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Variational multiscale methods lead to stable finite element approximations of theNavier–Stokes equations, dealing with both the indefinite nature of the system (pressure stability) andthe velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulationwith a subgrid component that is modeled. In fact, the effect of the subgrid scale on the capturedscales has been proved to dissipate the proper amount of energy needed to approximate the correctenergy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allowone to compute flows without the need to capture all the scales in the system. In this article, weconsider a dynamic subgrid model that enforces the subgrid component to be orthogonal to thefinite element space in the L2 sense. We analyze the long-term behavior of the algorithm, provingthe existence of appropriate absorbing sets and a compact global attractor. The improvementswith respect to a finite element Galerkin approximation are the long-term estimates for the subgridcomponent, which are translated to effective pressure and velocity stability. Thus, the stabilizationintroduced by the subgrid model into the finite element problem does not deteriorate for infinite timeintervals of computation

siam journal on numerical analysis

Long-Term Stability Estimates and Existence of a Global Attractor in a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling