Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: Space discretization and convergence

Numerische Mathematik

Volumn 107, Issue 1, 2007, Pages 39-77

  Burman, Erik ^a   Fernández, Miguel A ^b  

^a EPFL   (Switzerland)

^b INRIA   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34547273293     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-007-0070-5     Document Type: Article

References (35)

1. Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4), 173-199 (2001)
2. Becker, R., Braack, M.: A two-level stabilization scheme for the Navier-Stokes equations. In: Numerical mathematics and advanced applications, pages 123-130. Springer, Berlin (2004)
3. Bertoluzza, S.: The discrete commutator property of approximation spaces. C. R. Acad. Sci. Paris Sér. I Math. 329(12), 1097-1102 (1999)
4. p-norm. BIT 46(2), 249-260 (2006)
5. Braack, M., Burman, E.: Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SLAM J. Numer. Anal. (2005)
6. 1(Ω). Math. Comput. (Electronic) 71(237), 147-156 (2002)
7. Brezzi, F., Fortin, M.: A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89(3), 457-491 (2001)
8. Burman, E.: A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43(5), 2012-2033 (2005)
9. Burman, E., Fernández, M.A.: A finite element method with edge oriented stabilization for the time-dependent Navier-Stokes equations: space discretization and convergence. Technical Report RR-5630, INRIA (2005)
10. Burman, E., Fernández, M.A., Hansbo, P.: Continuous interior penalty finite element method for Oseen's equations. SIAM J. Numer. Anal. 44(3), 1248-1274 (2006)
11. Burman, E., Hansbo, P.: Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195(19-22), 2393-2410 (2006)
12. Burman, E., Hansbo, P.: Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193(15-16). 1437-1453 (2004)
13. Clément, P.: Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9(R-2), 77-84 (1975)
14. Codina, R.: Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput. Methods Appl. Mech. Engrg. 191(39-40), 4295-4321 (2002)
15. Codina, R., Blasco, J.: Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations. Numer. Math. 87(1), 59-81 (2000)
16. 2-projection onto finite element function spaces. Math. Comput. 48(178), 521-532 (1987)
17. Douglas J. Jr., Dupont, T.: Interior penalty procedures for elliptic and parabolic Galerkin methods, volume 58 of Lecture Notes in Physics. Springer, Berlin (1976)
18. Ern, A., Guermond, J.-L.: Theory and practice of finite elements, volume 159 of applied mathematical sciences. Springer, New York (2004)
19. Freund, J., Stenberg, R.: On weakly imposed boundary conditions for second order problems. In: Morandi Cecchi, M. et al. (ed.) Proceedings of the ninth international conference finite elements in fluids, pp. 327-336, Venice (1995)
20. Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations, vol. 5 of Springer series in computational mathematics. Springer, Berlin (1986)
21. Guermond, J.-L.: Stabilization of Galerkin approximations of transport equations by subgrid modeling. M2AN Math. Model. Numer. Anal. 33(6), 1293-1316 (1999)
22. Guermond, J.-L., Prudhomme, S.: Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows. M2AN Math. Model. Numer. Anal. 37(6), 893-908 (2003)
23. Hansbo, P., Szepessy, A.: A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 84(2), 175-192 (1990)
24. He, Y., Lin, Y., Sun, W.: Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete Contin. Dyn. Syst. Ser. (Electronic) B 6(1), 41-68 (2006)
25. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275-311 (1982)
26. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time. SIAM J. Numer. Anal. 23(4), 750-777 (1986)
27. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. III. Smoothing property and higher order error estimates for spatial discretization. SIAM. J. Numer. Anal. 25(3), 489-512 (1988)
28. Hoppe, R.H.W., Wohlmuth, B.: Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO Modél. Math. Anal. Numer. 30(2), 237-263 (1996)
29. Johnson, C., Szepessy, A., Hansbo, P.: On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws. Math. Comp. 54(189), 107-129 (1990)
30. Johnson, C., Saranen, J.: Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations. Math. Comp. 47(175), 1-18 (1986)
31. Oswald, P.: On a BPX-preconditioner for P1 elements. Computing 51(2), 125-133 (1993)
32. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483-493 (1990)
33. Temam, R.: Une méthode d'approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 96, 115-152 (1968)
34. Thomée, V.: Galerkin finite element methods for parabolic problems, volume 25 of Springer series in computational mathematics, 2nd edn. Springer, Berlin (2006)
35. Tobiska, L., Verfürth, R.: Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J. Numer. Anal. 33(1), 107-127 (1996)