Computer Methods in Applied Mechanics and Engineering
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Computer Methods in Applied Mechanics and Engineering - Special edition on the 20th Anniversary
A nonlinear Galerkin method for the Navier-Stokes equations
ICOSAHOM '89 Proceedings of the conference on Spectral and high order methods for partial differential equations
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Solution of the incompressible Navier-Stokes equations by the nonlinear Galerkin method
Journal of Scientific Computing
Attractors and Error Estimates for Discretizations of Incompressible Navier--Stokes Equations
SIAM Journal on Numerical Analysis
Robust A Posteriori Error Estimates for Stationary Convection-Diffusion Equations
SIAM Journal on Numerical Analysis
Applied Numerical Mathematics
SIAM Journal on Numerical Analysis
Unified Stabilized Finite Element Formulations for the Stokes and the Darcy Problems
SIAM Journal on Numerical Analysis
Approximation of the inductionless MHD problem using a stabilized finite element method
Journal of Computational Physics
A finite element dynamical nonlinear subscale approximation for the low Mach number flow equations
Journal of Computational Physics
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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 $L^2$ 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.