Some Numerical Results on the Finite-Element Approximation of the Navier-Stokes Equations Using Dynamic Subscales

Abstract

In this work we analyze a particular stabilized finite-element method for the incompressible Navier-Stokes equations which is based on the decomposition of the fluid velocity into a resolvable finite element component plus a sub-grid scale component. The main characteristic is that the sub-grid scale component is modeled to be dynamic and $L^2$-orthogonal to the finite element space. We are particularly interested in understanding the purest mathematical aspects for this method such as long- and short-time stability and convergence under mild regularity of the problem data. Stability analysis provides a control of the sub-grid scale component in term of the energy norms which is translated into an extra control of the gradient of the pressure and the convective term for both short- and long-time behavior. These estimates are independent of all viscosity sizes. This benefit contributes to think that this method is a good candidate to model turbulence. Estimates of the Dirichlet norm for the velocity are also proved for two-dimensional domains giving arise to the existence of appropriate absorbing sets and a compact global attractor. Moreover, compactness results show that one can recover a weak solution of the NavierStokes equations under minimal requirements on the initial condition, forcing terms, and the domain. Finally, we investigate if such a solution is suitable in such a way it holds an entropy energy inequality.