Solving incompressible Navier-Stokes equations: A nonlinear multiscale approach

In this work, we present a nonlinear variational multiscale finite element method for solving both stationary and transient incompressible Navier-Stokes equations. The method is founded on a two-level decomposition of the approximation space, where a nonlinear artificial viscosity operator is exclusively added to the unresolved scales. It can be regarded as a self-adaptive method, since the amount of subgrid viscosity is automatically introduced according to the residual of the equation, in its strong form, associated with the resolved scales. Two variants for the subgrid viscosity are presented: one considering only the residual of the momentum equation and the other also incorporating the residual of the conservation of mass. To alleviate the computational cost typical of two-scale methods, the microscale space is defined through polynomial functions that vanish on the boundary of the elements, known as bubble functions. We compared the numerical and computational performance of the method with the results obtained by the Streamline-Upwind/Petrov-Galerkin (SUPG) formulation combined with the Pressure Stabilizing/Petrov-Galerkin (PSPG) method through a set of 2D reference problems.