The paper presents finite element error estimates of a variational multiscale method (VMS) for the incompressible Navier–Stokes equations. The constants in these estimates do not depend on the Reynolds number but on a reduced Reynolds number or on the mesh size of a coarse mesh.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.A. Adams, Sobolev Spaces (Academic, New York, 1975).
I. Babuška and J. Osborn, Analysis of finite element methods for second order boundary value problems using mesh dependent norms, Numer. Math. 34 (1980) 41–62.
L.C. Berselli, C.R. Grisanti and V. John, On commutation errors in the derivation of the space averaged Navier–Stokes equations, Preprint 12, Università di Pisa, Dipartimento di Matematica Applicata “U.Dini,” 2004.
L.C. Berselli and V. John, On the comparison of a commutation error and the Reynolds stress tensor for flows obeying a wall law, Preprint 18, Università di Pisa, Dipartimento di Matematica Applicata “U.Dini,” 2004.
S.S. Collis, Monitoring unresolved scales in multiscale turbulence modeling, Phys. Fluids 13 (2001) 1800–1806.
A. Dunca, V. John and W.J. Layton, The commutation error of the space averaged Navier–Stokes equations on a bounded domain, in: Contributions to Current Challenges in Mathematical Fluid Mechanics, eds. J.G. Heywood, G.P. Galdi and R. Rannacher (Birkhäuser, 2004) pp. 53–78.
G.P. Galdi, An introduction to the Navier–Stokes initial-boundary value problem, in: Fundamental Directions in Mathematical Fluid Dynamics, eds. G.P. Galdi, J.G. Heywood and R. Rannacher (Birkhäuser, 2000) pp. 1–70.
V. Gravemeier, The variational multiscale method for laminar and turbulent incompressible flow, PhD thesis, Institute of Structural Mechanics, University of Stuttgart, 2003.
J.-L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling, M2AN 33 (1999) 1293–1316.
J.G. Heywood and R. Rannacher, 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 (1982) 275–311.
J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem III: Smoothing property and higher order estimates for spatial discretization, SIAM J. Num. Anal. 25 (1988) 489–512.
T.J. Hughes, L. Mazzei and K.E. Jansen, Large eddy simulation and the variational multiscale method, Comput. Vis. Sci. 3 (2000) 47–59.
T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origin of stabilized methods, Comput. Methods Appl. Mech. Eng. 127 (1995) 387–401.
V. John, Large Eddy Simulation of Turbulent Incompressible Flows. Analytical and Numerical Results for a Class of LES Models, volume 34 of Lecture Notes in Computational Science and Engineering, (Springer, Berlin Heidelberg New York, 2004).
V. John and S. Kaya, A finite element variational multiscale method for the Navier–Stokes equations, SIAM J. Sci. Comput. 26 (2005) 1485–1503.
V. John, S. Kaya and W. Layton, A two-level variational multiscale method for convection–diffusion equations, Comp. Meth. Appl. Math. Engrg. (2006) in press.
V. John and W.J. Layton, Analysis of numerical errors in large eddy simulation, SIAM J. Numer. Anal. 40 (2002) 995–1020.
W. Layton and W. Lenferink, A multilevel mesh independence principle for the Navier–Stokes equations, SIAM J. Numer. Anal. 33 (1996) 17–30.
W. Layton and L. Tobiska, A two-level method with backtracking for the Navier–Stokes equations, SIAM J. Numer. Anal. 35(5) (1998) 2035–2054.
P. Sagaut, Large Eddy Simulation for Incompressible Flows, 2nd edition (Springer, Berlin Heidelberg New York, 2003).
H. Sohr, The Navier-Stokes Equations, An Elementary Functional Analytic Approach. Birhäuser Advanced Texts (Birkhäuser Verlag Basel, Boston, Berlin, 2001).
R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, volume 2 of Studies in Mathematics and Its Applications (North-Holland, Amsterdam, New York, Oxford, 1977).
F. van der Bos and B.J. Geurts, Commutator erros in the filtering approach to large-eddy simulation, Phys. Fluids 17 (2005) 035108.
J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15(1) (1994) 231–237.
J. Xu, Two-grid finite element discretizations for nonlinear p.d.e.’s, SIAM J. Numer. Anal. 33(5) (1996) 1759–1777.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jinchao Xu
This work is partially supported by NSF grants DMS9972622, DMS20207627 and INT9814115.
Rights and permissions
About this article
Cite this article
John, V., Kaya, S. Finite element error analysis of a variational multiscale method for the Navier-Stokes equations. Adv Comput Math 28, 43–61 (2008). https://doi.org/10.1007/s10444-005-9010-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-005-9010-z
Keywords
- variational multiscale method
- finite element method
- error analysis
- incompressible Navier–Stokes equations