## Abstract

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 L^{2} 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.

Original language | English |
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Pages (from-to) | 1013-1037 |

Number of pages | 25 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 48 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sept 2010 |

Externally published | Yes |

## Keywords

- Absorbing set
- Global attractor
- Long-term stability
- Navier-Stokes problem
- Stabilized finite element methods
- Subgrid scales