Leo G. Rebholz

Large Eddy Simulation

At high Reynolds number the fluid velocity is exponentially sensitive to perturbations of the problem data. This sensitivity, however, is not uniform. The large structures (large eddies) evolve deterministically and are thus not sensitive [BFG02]. The small eddies are sensitive because they have a random character.

A Connection Between Scott-Vogelius and Grad-Div Stabilized Taylor-Hood FE Approximations of the Navier-Stokes Equations

This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier-Stokes equations using continuous velocity fields. With a particular mesh construction, the Scott-Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier-Stokes equations. We also prove that the limit of the grad-div stabilized Taylor-Hood solutions to the Navier-Stokes problem converges to the Scott-Vogelius solution as the stabilization parameter tends to infinity. That is, we provide theoretical justification that choosing the grad-div parameter large does not destroy the solution. Numerical tests are provided which verify the theory and show how both Scott-Vogelius and grad-div stabilized Taylor-Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier-Stokes approximations.

On the parameter choice in grad-div stabilization for the Stokes equations

Abstract Standard error analysis for grad-div stabilization of inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be 𝒪(1)

On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

\mathcal O(1)

Velocity-vorticity-helicity formulation and a solver for the Navier-Stokes equations

. This paper revisits this choice for the Stokes equations on the basis of minimizing the H1(Ω)

An Energy- and Helicity-Conserving Finite Element Scheme for the Navier-Stokes Equations

H^{1}(\Omega )

Efficient, Unconditionally Stable, and Optimally Accurate FE Algorithms for Approximate Deconvolution Models

error of the velocity and the L2(Ω)

ON THE HIGH ACCURACY NS-ALPHA-DECONVOLUTION TURBULENCE MODEL

L^{2}(\Omega )

On an Efficient Finite Element Method for Navier-Stokes-ω with Strong Mass Conservation

error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. In particular, the approximation property of the pointwise divergence-free subspace plays a key role. With such an optimal approximation property and with an appropriate choice of the stabilization parameter, estimates for the H1(Ω)

On conservation laws of Navier–Stokes Galerkin discretizations

H^{1}(\Omega )