A finite element variational multiscale method for incompressible flows based on two local gauss integrations

In this article, we present a finite element variational multiscale (VMS) method for incompressible flows based on two local Gauss integrations, and compare it with common VMS method which is defined by a low order finite element space L_h on the same grid as X_h for the velocity deformation tensor and a stabilization parameter α. The best algorithmic feature of our method is using two local Gauss integrations to replace projection operator. We theoretically discuss the relationship between our method and common VMS method for the Taylor-Hood elements, and show that the nonlinear system derived from our method by finite element discretization is much smaller than that of common VMS method computationally. Additionally we present numerical simulations to demonstrate the effectiveness, storage, computational complexity of our method. Finally, we give some numerical simulations of the nonlinear flow problems to show good stability and accuracy properties of the method.