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A comparison between the mini-element and the Petrov-Galerkin formulations for the generalized Stokes problem. (English) Zbl 0732.65100

The authors try an essay on the stability question of some finite element formulations for a Stokes-like problem where to the classical Stokes equation a term is added that is the product of the velocity and a positive parameter.
Unfortunately, the work contains some unrealistic assumptions on the physical unknowns and does not have the clarity needed for a work pretending to put in order such a complicated question.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
76M10 Finite element methods applied to problems in fluid mechanics
35Q30 Navier-Stokes equations
35J25 Boundary value problems for second-order elliptic equations
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References:

[1] Arnold, D. N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 337-344 (1984) · Zbl 0593.76039
[2] Hughes, T. J.R.; Franca, L. P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation for the Stokes problem accomodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59, 85-99 (1986) · Zbl 0622.76077
[3] Brezzi, F.; Douglas, J. J., Stabilized mixed methods for the Stokes problem, Numer. Math., 53, 225-235 (1988) · Zbl 0669.76052
[4] R.E. Bank and B.D. Welfert, A posteriori error estimates for the Stokes problem, SIAM J. Numer. Anal. (submitted).; R.E. Bank and B.D. Welfert, A posteriori error estimates for the Stokes problem, SIAM J. Numer. Anal. (submitted). · Zbl 0731.76040
[5] Verfürth, R., A posteriori error estimators for the Stokes equations, Preprint Nr. 445 (December 1987)
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