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

TL;DR

The paper develops the HWOPSIP method for the Poisson equation on anisotropic, convex domains, offering a simple, hybrid approach that maintains stability across penalty choices. It provides a rigorous anisotropic interpolation framework using a two-step affine mapping alongside CR and RT interpolants to bound consistency errors, and proves discrete Poincaré-type inequalities to ensure stability. Through Strang-type and duality arguments, it derives energy and $L^2$ error bounds, validated by numerical experiments that show optimal convergence on semi-regular anisotropic meshes without penalty tuning. The results advance robust finite element analysis on highly stretched meshes, with practical implications for problems exhibiting anisotropic solution features.

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.

Figures (7)

• Figure 1: Affine mapping $\Phi_{T}$ and vectors $r_i$, $i=1,2$
• Figure 2: (Type i) Vectors $r_i$, $i=1,2,3$
• Figure 3: (Type ii) Vectors $r_i$, $i=1,2,3$
• Figure 4: (I) Standard mesh
• Figure 5: (II) Shishkin mesh, $\delta = \frac{1}{128}$