A Simple Error Estimate of Discontinuous Galerkin Methods for Elliptic Equations with Low Regularity

Abstract

In this work, we develop a low-regularity error analysis for the interior-penalty discontinuous Galerkin (IPDG) method, incorporating numerical fluxes originally proposed by Brezzi et al. [Numer. Methods Partial Differential Equations, 16 (2000)]. Our analysis specifically addresses elliptic problems with solutions residing in the low-regularity space $H^s,$ where $0 ≤ s < 1/2.$ Notably, our error estimates hold under two critical settings: discontinuous coefficients and general Lipschitz domains, precisely capturing the essential features of practical applications. We establish error estimates in the energy norm and the $L^2$-norm, providing a complete theoretical framework for the IPDG method in low-regularity limitation. To systematically verify the theoretical results, we conduct some numerical experiments incorporating precision-controlled parameters that directly correspond to the analytical model’s constraints.