An extra-dof-free generalized finite element method for incompressible Navier-Stokes equations

The generalized finite element method (GFEM) without extra degrees of freedom (dof) is extended to solve incompressible Navier-Stokes (N-S) equations. Unlike the existing extra-dof-free GFEM, we propose a new approach to construct the nodal enrichments based on the weighted least-squares. As a result, the essential boundary conditions can be imposed more accurately. The Characteristic-Based Split (CBS) scheme is used to suppress oscillations due to the standard Galerkin discretization of the convective terms, and the pressure is further stabilized by the finite increment calculus (FIC) formulation. Hence, equal velocity-pressure interpolation and the incremental version of the split scheme can be used without inducing spurious oscillations. The developed extra-dof-free GFEM is very flexible and can achieve high-order spatial accuracy and convergence rates by adopting high-order polynomial enrichments. In particular, better accuracy could be obtained with special enrichments reflecting a-priori knowledge about the solution. This is demonstrated by numerical results. Benchmark examples such as the Lid-Driven Cavity flow and the flow past a circular cylinder are also presented to further verify the effectiveness of the proposed extra-dof-free GFEM for incompressible flow.