Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem


Author: J. Guzmán
Journal: Math. Comp. 77 (2008), 1293-1322
MSC (2000): Primary 65N30, 65N15
DOI: https://doi.org/10.1090/S0025-5718-08-02067-X
Published electronically: January 25, 2008
MathSciNet review: 2398769
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to the Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries.


References [Enhancements On Off] (What's this?)

  • 1. D.N. Arnold, F. Brezzi, B. Cockburn and D.L. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2001/02), 1749-1779. MR 1885715 (2002k:65183)
  • 2. D.N. Arnold and X. Liu, Local error estimates for finite element discretizations of Stokes equations, Math. Modeling Num. Anal., 29 (1995), 367-389. MR 1342712 (96d:76055)
  • 3. H. Chen, Pointwise error estimates for finite element solutions of the Stokes problem, SIAM J. Num. Anal., 44 (2006), 1-28. MR 2217368 (2007a:65190)
  • 4. H. Chen and Z. Chen, Pointwise estimates of discontinuous Galerkin methods with penalty for second order elliptic problems, SIAM J. Numer. Anal., 42 (2004), 1146-1166. MR 2113680 (2006a:65156)
  • 5. H. Chen, Local error estimates of mixed discontinuous Galerkin methods for elliptic problems, J. Num. Math., 12 (2004), 1-21. MR 2039367 (2005a:65118)
  • 6. B. Cockburn, G. Kanschat and D. Schötzau, Local discontinuous Galerkin method for Oseen equations, Math. Comp., 73 (2004), 569-593. MR 2031395 (2005g:65168)
  • 7. B. Cockburn, G. Kanschat and D. Shötzau, A locally conservative LDG Method for the incompressible Navier-Stokes Equations, Math. Comp., 74 (2004), 1067-1095. MR 2136994 (2006a:65157)
  • 8. B. Cockburn, G. Kanschat and D. Schötzau, Local discontinuous Galerkin method for linearized incompressible fluid flow: a review, Comput. and Fluids, 34 (2005), 491-506. MR 2136586
  • 9. B. Cockburn, G. Kanschat and D. Schötzau, A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput., to appear.
  • 10. B. Cockburn G. Kanschat, D. Shötzau and C. Schwab, Local discontinuous Galerkin methods for the Stokes problem, SIAM J. Num. Anal. 40 (2002), 319-343. MR 1921922 (2003g:65141)
  • 11. A. Demlow, Piecewise linear finite element methods are not localized, Math. Comp., 73 (2004), no. 247, 1195-1201. MR 2047084 (2005a:65129)
  • 12. R. Durán, R. Nochetto and J. Wang, Sharp maximum norm error estimates for finite element approximations of Stokes problem, Math. Comp., 51 (1988), 491-506. MR 935076 (89b:65261)
  • 13. R. Durán and R. Nochetto, Weighted inf-sup condition and pointwise error estimates for the Stokes problem, Math. Comp., 54 (1990), 63-79. MR 995211 (90f:65203)
  • 14. R. Durán and R. Nochetto, Pointwise Accuracy of Stable Petrov-Galerkin Approximation to the Stokes Problem, SIAM J. Num. Anal., 26 (1989), 1395-1406. MR 1025095 (90m:65197)
  • 15. R. Durán and A. Muschietti, An explicit right inverse of the divergence operator which is continuous in weighted norms, Studia Mathematica, 148 (2001), 207-219. MR 1880723 (2002m:42012)
  • 16. V. Girault, R. Nochetto and R. Scott, Stability of the finite element Stokes projection in $ W\sp {1,\infty}$, C. R. Math. Acad. Sci. Paris, 338 (2004), 957-962. MR 2066358
  • 17. J. Guzmán, Pointwise estimates for discontinuous Galerkin methods with lifting operators for elliptic problems, Math. Comp., 75 (2006), 1067-1085. MR 2219019 (2006m:65269)
  • 18. W. Hoffman, A.H. Schatz, L.B. Wahlbin and G. Wittum, Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part I: A smooth problem and gloablly quasi-uniform meshes, Math. Comp., 70 (2001), 897-909. MR 1826572 (2002a:65178)
  • 19. D. Leykekhman and L.B. Wahlbin, A posteriori error estimates by recovered gradients in parabolic finite element equations, preprint.
  • 20. J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp., 28 (1974), 937-958. MR 0373325 (51:9525)
  • 21. A.H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids : Part I. Global Estimates, Math. Comp., 67 (1998), 877-899. MR 1464148 (98j:65082)
  • 22. A.H. Schatz and L.B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp., 31 (1977), 414-442. MR 0431753 (55:4748)
  • 23. A.H. Schatz, I. Sloan and L.B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal., 33 (1996), 505-521. MR 1388486 (98f:65112)
  • 24. A.H. Schatz, Perturbation of forms and error estimates for the finite element method at a point, with an application to improved error estimates for subspaces that are symmetric with respect to a point, SIAM J. Numer. Anal. 42 (2005), 2342-2365. MR 2139396 (2006g:65194)
  • 25. D. Schötzau, C. Schwab and C. Toselli, hp-DGFEM for incompressible flows, SIAM J. Num. Anal., 40 (2003), 2171-2194.
  • 26. V.A. Solonnikov, On Green's matrices for elliptic boundary value problems. I, Proc. Steklov. Inst. Math., 110 (1970), 123-169. MR 0289935 (44:7120)
  • 27. R. Rannacher and L.R. Scott, Some optimal error estimates for linear finite element approximations, Math. Comp., 38 (1982), 437-445. MR 645661 (83e:65180)
  • 28. R. Temam, Navier-Stokes Equations, North-Holland Amsterdam, 1984. MR 769654 (86m:76003)
  • 29. L.B. Wahlbin, Local behavior in finite element methods, In Handbook of Numerical Analysis, Volume II, 353-522, North-Holland, Amsterdam, 1991. MR 1115238

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N15

Retrieve articles in all journals with MSC (2000): 65N30, 65N15


Additional Information

J. Guzmán
Affiliation: School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, Minnesota 55455
Email: guzma033@umn.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02067-X
Keywords: Finite elements, discontinuous Galerkin, Stokes problem
Received by editor(s): September 26, 2006
Received by editor(s) in revised form: April 30, 2007
Published electronically: January 25, 2008
Additional Notes: The author was supported by a National Science Foundation Mathematical Science Postdoctoral Research Fellowship (DMS-0503050)
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society