A locally conservative LDG method for the incompressible Navier-Stokes equations
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- by Bernardo Cockburn, Guido Kanschat and Dominik Schötzau HTML | PDF
- Math. Comp. 74 (2005), 1067-1095 Request permission
Abstract:
In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in $H(\mathrm {div};\Omega )$ is obtained by simple, element-by-element post-processing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers.References
- Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. MR 664882, DOI 10.1137/0719052
- Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
- Susanne C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal. 41 (2003), no. 1, 306–324. MR 1974504, DOI 10.1137/S0036142902401311
- Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Paul Castillo, Bernardo Cockburn, Ilaria Perugia, and Dominik Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), no. 5, 1676–1706. MR 1813251, DOI 10.1137/S0036142900371003
- B. Cockburn, G. Kanschat, and D. Schötzau, The local discontinuous Galerkin methods for linear incompressible flow: A review, Computers and Fluids (Special Issue: Residual based methods and discontinuous Galerkin schemes), to appear.
- Bernardo Cockburn, Guido Kanschat, and Dominik Schötzau, The local discontinuous Galerkin method for the Oseen equations, Math. Comp. 73 (2004), no. 246, 569–593. MR 2031395, DOI 10.1090/S0025-5718-03-01552-7
- Bernardo Cockburn, Guido Kanschat, Dominik Schötzau, and Christoph Schwab, Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal. 40 (2002), no. 1, 319–343. MR 1921922, DOI 10.1137/S0036142900380121
- 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.
- Peter Hansbo and Mats G. Larson, Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 17-18, 1895–1908. MR 1886000, DOI 10.1016/S0045-7825(01)00358-9
- G. Kanschat, Block preconditioners for LDG discretizations of linear incompressible flow problems, J. Sci. Comput., 25 (2003), 815–831.
- Ohannes A. Karakashian and Wadi N. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes equations, SIAM J. Numer. Anal. 35 (1998), no. 1, 93–120. MR 1618436, DOI 10.1137/S0036142996297199
- L. I. G. Kovasznay, Laminar flow behind two-dimensional grid, Proc. Cambridge Philos. Soc. 44 (1948), 58–62. MR 24282
- P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. MR 0658142
- Ilaria Perugia and Dominik Schötzau, An $hp$-analysis of the local discontinuous Galerkin method for diffusion problems, Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), 2002, pp. 561–571. MR 1910752, DOI 10.1023/A:1015118613130
- Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. MR 1299729
- W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
- Dominik Schötzau, Christoph Schwab, and Andrea Toselli, Mixed $hp$-DGFEM for incompressible flows, SIAM J. Numer. Anal. 40 (2002), no. 6, 2171–2194 (2003). MR 1974180, DOI 10.1137/S0036142901399124
- 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), no. 1, 111–143 (English, with French summary). MR 813691, DOI 10.1051/m2an/1985190101111
- Roger Temam, Sur l’approximation des solutions des équations de Navier-Stokes, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A219–A221 (French). MR 211059
- Roger Temam, Une méthode d’approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France 96 (1968), 115–152 (French). MR 237972
- Andrea Toselli, $hp$ discontinuous Galerkin approximations for the Stokes problem, Math. Models Methods Appl. Sci. 12 (2002), no. 11, 1565–1597. MR 1938957, DOI 10.1142/S0218202502002240
Additional Information
- Bernardo Cockburn
- Affiliation: School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, Minnesota 55455
- Email: cockburn@math.umn.edu
- Guido Kanschat
- Affiliation: Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 293/294, 69120 Heidelberg, Germany
- MR Author ID: 622524
- Email: kanschat@dgfem.org
- Dominik Schötzau
- Affiliation: Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada
- Email: schoetzau@math.ubc.ca
- Received by editor(s): June 10, 2003
- Received by editor(s) in revised form: March 12, 2004
- Published electronically: October 5, 2004
- Additional Notes: The first author was supported in part by the National Science Foundation (Grant DMS-0107609) and by the University of Minnesota Supercomputing Institute
This work was carried out in part while the authors were at the Mathematisches Forschungsinstitut Oberwolfach for the meeting on Discontinuous Galerkin Methods in April 21–27, 2002 and while the second and third authors visited the School of Mathematics, University of Minnesota, in September 2002. - © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 1067-1095
- MSC (2000): Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-04-01718-1
- MathSciNet review: 2136994