A posteriori error analysis of a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations

Abstract: We introduce a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The proposed method is based on a coarse grid and iteratively improves the solution's accuracy by solving local elliptic problems in refined subdomains. For purely diffusion problems, we already proved that this scheme converges under minimal regularity assumptions [A. Abdulle and G.Rosilho de Souza, ESAIM: M2AN, 53(4):1269--1303, 2019]. In this paper, we provide an algorithm for the automatic identification of the local elliptic problems' subdomains employing a flux reconstruction strategy. Reliable error estimators are derived for the local adaptive method. Numerical comparisons with a classical nonlocal adaptive algorithm illustrate the efficiency of the method.