An optimal adaptive finite element method for the stokes problem

SIAM Journal on Numerical Analysis

Volumn 46, Issue 2, 2008, Pages 747-775

  Kondratyuk, Yaroslav ^a   Stevenson, Rob ^b  

^a HUMBOLDT UNIVERSITY   (Germany)

^b UNIVERSITY OF AMSTERDAM   (Netherlands)

Author keywords

A posteriori error estimators; Adaptive finite element method; Computational complexity; Convergence rates; Stokes equations; Uzawa iteration

Indexed keywords

COMPUTATIONAL COMPLEXITY; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; PROBLEM SOLVING; VELOCITY;

A POSTERIORI ERROR ESTIMATORS; ADAPTIVE FINITE ELEMENT METHOD; CONVERGENCE RATES; STOKES EQUATIONS; UZAWA ITERATION;

FINITE ELEMENT METHOD;

EID: 55549114346     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/06066566X     Document Type: Article

References (17)

1. E. Bänsch, P. MORIN, AND R. H. NOCHETTO, An adaptive Uzawa FEM for the Stokes problem: Convergence without the inf-sup condition, SIAM J. Numer. Anal., 40 (2002), pp. 1207-1229.
2. P. BINEV, Private communication, 2007.
3. P. BINEV, W. DAHMEN, AND R. DEVORE, Adaptive finite element methods with convergence rates, Numer. Math., 97 (2004), pp. 219-268.
4. P. BINEV, W. DAHMEN, R. DEVORE, AND P. PETRUCHEV, Approximation classes for adaptive methods, Serdica Math. J., 28 (2002), pp. 391-416.
5. P. BINEV AND R. DEVORE, Fast computation in adaptive tree approximation, Numer. Math., 97 (2004), pp. 193-217.
6. S. DAHLKE, Besov regularity for the Stokes problem, in Advances in Multivariate Approximation, W. Haussmann, K. Jetter, and M. Reimer, eds., Math. Res. 107, Wiley-VCH, Berlin, 1999, pp. 129-138.
7. S. DAHLKE, W. DAHMEN, AND K. URBAN, Adaptive wavelet methods for saddle point problems-Optimal convergence rates, SIAM J. Numer. Anal., 40 (2002), pp. 1230-1262.
8. W. DÖRFLER, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106-1124.
9. Y. KONDRATYUK, Adaptive Finite Element Algorithms for the Stokes Problem: Convergence Rates and Optimal Computational Complexity, Preprint 1346, Department of Mathematics, Utrecht University, The Netherlands, 2006.
10. Y. KONDRATYUK AND R. P. STEVENSON, An Optimal Adaptive Finite Element Method for the Stokes Problem-Extended Version, Technical report, Department of Mathematics, Utrecht University, The Netherlands, 2007; available at http://staff.science. uva.nl/~rstevens/publ. html.
11. W. F. MITCHELL, A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Software, 15 (1989), pp. 326-347.
12. P. MORIN, R. H. NOCHETTO, AND K. G. SIEBERT, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), pp. 466-488.
13. R. NOCHETTO AND J. H. PYO, Optimal relaxation parameter for the Uzawa method, Numer. Math., 98 (2004), pp. 695-702.
14. R. P. STEVENSON, The completion of locally refined simplicial partitions created by bisection, Math. Comp., 77 (2008), pp. 227-241.
15. R. P. STEVENSON, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7 (2007), pp. 245-269.
16. R. VERFÜRTH, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, UK, 1996.
17. H. WU AND Z. CHEN, Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems, Sci. China Ser. A, 49 (2006), pp. 1405-1429.