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

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.