Stabilized hp-DGFEM for incompressible flow

Mathematical Models and Methods in Applied Sciences

Volumn 13, Issue 10, 2003, Pages 1413-1436

  Schotzau D ^a   Schwab, C ^b   Toselli, A ^b  

^a UNIVERSITY OF BASEL   (Switzerland)

^b ETH ZURICH   (Switzerland)

Author keywords

Anisotropic refinement; Discontinuous Galerkin methods; Hp FEM; Stokes problem

Indexed keywords

EID: 0344063693     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202503002970     Document Type: Article

References (30)

1. M. Ainsworth and P. Coggins, The stability of mixed hp-finite element methods for Stokes flow on high aspect ratio elements, SIAM J. Numer. Anal. 38 (2000) 1721-1761.
2. B. Andersson, U. Falk, I. Babuška and T. von Petersdorff, Reliable stress and fracture mechanics analysis of complex aircraft components using a hp-version FEM, Int. J. Numer. Methods Engrg. 38 (1995) 2135-2163.
3. D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982) 742-760.
4. 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.
5. I. Babuška and B. Guo, Approximation properties of the hp-version of the finite element methods, Comput. Methods Appl. Mech. Engrg. 133 (1996) 319-346.
6. 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.
7. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, in Springer Series in Computational Mathematics, Vol. 15 (Springer-Verlag, 1991).
8. C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982) 67-86.
9. B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, in High-Order Methods for Computational Physics, eds. T. Barth and H. Deconink (Springer-Verlag, 1999), Vol. 9, pp. 69-224.
10. B. Cockburn, G. Kanschat and D. Schötzau, The local discontinuous Galerkin method for the Oseen equations, Technical Report 02-05, Department of Mathematics, University of Basel, 2002, Math. Comp., in press.
11. 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.
12. Eds. B. Cockburn, G. E. Karniadakis and C.-W. Shu, Discontinuous Galerkin Methods. Theory, Computation and Applications, Lect. Notes Comput. Sci. Engrg. Vol. 11 (Springer-Verlag, 2000).
13. B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001) 173-261.
14. L. Franca, T. Hughes and R. Stenberg, Stabilized finite element methods, in Incompressible Computational Fluid Dynamics: Trends and Advances, eds. M. Gunzburger and R. Nicholaides (Cambridge Univ. Press, 1993), pp. 87-107.
15. L. Franca and R. Stenburg, Error analysis of some Galerkin least squares methods for the elasticity equations, SIAM J. Numer. Anal. 28 (1991) 1680-1697.
16. V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations (Springer-Verlag, 1986).
17. V. Girault, B. Rivière and M. F. Wheeler, A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier-Stokes problems, Technical Report 02-08, TICAM, UT Austin, 2002.
18. P. Hansbo and M. G. Larson, Discontinuous finite element methods for incompressible and nearly incompressible elasticity by use of Nitsche's method, Comput. Mech. Engrg. 191 (2002) 1895-1908.
19. 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.
20. 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.
21. J. M. Melenk and C. Schwab, hp-FEM for reaction-diffusion equations, I. Robust exponential convergence, SIAM J. Numer. Anal. 35 (1998) 1520-1557.
22. D. Schötzau and C. Schwab, Mixed hp-FEM on anisotropic meshes, Math. Models Methods Appl. Sci. 8 (1998) 787-820.
23. D. Schötzau, C. Schwab and R. Stenberg, Mixed hp-FEM on anisotropic meshes, II. Hanging nodes and tensor products of boundary layer meshes, Numer. Math. 83 (1999) 667-697.
24. D. Schötzau, C. Schwab and A. Toselli, Mixed hp-DGFEM for incompressible flows, Technical Report 02-10, Seminar for Applied Mathematics, ETH Zürich, 2000, SIAM J. Numer. Anal. 40 (2003) 2171-2194.
25. D. Schötzau, C. Schwab and A. Toselli, Mixed hp-DGFEM for incompressible flows, II. Geometric edge meshes, Technical Report 02-13, Department of Mathematics, University of Basel, 2002, submitted to IMA J. Numer. Anal.
26. C. Schwab, p- and hp-FEM - Theory and Application to Solid and Fluid Mechanics (Oxford Univ. Press, 1998).
27. C. Schwab and M. Suri, The p- and hp-version of the finite element method for problems with boundary layers, Math. Comp. 65 (1996) 1403-1429.
28. C. Schwab, M. Suri and C. A. Xenophontos, The hp-FEM for problems in mechanics with boundary layers, Comput. Methods Appl. Mech. Engrg. 157 (1998) 311-333.
29. A. Toselli, hp-discontinuous Galerkin approximations for the Stokes problem, Math. Models Methods Appl. Sci. 12 (2002) 1565-1616.
30. A. Toselli and C. Schwab, Mixed hp-finite element approximations on geometric edge and boundary layer meshes in three dimensions, Numer. Math. 94 (2003) 771-801.