Variational multiscale modeling with discretely divergence-free subscales

Abstract

We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with H 1 -conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier-Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence.

Variational multiscale modeling with discretely divergence-free subscales: Non-divergence-conforming discretizations

“…Our point of departure in the present work is [1], which proposed a new stabilized finite element formulation for the incompressible Navier-Stokes equations. The novelty of this formulation was its ability to combine residual-based stabilization of advection with energetic stability and satisfaction of a discrete incompressibility condition for a discrete velocity solution u h , i.e.,…”

“…where Ω is the problem domain and Q h is the discrete pressure space. The basic Galerkin discretization of incompressible flow obviously satisfies (1), but is unstable for the high Reynolds numbers ubiquitous in applications. Widely-used residual-based stabilized methods such as Galerkin least squares (GLS) [2] or the combination of streamline/upwind-Petrov-Galerkin (SUPG) [3] and pressure-stabilizing Petrov-Galerkin (PSPG) [4] typically fail to satisfy (1), due to an inner product of the strong problem's momentum-balance residual with the gradient of the pressure test function.…”

“…The basic Galerkin discretization of incompressible flow obviously satisfies (1), but is unstable for the high Reynolds numbers ubiquitous in applications. Widely-used residual-based stabilized methods such as Galerkin least squares (GLS) [2] or the combination of streamline/upwind-Petrov-Galerkin (SUPG) [3] and pressure-stabilizing Petrov-Galerkin (PSPG) [4] typically fail to satisfy (1), due to an inner product of the strong problem's momentum-balance residual with the gradient of the pressure test function. The method of [1] circumvents this difficulty through a novel application of the variational multiscale (VMS) [5][6][7][8][9] concept: a second discrete pressure field is introduced, formally taking the place of the fine-scale pressure in a VMS scale separation.…”

“…Widely-used residual-based stabilized methods such as Galerkin least squares (GLS) [2] or the combination of streamline/upwind-Petrov-Galerkin (SUPG) [3] and pressure-stabilizing Petrov-Galerkin (PSPG) [4] typically fail to satisfy (1), due to an inner product of the strong problem's momentum-balance residual with the gradient of the pressure test function. The method of [1] circumvents this difficulty through a novel application of the variational multiscale (VMS) [5][6][7][8][9] concept: a second discrete pressure field is introduced, formally taking the place of the fine-scale pressure in a VMS scale separation. The analysis of [1] showed that the proposed method satisfied energetic stability [1, Lemma 2], but only proved convergence of a simplified method for the Oseen equations-a linearized model of Navier-Stokes flow-under the restrictive assumption of a divergence-conforming discretization, i.e.,…”

“…The method of [1] circumvents this difficulty through a novel application of the variational multiscale (VMS) [5][6][7][8][9] concept: a second discrete pressure field is introduced, formally taking the place of the fine-scale pressure in a VMS scale separation. The analysis of [1] showed that the proposed method satisfied energetic stability [1, Lemma 2], but only proved convergence of a simplified method for the Oseen equations-a linearized model of Navier-Stokes flow-under the restrictive assumption of a divergence-conforming discretization, i.e.,…”