A multiscale finite element method for the incompressible Navier-Stokes equations

A. Masud*, Rooh Ul Amin Khurram

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    115 Scopus citations


    This paper presents a new multiscale finite element method for the incompressible Navier-Stokes equations. The proposed method arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. Modeling of the unresolved scales corrects the lack of stability of the standard Galerkin formulation and yields a method that possesses superior properties like that of the streamline upwind/Petrov-Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The multiscale method allows arbitrary combinations of interpolation functions for the velocity and the pressure fields, specifically the equal order interpolations that are easy to implement but violate the celebrated Babuska-Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor τ appears naturally via the solution of the fine-scale problem. A family of 2-D elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed. Convergence studies for the method on uniform, skewed as well as composite meshes are presented. Numerical simulations of the nonlinear steady and transient flow problems are shown that exhibit the good stability and accuracy properties of the method.

    Original languageEnglish (US)
    Pages (from-to)1750-1777
    Number of pages28
    JournalComputer Methods in Applied Mechanics and Engineering
    Issue number13-16
    StatePublished - Feb 15 2006


    • Arbitrary pressure-velocity interpolations
    • HVM formulation
    • Multiscale finite elements
    • Navier-Stokes equations
    • Stabilized methods

    ASJC Scopus subject areas

    • Computational Mechanics
    • Mechanics of Materials
    • Mechanical Engineering
    • Physics and Astronomy(all)
    • Computer Science Applications


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