An anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation

We investigate an anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation. The method is one of the discontinuous Galerkin methods, simple and similar to the Crouzeix--Raviart finite element method. The main contributions of this paper are to show new proof for the consistency term. It allows us to obtain an anisotropic consistency error estimate. The idea of the proof is to use the relation between the Raviart--Thomas finite element space and the discontinuous space. In many papers, the inf-sup stable schemes of the discontinuous Galerkin method have been discussed on shape-regular mesh partitions. Our result shows that the Stokes pair, which is treated in this paper, satisfies the inf-sup condition on anisotropic meshes. Furthermore, we show an error estimate in an energy norm on anisotropic meshes. In numerical experiments, we have compared the calculation results for standard and anisotropic mesh partitions. The effectiveness of using anisotropic meshes can be confirmed for problems with boundary layers.