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

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.

11 pages, 7 figures, 1 table