Robust A Posteriori Error Estimates for Stationary Convection-Diffusion Equations

Abstract

We analyze a posteriori error estimators for finite element discretizations of convection-dominated stationary convection-diffusion equations using locally refined, isotropic meshes. The estimators are based on either the evaluation of local residuals or the solution of discrete local problems with Dirichlet or Neumann boundary conditions. All estimators yield global upper and lower bounds for the error measured in a norm that incorporates the standard energy norm and a dual norm of the convective derivative. They are fully robust in the sense that the ratio of the upper and lower bounds is uniformly bounded with respect to the size of the convection. The estimates are also uniform with respect to the size of the zero-order reaction term and also hold for the limit case of vanishing reaction.