Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations

We consider variational multiscale (VMS) methods with h-adaptive technique for the stationary incompressible Navier-Stokes equations. The natural combination of VMS with adaptive strategy retains the best features of both methods and overcomes many of their deficits. A reliable a posteriori projection error estimator is derived, which can be computed by two local Gauss integrations at the element level. Finally, some numerical tests are presented to illustrate the method's efficiency.