The local discontinuous Galerkin method for the Oseen equations

Mathematics of Computation

Volumn 73, Issue 246, 2004, Pages 569-593

  Cockburn, Bernardo ^a   Kanschat, Guido ^b   Schötzau, Dominik ^c  

^a UNIVERSITY OF MINNESOTA   (United States)

^b UNIVERSITY OF HEIDELBERG   (Germany)

^c UNIVERSITY OF BASEL   (Switzerland)

Author keywords

Discontinuous Galerkin methods; Finite elements; Oseen equations

Indexed keywords

EID: 1642602854     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01552-7     Document Type: Article

References (28)

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. Babuška, C.E. Baumann, and J.T. Oden, A discontinuous hp-finite element method for diffusion problems: 1-D analysis, Comput. Math. Appl. 37 (1999), 103-122. MR2000a: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 hp-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 hp-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 20021:65099
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 hp-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 SD 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 hp-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.