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A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems


Authors: Vivette Girault, Béatrice Rivière and Mary F. Wheeler
Journal: Math. Comp. 74 (2005), 53-84
MSC (2000): Primary 35Q30; Secondary 76D05, 76D07
Published electronically: March 23, 2004
MathSciNet review: 2085402
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Abstract: A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and $L^2$ estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.


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  • 1. R. A. Adams, Sobolev Spaces, Academic Press, New York, NY (1975). MR 56:9247
  • 2. G. A. Baker, W. N. Jureidini and O. A. Karakashian, Piecewise solenoidal vector fields and the Stokes problem, SIAM J. Numer. Anal. 27 (1987), pp. 1466-1485. MR 91m:65246
  • 3. R. Becker, P. Hansbo and R. Stenberg, A finite element method for domain decomposition with nonmatching grids, M2AN 37 (2003), pp. 209-225.
  • 4. P. Ciarlet, The finite element methods for elliptic problems, North-Holland, Amsterdam (1978). MR 58:25001
  • 5. B. Cockburn, G. Kanschat, D. Schotzau and C. Schwab, Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal. 40 (2002) pp. 319-343. MR 2003g:65141
  • 6. M. Crouzeix and R. S. Falk, Nonconforming finite elements for the Stokes problem, Math. Comp. 52 (186) (1989), pp. 437-456. MR 89i:65113
  • 7. M. Crouzeix and P. A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, R.A.I.R.O. Numerical Analysis R3 (1973), pp. 33-76. MR 49:8401
  • 8. M. Fortin, An analysis of the convergence of mixed finite element methods, R.A.I.R.O. Numerical Analysis 11 (1977), pp. 341-354. MR 57:4473
  • 9. M. Fortin and M. Soulié, A nonconforming piecewise quadratic finite element on triangles, International Journal for Numerical Methods in Engineering 19 (1983), pp. 505-520. MR 84g:76004
  • 10. V. Girault and J.-L. Lions, Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra, Port. Math. (N.S.) 58 (2001), no. 1, pp. 25-57. MR 2002m:65117
  • 11. V. Girault, R. Glowinski, H. López and J.-P. Vila, A boundary multiplier/fictitious domain method for the steady incompressible Navier-Stokes equations, Numer. Math. 88 (2001), no. 1, pp. 75-103. MR 2002b:65166
  • 12. V. Girault and P. A. Raviart, Finite element methods for Navier-Stokes equations: theory and algorithms, Springer Series in Computational Mathematics 5 (1986). MR 88b:65129
  • 13. V. Girault and R. L. Scott, A quasi-local interpolation operator preserving the discrete divergence, Calcolo 40 (2003), pp. 1-19.
  • 14. P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Monographs and Studies in Mathematics 24, Pitman, Boston, MA (1985). MR 86m:35044
  • 15. P. Houston, C. Schwab and E. Suli, Discontinuous hp-finite element methods for advection-diffusion problems, SIAM J. Numer. Anal. 39 (2002), no. 6, pp. 2133-2163. MR 2003d:65108
  • 16. O. A. Karakashian and W. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes equations, SIAM J. Numer. Anal. 35 (1998), pp. 93-120. MR 99d:65320
  • 17. P. Lesaint and P. A. Raviart, On a finite element method for solving the neutron transport equation, In: Mathematical Aspects of Finite Element Methods in Partial Differential Equations, C. A. de Boor (Ed.), Academic Press, (1974) pp. 89-123. MR 58:31918
  • 18. J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, I, Dunod, Paris (1968). MR 40:512
  • 19. J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris (1969). MR 41:4326
  • 20. J. T. Oden, I. Babuska, and C. E. Baumann, A discontinous hp finite element method for diffusion problems, Journal of Computational Physics 146 (1998) pp. 491-519. MR 99m:65173
  • 21. O. Pironneau, Finite Elements for Fluids, Wiley, Chichester (1989). MR 90j:76016
  • 22. B. Rivière, M. F. Wheeler and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I, Computational Geosciences 3 (1999) pp. 337-360. MR 2001d:65145
  • 23. B. Rivière, M. F. Wheeler and V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal., 39 (3) (2001) pp. 902-931. MR 2002g:65149
  • 24. B. Rivière and M. F. Wheeler, Nonconforming methods for transport with nonlinear reaction, Proceedings of the Joint Summer Research Conference on Fluid Flow and Transport in Porous Media (2001), Contemp. Math., vol. 295, Amer. Math. Soc., Providence, RI, 2002, pp. 421-432.
  • 25. R. Temam, Navier-Stokes equations. Theory and numerical analysis. Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001. MR 2002j:76001
  • 26. M. F. Wheeler, A priori $L^2$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (4) (1973) pp. 723-759. MR 50:3613
  • 27. M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1) (1978) pp. 152-161. MR 57:11117

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Additional Information

Vivette Girault
Affiliation: Université Pierre et Marie Curie, Paris VI, Laboratoire Jacques-Louis Lions, $4$, place Jussieu, F-75230 Paris Cedex 05, France
Email: girault@ann.jussieu.fr

Béatrice Rivière
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray, Pittsburgh, Pennsylvania 15260
Email: riviere@math.pitt.edu

Mary F. Wheeler
Affiliation: The Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas, 201 E. 24th St., Austin, Texas 78712
Email: mfw@ices.utexas.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-04-01652-7
Keywords: Discontinuous finite element methods, Navier-Stokes, domain decomposition, nonconforming grids, local mass conservation
Received by editor(s): March 26, 2002
Received by editor(s) in revised form: May 9, 2003
Published electronically: March 23, 2004
Additional Notes: Each author was supported in part by DOD Pet2 Grant and NSF Grants KDI#DMS-9873326 and ITR#EIA-0121523.
Article copyright: © Copyright 2004 American Mathematical Society