A Multiscale Finite Element Formulation for the Incompressible Navier-Stokes Equations

In this work we present a variational multiscale finite element method for solving the incompressible Navier-Stokes equations. The method is based on a two-level decomposition of the approximation space and consists of adding a residual-based nonlinear operator to the enriched Galerkin formulation, following a similar strategy of the method presented in [1,2] for scalar advection-diffusion equation. The artificial viscosity acts adaptively only onto the unresolved mesh scales of the discretization. In order to reduce the computational cost typical of two-scale methods, the subgrid scale space is defined using bubble functions whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Accuracy comparisons with the streamline-upwind/Petrov-Galerkin (SUPG) formulation combined with the pressure stabilizing/Petrov-Galerkin (PSPG) method are conducted based on 2D benchmark problems.