Improved error estimates of hybridizable interior penalty methods using a variable penalty for highly anisotropic diffusion problems

TL;DR

This paper improves error estimates for hybridizable interior penalty methods applied to anisotropic diffusion problems by introducing a variable penalty function, enhancing convergence analysis and supported by numerical validation.

Contribution

It introduces a variable penalty function in hybridizable interior penalty methods, leading to sharper error estimates for anisotropic diffusion problems.

Findings

Enhanced a priori error estimates derived for the methods.

The variable penalty impacts convergence, consistency, coercivity, and boundedness.

Numerical experiments confirm theoretical improvements.

Abstract

In this paper, we derive improved a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a variable penalty for second-order elliptic problems. The strategy is to use a penalization function of the form $\mathcal{O}(1/h^{1+\delta})$, where $h$ denotes the mesh size and $\delta$ is a user-dependent parameter. We then quantify its direct impact on the convergence analysis, namely, the (strong) consistency, discrete coercivity, and boundedness (with $h^{\delta}$-dependency), and we derive updated error estimates for both discrete energy- and $L^{2}$-norms. The originality of the error analysis relies specifically on the use of conforming interpolants of the exact solution. All theoretical results are supported by numerical evidence.