Mathematics of Computation

Author(s): Bernardo Cockburn; Guido Kanschat; Dominik Schötzau.
Journal: Math. Comp. 73 (2004), 569-593.
MSC (2000): Primary 65N30
Posted: May 21, 2003
Retrieve article in: PDF

Abstract: We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shape-regular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in

- and negative-order norms. Numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of Reynolds numbers.

1.: 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 (2001), 1749-1779. MR 2002k:65183
2.: I. Babuska, C.E. Baumann, and J.T. Oden, A discontinuous -finite element method for diffusion problems: 1-D analysis, Comput. Math. Appl. 37 (1999), 103-122. MR 2000a:65118
3.: G.A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), 45-59. MR 55:4737
4.: 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 91m:65246
5.: C.E. Baumann and J.T. Oden, A discontinuous -finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 175 (1999), 311-341. MR 2000d:65171
6.: C.E. Baumann and T.J. Oden, A discontinuous -finite element method for the solution of the Euler and Navier-Stokes equations, Internat. J. Numer. Methods in Fluids 31 (1999), 79-95. MR 2000g:76072
7.: F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer, New York, 1991. MR 92d:65187
8.: P. Castillo, Performance of discontinuous Galerkin methods for elliptic partial differential equations, SIAM J. Sci. Comput., to appear.
9.: P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), 1676-1706. MR 2002k:65175
10.: P.G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978. MR 58:25001
11.: 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 2002g:65140
12.: 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.
13.: B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), 2440-2463. MR 99j:65163
14.: -, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001), 173-261. MR 2002i:65099
15.: V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Springer, New York, 1986. MR 88b:65129
16.: P. Hansbo and M.G. Larson, Discontinuous finite element methods for incompressible and nearly incompressible elasticity by use of Nitsche's method, Comput. Methods Appl. Mech. Engrg. 191 (2002), 1895-1908.
17.: P. Houston, C. Schwab, and E. Süli, Discontinuous -finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal. 39 (2002), 2133-2163.
18.: 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 99d:65320
19.: L. I. G. Kovasznay, Laminar flow behind a two-dimensional grid, Proc. Camb. Philos. Soc. 44 (1948), 58-62. MR 9:476d
20.: P. LeSaint and P.A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed.), Academic Press, New York, 1974, pp. 89-145. MR 58:31918
21.: J.-G. Liu and C.-W. Shu, A high order discontinuous Galerkin method for 2D incompressible flows, J. Comput. Phys. 160 (2000), 577-596. MR 2000m:76079
22.: -, A numerical example on the performance of high-order discontinuous Galerkin method for 2D incompressible flows, Discontinuous Galerkin Methods: Theory, Computation and Applications, Lect. Notes Comput. Sci. Eng., vol. 11, Springer, 2000, pp. 369-374.
23.: J.-G. Liu and Z.-P. Xin, Convergence of a Galerkin method for 2D discontinuous Euler flows, Comm. Pure Appl. Math. 53 (2000), 786-798. MR 2000m:76028
24.: J.T. Oden, I. Babuska, and C.E. Baumann, A discontinuous -finite element method for diffusion problems, J. Comput. Phys. 146 (1998), 491-519. MR 99m:65173
25.: T.J. Oden and C.E. Baumann, A conservative DGM for convection-diffusion and Navier-Stokes problems, Discontinuous Galerkin Methods: Theory, Computation and Applications (B. Cockburn, G.E. Karniadakis, and C.-W. Shu, eds.), Lect. Notes Comput. Sci. Eng., vol. 11, Springer, 2000, pp. 179-196. MR 2002d:65128
26.: W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation, Tech. Report Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
27.: R. Témam, Navier-Stokes equations. Theory and numerical analysis, North-Holland, 1979. MR 82b:35133
28.: A. Toselli, hp-discontinuous Galerkin approximations for the Stokes problem, Tech. Report 2002-02, Seminar for Applied Mathematics, ETH Zürich, 2002, to appear in Math. Models Methods Appl. Sci.

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
Email: kanschat@dgfem.org

Dominik Schötzau
Affiliation: Department of Mathematics, University of Basel, Rheinsprung~21, 4051 Basel, Switzerland
Email: schotzau@math.unibas.ch

DOI: 10.1090/S0025-5718-03-01552-7
PII: S 0025-5718(03)01552-7
Keywords: Finite elements, discontinuous Galerkin methods, Oseen equations
Received by editor(s): February 14, 2002
Received by editor(s) in revised form: August 21, 2002
Posted: May 21, 2003
Additional Notes: This work was carried out while the third author was a Dunham Jackson Assistant Professor at the School of Mathematics, University of Minnesota.
The first and third authors were supported in part by the National Science Foundation (Grant DMS-0107609) and by the University of Minnesota Supercomputing Institute
The second author was supported in part by ``Deutsche Forschungsgemeinschaft'' through SFB~359 and Schwerpunktprogramm ANumE)
Copyright of article: Copyright 2003, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS