Anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation
Hiroki Ishizaka
TL;DR
This work develops and analyzes an anisotropic WOPSIP scheme for the Stokes equations, incorporating a DG framework that mirrors the Crouzeix--Raviart element while leveraging Raviart--Thomas spaces to manage consistency on anisotropic meshes. It proves uniform inf-sup stability and derives energy-norm error estimates on semi-regular anisotropic meshes, along with a discrete Poincaré inequality under convexity. A novel consistency-term analysis using RT-discontinuous-space relations yields sharp anisotropic error bounds, and numerical experiments demonstrate robust performance on anisotropic and boundary-layer problems, highlighting the method’s stability without penalty tuning. The results extend WOPSIP applicability to non-shape-regular meshes and provide practical guidance on penalty parameters and mesh design for Stokes-type problems.
Abstract
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 provide an error estimate in an energy norm on anisotropic meshes. In numerical experiments, we compare calculation results for standard and anisotropic mesh partitions.