Hybrid weakly over-penalised symmetric interior penalty method on anisotropic meshes

TL;DR

This paper introduces a new hybrid weakly over-penalised symmetric interior penalty method for solving the Poisson equation on anisotropic meshes, demonstrating its simplicity, consistency, and effectiveness through theoretical analysis and numerical experiments.

Contribution

It proposes a novel hybrid scheme for anisotropic meshes and provides a proof of its consistency, enhancing the understanding of anisotropic finite element methods.

Findings

The new scheme is simpler and easier to implement than existing methods.

The proof of consistency allows for better error estimation on anisotropic meshes.

Numerical results show improved performance on anisotropic mesh partitions.

Abstract

In this study, we investigate a hybrid-type anisotropic weakly over-penalised symmetric interior penalty method for the Poisson equation on convex domains. Compared with the well-known hybrid discontinuous Galerkin methods, our approach is simple and easy to implement. Our primary contributions are the proposal of a new scheme and the demonstration of a proof for the consistency term, which allows us to estimate 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. In numerical experiments, we compare the calculation results for standard and anisotropic mesh partitions.