Computational Mechanics

, Volume 44, Issue 2, pp 145–160

A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations

Original Paper

DOI: 10.1007/s00466-008-0362-3

Cite this article as:
Masud, A. & Calderer, R. Comput Mech (2009) 44: 145. doi:10.1007/s00466-008-0362-3


This paper presents a variational multiscale residual-based stabilized finite element method for the incompressible Navier–Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation furnished by the variational multiscale framework. A significant feature of the new method is that the fine scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the advection part of the stabilization tensor. The new method circumvents the Babuska–Brezzi (inf–sup) condition and yields a stable formulation for high Reynolds number flows. A family of equal-order pressure-velocity elements comprising 4-and 10-node tetrahedral elements and 8- and 27-node hexahedral elements is developed. Convergence rates are reported and accuracy properties of the method are presented via the lid-driven cavity flow problem.


Multiscale finite element methods Navier–Stokes equations Convergence rates Equal order interpolation functions Tetrahedral elements Hexahedral elements 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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