## A finite element variational multiscale method for incompressible flows based on two local Gauss integrations.(English)Zbl 1168.76028

Summary: 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 stabilization parameter $$\alpha$$. 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 Taylor-Hood elements, and show that the nonlinear system derived from our method by finite element discretization is computationally much smaller than that of common VMS method. Additionally, we present numerical simulations to demonstrate the effectiveness, storage, and computational complexity of our method. Finally, we give some numerical simulations of nonlinear flow problems to show good stability and accuracy of the method.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids

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