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.

翻译：我们建议、分析和测试一个新的适应性惩罚计划,该计划将惩罚参数 $\ epsilon$ 的元素按小元素选择, 在$\nabla\cdot u ⁇ h$是大的地方。 我们首先在最简单但有趣的环境, Stokes 问题上分析和测试新计划。 最后, 我们扩展并测试关于复杂流量问题的无法压缩的 Navier Stokes 方程式的算法。 测试表明, 新的适应性- $\ epsilon$ 惩罚方法算法精确地预测了流动行为。 该计划是在惩罚方法中开发的, 但也可用于选择 grad- div 稳定参数 。