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首页> 外文期刊>Computer Methods in Applied Mechanics and Engineering >The analogue of grad-div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes
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The analogue of grad-div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes

机译：不可压缩流的DG方法中的div-div稳定化的类似方法：限制行为和张量积网格的扩展

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摘要

grad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad-div stabilization for Discontinuous Galerkin methods is studied. Here, the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Our contribution is twofold: first, we characterize the limit for arbitrarily large penalization parameters, which shows that the stabilized nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit; second, we extend these ideas to the case of non-simplicial meshes; here, broken grad-div stabilization must be used in addition to the normal velocity jump penalization, in order to get the desired pressure robustness effect. The analysis is performed for the Stokes equations, and more complex flows and Crouzeix-Raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings. (C) 2018 Elsevier B.V. All rights reserved.
机译：grad-div稳定是一种经典的方法，可以解决不可压缩的流动问题，采用混合有限元方法来减轻有时被称为不良质量守恒的速度误差。这种误差是由于经典混合方法中散度约束的放松而产生的，每当空间离散化必须处理相对较大和复杂的压力时，这些误差就会被激发。在此贡献中，研究了间断Galerkin方法的grad-div稳定化类似物。在这里，关键是对三角剖分面上法向速度的跳跃进行惩罚，它控制离散速度解的分布散度的量度值部分。我们的贡献是双重的：首先，我们描述了任意大的惩罚参数的极限，这表明稳定的非一致性不连续Galerkin方法在此极限下仍保持稳健和准确。其次，我们将这些思想扩展到非简单网格的情况。在这里，除了正常速度跳变罚分之外，还必须使用破碎的div-div稳定性，以便获得所需的压力鲁棒性效果。对Stokes方程进行了分析，并在数值示例中考虑了更复杂的流和Crouzeix-Raviart元素，这些示例也显示了该理论在实际环境中的相关性。 （C）2018 Elsevier B.V.保留所有权利。

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