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

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.