On Adaptive Grad-Div Parameter Selection

TL;DR

This paper introduces an adaptive penalty scheme for selecting the grad-div parameter in fluid flow simulations, improving accuracy in complex flow problems by adjusting the penalty locally based on divergence magnitude.

Contribution

The paper presents a novel adaptive penalty method for grad-div parameter selection, extending it from the Stokes problem to complex Navier-Stokes flows.

Findings

Accurately predicts flow behavior in complex simulations

Effective adaptive penalty parameter selection

Applicable to both penalty method and grad-div stabilization

Abstract

We propose, analyze and test a new adaptive penalty scheme that picks the penalty parameter $\epsilon$ element by element small where $\nabla\cdot u^h$ is large. We start by analyzing and testing the new scheme on the most simple but interesting setting, the Stokes problem. Finally, we extend and test the algorithm on the incompressible Navier Stokes equation on complex flow problems. Tests indicate that the new adaptive-$\epsilon$ penalty method algorithm predicts flow behavior accurately. The scheme is developed in the penalty method but also can be used to pick a grad-div stabilization parameter.