Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Hybridized globally divergence-free LDG methods. Part I: The Stokes problem

Authors: Jesús Carrero, Bernardo Cockburn and Dominik Schötzau
Journal: Math. Comp. 75 (2006), 533-563
MSC (2000): Primary 65N30
Published electronically: December 16, 2005
MathSciNet review: 2196980
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. This optimally convergent method is obtained by using an LDG method to discretize a vorticity-velocity formulation of the Stokes equations and by applying a new hybridization to the resulting discretization. One of the main features of the hybridized method is that it provides a globally divergence-free approximate velocity without having to construct globally divergence-free finite-dimensional spaces; only elementwise divergence-free basis functions are used. Another important feature is that it has significantly less degrees of freedom than all other LDG methods in the current literature; in particular, the approximation to the pressure is only defined on the faces of the elements. On the other hand, we show that, as expected, the condition number of the Schur-complement matrix for this approximate pressure is of order $ h^{-2}$ in the mesh size $ h$. Finally, we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates.

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

  • 1. D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), 7-32. MR 0813687 (87g:65126)
  • 2. D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002), 1749-1779. MR 1885715 (2002k:65183)
  • 3. 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. MR 1080332 (91m:65246)
  • 4. -, A discontinuous $ hp$-finite element method for the Euler and the Navier-Stokes equations, Int. J. Numer. Meth. Fluids (Special Issue: Tenth International Conference on Finite Elements in Fluids, Tucson, Arizona) 31 (1999), 79-95. MR 1714511 (2000g:76072)
  • 5. J. H. Bramble, J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), 1-22.MR 1023042 (90k:65170)
  • 6. P. Brenner and L. R. Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, 1994.MR 1278258 (95f:65001)
  • 7. S. Brenner, Poincaré-Friedrichs inequalities for piecewise H$ ^1$ functions, SIAM J. Numer. Anal. 41 (2003), 306-324.MR 1974504 (2004d:65140)
  • 8. F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), 217-235. MR 0799685 (87g:65133)
  • 9. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, 1991. MR 1115205 (92d:65187)
  • 10. P. Causignac, Explicit basis functions of quadratic and improved quadratic finite element spaces for the Stokes problem, Comm. Appl. Numer. Methods 2 (1986), 205-211.
  • 11. -, Computation of pressure from the finite element vorticity stream-function approximation for the Stokes problem, Comm. Appl. Numer. Methods 3 (1987), 287-295.
  • 12. B. Cockburn and J. Gopalakrishnan, A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal. 42 (2004), 283-301. MR 2051067 (2005e:65183)
  • 13. B. Cockburn, G. Kanschat, I. Perugia, and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal. 39 (2001), 264-285. MR 1860725 (2002g:65140)
  • 14. B. Cockburn, G. Kanschat, and D. Schötzau, The local discontinuous Galerkin methods for linear incompressible flow: A review, Computer and Fluids (Special Issue: Residual based methods and discontinuous Galerkin schemes) 34 (2005), 491-506.MR 2136586
  • 15. -, A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp. 74 (2005), 1067-1095. MR 2136994
  • 16. -, Local discontinuous Galerkin methods for the Oseen equations, Math. Comp. 73 (2004), 569-593.MR 2031395 (2005g:65168)
  • 17. B. Cockburn, G. Kanschat, D. Schötzau, and C. Schwab, Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal. 40 (2002), 319-343. MR 1921922 (2003g:65141)
  • 18. M. Crouzeix and P. A. Raviart, Conforming and nonconforming finite element methods for solving the stationary stokes equations, RAIRO Modél. Math. Anal. Numér. 7 (1973), 33-75. MR 0343661 (49:8401)
  • 19. W. Dörfler, The conditioning of the stiffness matrix for certain elements approximating the incompressibility condition in fluid dynamics, Numer. Math. 58 (1990), 203-214. MR 1069279 (91k:65142)
  • 20. M. Fortin, Calcul numérique des écoulements des fluides de Bingham et des fluides Newtoniens incompressibles par la méthode des élements finis, Ph.D. thesis, Université de Paris VI, 1972.
  • 21. -, Utilization de la méthode des éléments finis en mécanique des fluides, Calcolo 12 (1975), 405-441. MR 0421339 (54:9344a)
  • 22. V. Girault and P. A. Raviart, Finite element approximations of the Navier-Stokes equations, Springer-Verlag, New York, 1986.MR 0548867 (83b:65122)
  • 23. V. Girault, B. Rivière, and M. F. Wheeler, A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier-Stokes problems, Math. Comp. 74 (2005), 53-84. MR 2085402 (2005f:65149)
  • 24. J. Gopalakrishnan, A Schwarz preconditioner for a hybridized mixed method, Comput. Methods Appl. Math. 3 (2003), 116-134. MR 2002260 (2004g:65033)
  • 25. D.F. Griffiths, Finite elements for incompressible flow, Math. Methods Appl. Sci. 1 (1979), 16-31. MR 0548403 (80j:76027)
  • 26. M. D. Gunzburger, Finite element methods for viscous incompressible flows: A guide to theory, practice and algorithms, Academic Press, 1989.MR 1017032 (91d:76053)
  • 27. -, The $ {\mathrm{inf-sup}}$ condition in mixed finite element methods with application to the Stokes system, Collected Lectures on the Preservation of Stability under Discretization (D. Estep and S. Tavener, eds.), SIAM, 2002, pp. 93-121.MR 2026665
  • 28. F. Hecht, Construction d'une base $ P_1$ non conforme à divergence nulle, RAIRO Modél. Math. Anal. Numér. 15 (1981), 119-150.MR 0618819 (83j:65023)
  • 29. 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. MR 1618436 (99d:65320)
  • 30. O. A. Karakashian and T. Katsaounis, A discontinuous Galerkin method for the incompressible Navier-Stokes equations, Discontinuous Galerkin Methods. Theory, Computation and Applications (Berlin) (B. Cockburn, G.E. Karniadakis, and C.-W. Shu, eds.), Lect. Notes Comput. Sci. Engrg., vol. 11, Springer-Verlag, February 2000, pp. 157-166. MR 1842171
  • 31. O. A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of a second order elliptic problems, SIAM J. Numer. Anal. 41 (2003), 2374-2399. MR 2034620 (2005d:65192)
  • 32. L. I. G. Kovasznay, Laminar flow behind a two-dimensional grid, Proc. Camb. Philos. Soc. 44 (1948), 58-62. MR 0024282 (9:476d)
  • 33. I. Perugia and D. Schötzau, An $ hp$-analysis of the local discontinuous Galerkin method for diffusion problems, J. Sci. Comput. (Special Issue: Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01), Uppsala, Sweden) 17 (2002), 561-571. MR 1910752
  • 34. I. Perugia and D. Schötzau, The $ hp$-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations, Math. Comp. 72 (2003), 1179-1214. MR 1972732 (2004b:65190)
  • 35. A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, Springer-Verlag, New York, 1994. MR 1299729 (95i:65005)
  • 36. D. Schötzau, C. Schwab, and A. Toselli, hp-DGFEM for incompressible flows, SIAM J. Numer. Anal. 40 (2003), 2171-2194. MR 1974180 (2004k:65161)
  • 37. L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér. 19 (1985), 111-143. MR 0813691 (87i:65190)
  • 38. F. Thomasset, Implementation of finite element methods for Navier-Stokes equations, Springer Series in Computational Physics, Springer-Verlag, New York, 1981. MR 0720192 (84k:76015)

Similar Articles

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

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

Additional Information

Jesús Carrero
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Dominik Schötzau
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

Keywords: Divergence-free elements, local discontinuous Galerkin methods, hybridized methods, Stokes equations
Received by editor(s): June 9, 2004
Received by editor(s) in revised form: February 9, 2005
Published electronically: December 16, 2005
Additional Notes: The second author was supported in part by the National Science Foundation (Grant DMS-0411254) and by the University of Minnesota Supercomputing Institute. The third author was supported in part by the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society