Adaptive discontinuous Galerkin methods for nonstationary convection–diffusion problems

Abstract

This work is concerned with the derivation of a robust a posteriori error estimator for a discontinuous Galerkin (dG) method discretization of a linear nonstationary convection–diffusion initial/boundary value problem and with the implementation of a corresponding adaptive algorithm. More specifically, we derive a posteriori bounds for the error in the L²(H¹) + L^∞(L²)-type norm for an interior penalty dG discretization in space and a backward Euler discretization in time. Finally, an adaptive algorithm is proposed utilizing the error estimator. Optimal rate of convergence of the adaptive algorithm is observed in a number of test problems and for various Pèclet numbers.

discontinuous Galerkin, unsteady convection–diffusion, a posteriori error estimation, adaptive finite element methods