formulation for efficiently solving a two-dimensional Stokes problem in the case of nonstandard boundary conditions. More specifically, we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary. The proposed computational methodology consists in reformulating the considered boundary value problem via a mixed-type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the Lagrange multiplier. The obtained formulation is then discretized and a convergence analysis is performed. A priori error estimates are established, and some numerical results are presented to highlight the perfomance of the proposed computational methodology.

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
M. Amara and F. Dabaghi, An optimal $\textbf {C}^0$ finite element algorithm for the 2D biharmonic problem: theoretical analysis and numerical results, Numer. Math. 90 (2001), no. 1, 19–46. MR 1868761, DOI 10.1007/s002110100284
Amara, M., Chacon Vera, E. and Trujillo, D., Stokes equations with non standard boundary conditions, Prépubli. du Labo. de Math. Appli. de Pau, n. 13, (2001).
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
Christine Bernardi, Vivette Girault, and Yvon Maday, Mixed spectral element approximation of the Navier-Stokes equations in the stream-function and vorticity formulation, IMA J. Numer. Anal. 12 (1992), no. 4, 565–608 (English, with English and French summaries). MR 1186736, DOI 10.1093/imanum/12.4.565
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
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
François Dubois, Michel Salaün, and Stéphanie Salmon, Discrete harmonics for stream function-vorticity Stokes problem, Numer. Math. 92 (2002), no. 4, 711–742. MR 1935807, DOI 10.1007/s002110100369
Fritz John, Partial differential equations, 3rd ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York-Berlin, 1978. MR 514404, DOI 10.1007/978-1-4684-0059-5
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
Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, DOI 10.1016/0045-7825(86)90152-0
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 247243