A finite element variational multiscale method for incompressible flows based on the construction of the projection basis functions

Abstract

SUMMARY In this article, we present a finite element variational multiscale (VMS) method for incompressible flows based on the construction of projection basis functions and compare it with common VMS method, which is defined by a low‐order finite element space Lh on the same grid as Xh for the velocity deformation tensor and a stabilization parameter α. The best algorithmic feature of our method is to construct the projection basis functions at the element level with minimal additional cost to replace the global projection operator. Finally, we give some numerical simulations of the nonlinear flow problems to show good stability and accuracy properties of the method. Copyright © 2011 John Wiley & Sons, Ltd.

A Multilevel Finite Element Variational Multiscale Method for Incompressible Navier‐Stokes Equations Based on Two Local Gauss Integrations

“…Then the approximate solution ( ℎ 2 , ℎ 2 ) ∈ ( ℎ 2 , ℎ 2 ) given by the multilevel finite element method (23)…”

“…1 (Ω) ∩ ) be a nonsingular solution of (6). If ℎ > 0 is sufficient small, then the approximate solution ( ℎ , ℎ ) ∈ ( ℎ , ℎ ) given by the multilevel finite element method (23) satisfies…”

“…Under the circumstance, it is necessary to define an appropriate operator, function spaces, and some additional dependent variables. Recently, a finite element VMS method based on two local Gauss integrations was proposed and developed by the authors of [21][22][23][24][25]. This finite element VMS method is simpler than the common finite element VMS method, 2 Mathematical Problems in Engineering which does not need additional dependent variables and the projection operator but keeps the same efficiency.…”