Sharp $L^\infty$ estimates of HDG methods for Poisson equation II: 3D

TL;DR

This paper extends sharp $L^ Infty$ estimates for HDG methods solving Poisson's equation from 2D to 3D, eliminating logarithmic factors and confirming results with numerical experiments.

Contribution

It provides the first sharp $L^ Infty$ estimates for HDG methods in 3D, improving upon previous 2D-only results.

Findings

Sharp $L^ Infty$ estimates achieved in 3D

Logarithmic factors eliminated in estimates

Numerical experiments confirm theoretical results

Abstract

In [SIAM J. Numer. Anal., 59 (2), 720-745], we proved quasi-optimal $L^\infty$ estimates (up to logarithmic factors) for the solution of Poisson's equation by a hybridizable discontinuous Galerkin (HDG) method. However, the estimates only work in 2D. In this paper, we obtain sharp (without logarithmic factors) $L^\infty$ estimates for the HDG method in both 2D and 3D. Numerical experiments are presented to confirm our theoretical result.