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Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations. (English) Zbl 0887.76038
We discuss the stabilization, via bubble functions, of a finite element method for stationary linearized incompressible Navier-Stokes equations. It is shown that the upwind bubbles can be used to obtain the SUPG stabilization. Furthermore, the diffusion-dominated case is discussed in more detail, and it is also shown that in this regime the bubbles reproduce the right parameter of SUPG.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Baiocchi, C.; Brezzi, F.; Franca, L.P., Virtual bubbles and gals, Comput. methods appl. mech. engrg., 105, 125-142, (1993)
[2] Brezzi, F.; Russo, A., Choosing bubbles for advection-diffusion problems, Math. models methods appl. sci., 4, 571-587, (1994) · Zbl 0819.65128
[3] Bank, R.E.; Welfert, B.D., A comparison between the mini-element and the Petrov-Galerkin formulations for the generalized Stokes problem, Comput. methods appl. mech. engrg., 83, 61-68, (1990) · Zbl 0732.65100
[4] Gresho, P.M., Some current CFD issues relevant to the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 87, 201-252, (1991) · Zbl 0760.76018
[5] Temam, R., Navier-Stokes equations, (1977), North-Holland Amsterdam · Zbl 0335.35077
[6] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[7] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO ser. rouge, 8, 129-151, (1974) · Zbl 0338.90047
[8] Brezzi, F.; Fortan, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag New York
[9] Johnson, C.; Nävert, U.; Pitkaranta, J., Finite element methods for linear hyperbolic problems, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[10] Franca, L.P.; Frey, S.L., Stabilized finite element methods: II. the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 99, 209-233, (1992) · Zbl 0765.76048
[11] Franca, L.P.; Frey, S.L.; Hughes, T.J.R., Stabilized finite element methods: I. application to the advective-diffusive model, Comput. methods appl. mech. engrg., 95, 253-276, (1992) · Zbl 0759.76040
[12] Vishik, M.I.; Lyusternik, L.A.; Vishik, M.I.; Lyusternik, L.A., Regular degeneration and boundary layer for linear differential equations with a small parameter, Uspekki mat. nauk., Am. math. soc. transl. ser., 2, 20, 239-364, (1962) · Zbl 0122.32402
[13] A. Russo, A posteriori error estimators via bubble functions. Math. Models Methods Appl. Sci., in press. · Zbl 0853.65109
[14] Russo, A., A posteriori error estimators for the Stokes problem, Appl. math. lett., 8, 1-4, (1995) · Zbl 0821.65066
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