In this study, we investigate an anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation. 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, and the results show the effectiveness of using anisotropic meshes for problems with boundary layers.

翻译：本研究探讨了用于Stokes方程的各向异性弱过惩罚对称内罚法。该方法是一种类似于Crouzeix-Raviart有限元法的简单间断Galerkin方法。作为主要贡献，我们提出了关于一致性项的新证明，从而能够估计各向异性一致性误差。该证明的关键思想是运用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剖分上间断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，我们还给出了各向异性网格上能量范数的误差估计。在数值实验中，我们比较了标准网格与各向异性网格剖分的计算结果，结果表明在处理含边界层的问题时，采用各向异性网格具有有效性。