An adaptive algorithm for the Stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio

Applied Numerical Mathematics

Volumn 54, Issue 3-4, 2005, Pages 470-490

  Picasso, M ^a  

^a EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

ADAPTIVE ALGORITHMS; APPROXIMATION THEORY; ERRORS; FINITE ELEMENT METHOD; LINEAR EQUATIONS; PIECEWISE LINEAR TECHNIQUES; ROBUSTNESS (CONTROL SYSTEMS); VELOCITY MEASUREMENT;

ANISOTROPIC ERRORS; PRESSURE ERROR; STOKES PROBLEM; VELOCITY ERRORS;

PROBLEM SOLVING;

EID: 20444457092     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2004.09.014     Document Type: Conference Paper

References (42)

1. M. Ainsworth, and P. Coggins The stability of mixed hp-finite element methods for Stokes flow on high aspect ratio elements SIAM J. Numer. Anal. 38 5 2000 1721 1761 (electronic)
2. M. Ainsworth, and J.T. Oden A posteriori error estimation in finite element analysis Comput. Methods Appl. Mech. Engrg. 142 1-2 1997 1 88
3. D. Ait-Ali-Yahia, G. Baruzzi, W. Habashi, M. Fortin, J. Dompierre, and M.-G. Vallet Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. II. Structured grids Internat. J. Numer. Methods Fluids 39 8 2002 657 673
4. T. Apel Interpolation of non-smooth functions on anisotropic finite element meshes Modél. Math. Anal. Numér. 33 6 1999 1149 1185
5. T. Apel, and G. Lube Anisotropic mesh refinement in stabilized Galerkin methods Numer. Math. 74 3 1996 261 282
6. T. Apel, and G. Lube Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem Appl. Numer. Math. 26 4 1998 415 433
7. T. Apel, and S. Nicaise The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges Math. Methods Appl. Sci. 21 6 1998 519 549
8. T. Apel, S. Nicaise, and J. Schöberl A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges IMA J. Numer. Anal. 21 4 2001 843 856
9. T. Apel, and H. Maharavo Randrianarivony Stability of discretizations of the Stokes problem on anisotropic meshes MODELLING, 2001, Pilsen Math. Comput. Simulation 61 3-6 2003 437 447
10. R. Bank, and B. Welfert A posteriori error estimates for the Stokes problem SIAM J. Numer. Anal. 28 3 1991 591 623
11. R. Becker, and R. Rannacher Finite element solution of the incompressible Navier-Stokes equations on anisotropically refined meshes Fast Solvers for Flow Problems, Kiel, 1994 Notes on Numerical Fluid Mechanics vol. 49 1995 Vieweg Braunschweig 52 62
12. H. Borouchaki, P. Laug, The bl2d mesh generator: Beginner's guide, user's and programmer's manual, Technical Report RT-0194, Institut National de Recherche en Informatique et Automatique (INRIA), Rocquencourt, 78153 Le Chesnay, France, 1996
13. F. Brezzi, and J. Douglas Stabilized mixed methods for the Stokes problem Numer. Math. 53 1-2 1988 225 235
14. E. Burman, A. Jacot, and M. Picasso Adaptive finite elements with high aspect ratio for the computation of coalescence using a phase-field model J. Comput. Phys. 195 2004 154 174
15. E. Burman, and M. Picasso Anisotropic, adaptive finite elements for the computation of a solutal dendrite J. Interfaces Free Boundaries 5 2003 103 127
16. M.J. Castro Díaz, and F. Hecht Anisotropic mesh generation. Minimization of interpolation error XV Congress on Differential Equations and Applications/V Congress on Applied Mathematics, Vol. I, II (Spanish) (Vigo, 1997) Colecc. Congr. vol. 9 1998 Univ. Vigo Vigo 647 653
17. P. Clément Approximation by finite element functions using local regularization RAIRO Anal. Numér. 9 1975 77 84
18. J. Dompierre, M.-G. Vallet, Y. Bourgault, M. Fortin, and W. Habashi Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. III. Unstructured meshes Internat. J. Numer. Methods Fluids 39 8 2002 675 702
19. H. Elman, D. Silvester, and A. Wathen Block preconditioners for the discrete incompressible Navier-Stokes equations ICFD Conference on Numerical Methods for Fluid Dynamics, Part II, Oxford, 2001 Internat. J. Numer. Methods Fluids 40 3-4 2002 333 344
20. H. Elman, D. Silvester, and A. Wathen Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations Numer. Math. 90 4 2002 665 688
21. L. Formaggia, S. Micheletti, S. Perotto, Anisotropic mesh adaptation in computational fluid dynamics: Application to the advection-diffusion-reaction and the stokes problems, Technical Report 15, MOCS, Politecnico di Milano, Italy, 2003, Mathematics and Computers in Simulation, submitted for publication
22. L. Formaggia, and S. Perotto New anisotropic a priori error estimates Numer. Math. 89 4 2001 641 667
23. L. Formaggia, and S. Perotto Anisotropic error estimates for elliptic problems Numer. Math. 94 1 2003 67 92
24. L. Franca, and S. Frey Stabilized finite element methods. II. The incompressible Navier-Stokes equations Comput. Methods Appl. Mech. Engrg. 99 2-3 1992 209 233
25. L. Franca, and R. Stenberg Error analysis of some GLS methods for elasticity equations SIAM. J. Numer. Anal. 28 6 1991 1680 1697
26. W. Habashi, J. Dompierre, Y. Bourgault, D. Ait-Ali-Yahiar, M. Fortin, and M.-G. Vallet Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. I. General principles Internat. J. Numer. Methods Fluids 32 6 2000 725 744
27. T.J.R. Hughes, L. Franca, and M. Balestra A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations Comput. Methods Appl. Mech. Engrg. 59 1 1986 85 99
28. G. Kunert An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes Numer. Math. 86 3 2000 471 490
29. G. Kunert Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes Modél. Math. Anal. Numér. 35 6 2001 1079 1109
30. G. Kunert Toward anisotropic mesh construction and error estimation in the finite element method Numer. Methods Partial Differential Equations 18 5 2002 625 648
31. S. Micheletti, S. Perotto, and M. Picasso Stabilized finite elements on anisotropic meshes: A priori error estimates for the advection-diffusion and stokes problem SIAM J. Numer. Anal. 41 2003 1131 1162
32. M. Picasso An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: Application to elliptic and parabolic problems SIAM J. Sci. Comput. 24 4 2003 1328 1355
33. M. Picasso Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz-Zhu error estimator Comm. Numer. Methods Engrg. 19 1 2003 13 23
34. M. Picasso, and J. Rappaz Stability of time-splitting schemes for the Stokes problem with stabilized finite elements Numer. Methods Partial Differential Equations 17 2001 632 656
35. W. Rachowicz An anisotropic h-adaptive finite element method for compressible Navier-Stokes equations Comput. Methods Appl. Mech. Engrg. 146 3-4 1997 231 252
36. R. Rannacher Error control in finite element computations. An introduction to error estimation and mesh-size adaptation Error Control and Adaptivity in Scientific Computing, Antalya, 1998 NATO Adv. Sci. Ser. C Math. Phys. Sci. vol. 536 1999 Kluwer Academic Dordrecht 247 278
37. J.-F. Remacle, Transient adaptive discontinuous Galerkin method with anisotropic meshes, Technical Report, Univ. Cath. Louvain, Dept. Génie Civil, 1348 Louvain-La-Neuve, Belgium, 2003
38. R. Rodríguez Some remarks on Zienkiewicz-Zhu estimator Numer. Methods Partial Differential Equations 10 5 1994 625 635
39. K. Siebert An a posteriori error estimator for anisotropic refinement Numer. Math. 73 3 1996 373 398
40. R. Verfürth A posteriori error estimators for the Stokes equations Numer. Math. 55 1989 309 325
41. R. Verfürth A posteriori error estimates for nonlinear problems. finite element discretizations of elliptic equations Math. Comp. 62 206 1994 445 475
42. J. Wang, and X. Ye Superconvergence of finite element approximations for the stokes problem by projection methods SIAM J. Numer. Anal. 39 3 2001 1001 1013