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

Arif Masud, Ramon Calderer

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)145-160
Number of pages16
JournalComputational Mechanics
Issue number2
StatePublished - Jul 2009


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

ASJC Scopus subject areas

  • Computational Mechanics
  • Ocean Engineering
  • Mechanical Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

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