A posteriori error estimators for the Stokes equations. (English) Zbl 0674.65092

The paper deals with an error analysis for the Stokes equation. Two methods are discussed, the first one based on a suitable evaluation of the residual of the finite element solution, and the second one based on solution of the local Stokes problem involving the residual of the finite element solution. Numerical examples show the efficiency of both methods for estimation of the error and for controlling self-adaptive mesh refinement process.
Reviewer: D.Fagé


65Z05 Applications to the sciences
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Abdalass, E.M.: Resolution performance du probl?me de Stokes par mini-?l?ments, maillages auto-adaptifs et m?thodes multigrilles-applications. Th?se de 3me cycle, Ecole Centrale de Lyon 1987
[2] Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo21, 337-344 (1984) · Zbl 0593.76039 · doi:10.1007/BF02576171
[3] Babuska, I.: The finite element method with Lagrange multipliers. Numer. Math.20, 179-192 (1973) · Zbl 0258.65108 · doi:10.1007/BF01436561
[4] Babuska, I., Rheinboldt, W.C.: A posteriori error estimates for the finite element method. Int. J. Numer. Methods Eng.12, 1597-1615 (1978) · Zbl 0396.65068 · doi:10.1002/nme.1620121010
[5] Babuska, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal.15, 736-754 (1978) · Zbl 0398.65069 · doi:10.1137/0715049
[6] Bank, R.E., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Konrad Zuse Zentrum, Berlin, Preprint SC-87-2 (1987) · Zbl 0645.65074
[7] Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comput.44, 283-301 (1985) · Zbl 0569.65079 · doi:10.1090/S0025-5718-1985-0777265-X
[8] Brezzi, F.: On the existence, uniqueness, and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Num?r.8, 129-151 (1974) · Zbl 0338.90047
[9] Buckley, A., Lenir, A.:QN-like variable storage conjugate gradients. Math. Program.27, 155-175 (1983) · Zbl 0519.65038 · doi:10.1007/BF02591943
[10] Ciarlet, P.G.: The finite element method for elliptic problems, 2nd Ed. Amsterdam: North Holland 1978 · Zbl 0383.65058
[11] Girault, V., Raviart, P.A.: Finite element approximation of the Navier-Stokes equations. Series in Computational Mathematics. Berlin Heidelberg New York: Springer 1986 · Zbl 0585.65077
[12] Verf?rth, R.: A combined conjugate gradient ? multi-grid algorithm for the numerical solution of the Stokes problem. IMA J. Numer. Anal.4, 441-455 (1984) · Zbl 0563.76028
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