Anisotropic a posteriori error estimation for the mixed discontinuous Galerkin approximation of the stokes problem

Numerical Methods for Partial Differential Equations

Volumn 22, Issue 2, 2006, Pages 449-483

  Creusé, Emmanuel ^a   Nicaise, Serge ^a  

^a UNIVERSITÉ DE VALENCIENNES   (France)

Author keywords

Anisotropic solution; DG method; Error estimator; Stokes problem; Stretched elements

Indexed keywords

EID: 33645006151     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20107     Document Type: Article

References (44)

1. 1 H.‐G. Roos, M. Stynes, and L. Tobiska, Numerical methods for singularly perturbed differential equations: convection‐diffusion and flow problems, Springer, Berlin, 1996.
2. 2 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 (2001), 843–856.
3. 3 D. Schötzau and C. Schwab, Mixed hp ‐FEM on anisotropic meshes, Math Models Meth Appl Sci 8 (1998), 787–820.
4. 4 D. Schötzau, C. Schwab, and R. Stenberg, Mixed hp ‐FEM on anisotropic meshes II: Hanging nodes and tensor products of boundary layer meshes, Numer Math 83 (1999), 667–697.
5. 5 A. Toselli and C. Schwab, Mixed hp ‐finite element approximations on geometric boundary layer meshes in R 3, C R Acad Paris Série I, 332 (2001), 857–862.
6. 6 G. Kunert, A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes. Logos Verlag, Berlin, 1999. Also PhD thesis, TU Chemnitz, http://archiv.tu‐chemnitz.de/pub/1999/0012/index.html.
7. 7 L. Formaggia, S. Perotto, and P. Zunino, An anisotropic a‐posteriori error estimate for a convection‐diffusion problem, Comput Vis Sci 4 (2) (2001), 99–104.
8. 8 M. J. Castro‐Díaz, F. Hecht, and B. Mohammadi, New progress in anisotropic grid adaption for inviscid and viscous flow simulations. In Proceedings of the 4 th Annual International Meshing Roundtable, Albuquerque, NM, Sandia National Laboratories, 1995, pp. 73–85.
9. 9 B. Cockburn, G. E. Karniadakis, and C.‐W. Shu, The development of discontinuous Galerkin methods, Vol. 11, Lect Notes Comput Sci Eng, Springer Verlag, Berlin, 2000.
10. 10 D. G. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal 39 (2001), 1749–1779.
11. 11 G. A. Baker, W. N. Jureidini, and O. A. Karakashian, Piecewise solenoidal vector fields and the Stokes problem, SIAM J Numer Anal 27 (1990), 1466–1485.
12. 12 O. A. Karakashian and W. N. Jureidini, A nonconforming finite element method for the stationary Navier‐Stokes equations, SIAM J Numer Anal 35 (1998), 93–120.
13. 13 B. Cockburn, G. Kanschat, D. Schötzau, and C. Schwab, Local discontinuous Galerkin methods for the Stokes problem, SIAM J Numer Anal 40 (2002), 319–343.
14. 14 A. Toselli, hp ‐Discontinuous Galerkin approximations for the Stokes problem, Math Models Meth Appl Sci 12 (2002), 1565–1616.
15. 15 D. Schötzau, C. Schwab, and A. Toselli, Mixed hp ‐DGFEM for incompressible flows, SIAM J Numer Anal 40 (2003), 2171–2194.
16. 16 R. Verfürth, A posteriori error estimators for the Stokes equation, Numer Math 55 (1989), 309–325.
17. 17 R. E. Bank and B. D. Welfert, A posteriori error estimates for the Stokes equations: a comparison, Comput Meth Appl Mech Engrg 82 (1990), 323–390.
18. 18 E. Dari, R. Durán, and C. Padra, Error estimators for nonconforming finite element approximations of the Stokes problem, Math Comput 64 (211) (1995), 1017–1033.
19. 19 M. Ainsworth and J. Oden, A posteriori error estimators for the Stokes and Oseen equations, SIAM J Numer Anal 34 (1) (1997), 228–245.
20. 20 D. Kay and D. Silvester, A posteriori error estimation for stabilized mixed approximations of the Stokes equations, SIAM J Sci Comput 21 (4) (2000), 1321–1336.
21. 21 C. Carstensen and S. Funken, A posteriori error control in low‐order finite element discretisations of incompressible stationary flow problems, Math Comp 70 (236) (2001), 1353–1381.
22. 22 E. Creusé, G. Kunert, and S. Nicaise, A posteriori error estimation for the Stokes problem: anisotropic and isotropic discretizations, Math Models Meth Appl Sci 14 (2004), 1297–1341.
23. 23 M. Picasso, An adaptive algorithm for the Stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio, Appl Numer Math, to appear.
24. 24 R. Becker, P. Hansbo, and M. G. Larson, Energy norm a posteriori error estimation for discontinuous Galerkin methods, Comput Meth Appl Mech Engrg 192 (2003), 723–733.
25. 25 R. Becker, P. Hansbo, and R. Stenberg, A finite element method for domain decomposition with non‐matching grids, Modél Math Anal Numér 37 (2003), 209–225.
26. 26 O. A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second‐order problems, SIAM J Numer Anal 41 (2003), 2374–2399.
27. 27 P. Houston, D. Schötzau, and T. Wihler, Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem, J Sci Comput 22 (1) (2005), 357–380.
28. 28 V. Girault and P.‐A. Raviart, Finite element methods for Navier‐Stokes equations: theory and algorithms, Springer Series in Computational Mathematics, Vol. 5, Springer, Berlin, 1986.
29. 29 G. Kunert, An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes, Numer Math 86 (3) (2000), 471–490.
30. 30 G. Kunert, Robust a posteriori error estimation for a singularly perturbed reaction‐diffusion equation on anisotropic tetrahedral meshes, Adv Comp Math 15 (1–4) (2001), 237–259.
31. 31 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.
32. 32 M. Picasso, Anisotropic, adaptive, stabilized finite elements for Stokes problem, Comput Meth Appl Mech Engrg 2003, submitted.
33. 33 P. Houston, I. Perugia, and D. Schötzau, Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator, Comput Meth Appl Mech Engrg 194 (2005), 499–510.
34. 34 P. Hansbo and M. G. Larson, Discontinuous finite element methods for incompressible and nearly incompressible elasticity by use of Nitsche's method, Comput Meth Appl Mech Engrg 191 (2002), 1895–1908.
35. 35 P. G. Ciarlet, The finite element method for elliptic problems, North‐Holland, Amsterdam, 1978.
36. 36 R. Verfürth, A review of a posteriori error estimation and adaptive mesh‐refinement techniques, Wiley and Teubner, Chichester and Stuttgart, 1996.
37. 37 F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer‐Verlag, New York, 1991.
38. 38 M. Dauge, Elliptic boundary value problems on corner domains—smoothness and asymptotics of solutions, Lecture Notes in Mathematics, Vol. 1341, Springer, Berlin, 1988.
39. 39 I. Perugia and D. Schötzau, An hp‐analysis of the local discontinuous Galerkin method for diffusion problems, J Sci Comp 17 (2002), 561–571.
40. 40 P. Houston, I. Perugia, and D. Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM J Numer Anal 42 (2004), 434–459.
41. 41 G. Kunert, A local problem error estimator for anisotropic tetrahedral finite element meshes, SIAM J Numer Anal 39 (2) (2001), 668–689.
42. 42 G. Kunert, Towards anisotropic mesh construction and error estimation in the finite element method, Numer Meth PDE 18 (6) (2002), 625–648.
43. 43 M. Picasso, Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz‐Zhu error estimator, Commun Numer Meth Engrg 19 (1) (2003), 13–23.
44. 44 R. Becker, P. Hansbo, and M. L. Larson, Energy norm a posteriori error estimation for discontinuous Galerkin methods, Calcolo, 2003, submitted.