论文标题
多边形上的差异性重建和Stokes问题的真正压力稳定虚拟元素方法
Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
论文作者
论文摘要
不可压缩的Stokes问题的无差异离散性可能会遭受缺乏压力的特征，其特征是在动量平衡中由于无关力而导致的较大离散误差。本文认为，只要右侧不以仔细的方式离散，多边形网格上的无差异虚拟元素方法（VEM）并不是真正的压力。为了能够评估测试功能的右侧，需要对虚拟测试功能进行一些明确的插值，这些插值在任何地方都可以在任何地方进行评估。通过$ l^2 $ - 最佳近似的标准离散不能保留差异，因此破坏了无差异测试功能与右侧可能是杰出的梯度力之间的正交性。为了修复这种正交性并恢复压力量，建议基于对多边形的局部亚三角态的Raviart-Thomas近似，建议了另一种具有差异的重建。所有发现在理论上均得到证明，并在两个维度上进行了数值证明。该结构对于多面部或多面体网格的混合高阶方法也很有趣。
Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the test functions, some explicit interpolation of the virtual test functions is needed that can be evaluated pointwise everywhere. The standard discretisation via an $L^2$-best approximation does not preserve the divergence and so destroys the orthogonality between divergence-free test functions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart-Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.
