Optimal error estimates for the pseudostress formulation of the Navier-Stokes equations. (English) Zbl 1457.65200
Summary: In this paper, we prove optimal a priori error estimates for the pseudostress-velocity mixed finite element formulation of the incompressible Navier-Stokes equations, thus improve the result of Z. Cai et al. [Numer. Methods Partial Differ. Equations 26, No. 4, 957–978 (2010; Zbl 1267.76059)]. This is achieved by applying Petrov-Galerkin type Brezzi-Rappaz-Raviart theory.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 35Q30 | Navier-Stokes equations |
Citations:
Zbl 1267.76059References:
| [1] | Brezzi, F.; Rappaz, J.; Raviart, P. A., Finite dimensional approximation of nonlinear problems, Numer. Math., 36, 1-25 (1980) · Zbl 0488.65021 |
| [2] | Girault, V.; Raviart, P. A., Finite Element Methods for Navier-Stokes Equations, 27-28 (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0585.65077 |
| [3] | Caloz, G.; Rappaz, J., Numerical analysis for nonlinear and bifurcation problems, (Ciarlet, P. G.; Lions, J. L., Handbook of Numerical Analysis, Vol. V (1997), North-Holland: North-Holland Amsterdam), 487-637 |
| [4] | Kim, D.; Park, E.-J., A priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations, SIAM J. Numer. Anal., 48, 3, 1186-1207 (2010) · Zbl 1214.65060 |
| [5] | Arnold, D. N.; Falk, R. S., A new mixed formulation for elasticity, Numer. Math., 53, 13-30 (1988) · Zbl 0621.73102 |
| [6] | Cai, Z.; Tong, C.; Vassilevski, P. S.; Wang, C., Mixed finite element methods for incompressible flow: stationary Stokes equations, Numer. Methods Partial Differential Equations, 26, 4, 957-978 (2010) · Zbl 1267.76059 |
| [7] | Carstensen, C.; Kim, D.; Park, E. J., A priori and a posteriori pseudostress-velocity mixed finite element error analysis for the Stokes problem, SIAM J. Numer. Anal., 49, 6, 2501-2523 (2011) · Zbl 1232.65098 |
| [8] | Gatica, G. N.; Marquez, A.; Sanchez, M. A., Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations, Comput. Methods Appl. Mech. Engrg., 199, 17-20, 1064-1079 (2010) · Zbl 1227.76030 |
| [9] | Cai, Z.; Wang, C.; Zhang, S., Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations, SIAM J. Numer. Anal., 48, 1, 79-94 (2010) · Zbl 1410.76160 |
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