A posteriori error analysis of the fully discretized time-dependent Stokes equations
Bernardi, Christine ; Verfürth, Rüdiger
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004), p. 437-455 / Harvested from Numdam
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The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

Publié le : 2004-01-01
DOI : https://doi.org/10.1051/m2an:2004021
Classification:  65N30,  65N15,  65J15 
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     author = {Bernardi, Christine and Verf\"urth, R\"udiger},
     title = {A posteriori error analysis of the fully discretized time-dependent Stokes equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {38},
     year = {2004},
     pages = {437-455},
     doi = {10.1051/m2an:2004021},
     mrnumber = {2075754},
     zbl = {1079.76042},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2004__38_3_437_0}
}

Bernardi, Christine; Verfürth, Rüdiger. A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 437-455. doi : 10.1051/m2an:2004021. http://gdmltest.u-ga.fr/item/M2AN_2004__38_3_437_0/

[1] A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comput. (to appear). | Zbl 1072.65124

[2] C. Bernardi and B. Métivet, Indicateurs d'erreur pour l'équation de la chaleur. Rev. Européenne Élém. Finis 9 (2000) 425-438. | Zbl 0959.65106

[3] C. Bernardi, B. Métivet and R. Verfürth, Analyse numérique d'indicateurs d'erreur, in Maillage et adaptation. P.-L. George Ed., Hermès (2001) 251-278.

[4] M. Bieterman and I. Babuška, The finite element method for parabolic equations. I. A posteriori error estimation. Numer. Math. 40 (1982) 339-371. | Zbl 0534.65072

[5] M. Bieterman and I. Babuška, The finite element method for parabolic equations. II. A posteriori error estimation and adaptive approach. Numer. Math. 40 (1982) 373-406. | Zbl 0534.65073

[6] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | Zbl 0368.65008

[7] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28 (1991) 43-77. | Zbl 0732.65093

[8] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729-1749. | Zbl 0835.65116

[9] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Lect. Notes Math. 749 (1979). | Zbl 0413.65081

[10] J.G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990) 353-384. | Zbl 0850.76350

[11] C. Johnson, Y.-Y. Nie and V. Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem. SIAM J. Numer. Anal. 27 (1990) 277-291. | Zbl 0701.65063

[12] M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998) 223-237. | Zbl 0935.65105

[13] J. Pousin and J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213-231. | Zbl 0822.65034

[14] R. Temam, Theory and Numerical Analysis of the Navier-Stokes Equations. North-Holland (1977). | Zbl 0383.35057

[15] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996). | Zbl 0853.65108

[16] R. Verfürth, A posteriori error estimates for nonlinear problems: L r (0,T;W 1,ρ (Ω))-error estimates for finite element discretizations of parabolic equations. Numer. Methods Partial Differential Equations 14 (1998) 487-518. | Zbl 0974.65087

[17] R. Verfürth, A posteriori error estimates for nonlinear problems 67 (1998) 1335-1360. | Zbl 0907.65090

[18] R. Verfürth, Error estimates for some quasi-interpolation operators. ESAIM: M2AN 33 (1999) 695-713. | Numdam | Zbl 0938.65125

[19] R. Verfürth, A posteriori error estimation techniques for non-linear elliptic and parabolic pdes, Rev. Européenne Élém. Finis 9 (2000) 377-402. | Zbl 0948.65092