In this study, we investigate an anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation {on convex domains}. Our approach is a simple discontinuous Galerkin method similar to the Crouzeix--Raviart finite element method. As our primary contribution, we show a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart--Thomas finite element space and a discontinuous space. While inf-sup stable schemes of the discontinuous Galerkin method on shape-regular mesh partitions have been widely discussed, our results show that the Stokes element satisfies the inf-sup condition on anisotropic meshes. Furthermore, we also provide an error estimate in an energy norm on anisotropic meshes. In numerical experiments, we compare calculation results for standard and anisotropic mesh partitions.

翻译：本研究提出一种各向异性弱过罚对称内部惩罚方法，用于求解凸区域上的Stokes方程。该方法是类似于Crouzeix-Raviart有限元法的简单间断Galerkin方法。作为主要贡献，我们提出了一致性项的新证明，从而获得各向异性一致性误差的估计。该证明的关键思想在于应用Raviart-Thomas有限元空间与间断空间之间的关系。尽管在形状正则网格划分上，间断Galerkin方法的inf-sup稳定格式已被广泛讨论，但我们的结果表明，Stokes单元在各向异性网格上仍满足inf-sup条件。此外，我们还提供了各向异性网格上能量范数的误差估计。在数值实验中，我们对标准网格划分和各向异性网格划分的计算结果进行了比较。