The reliability of local error estimators for convection-diffusion equations

We assess the reliability of a simple a posteriori error estimator for steady-state convection-diffusion equations in cases where convection dominates. Our estimator is computed by solving a local Poisson problem with Neumann boundary conditions. It gives global upper and local lower bounds on the error measured in the H1 semi-norm. However, the error may be overestimated locally within boundary layers if these are not resolved by the mesh, that is, when the local mesh Péclet number is significantly greater than unity. We discuss the implications of this overestimation in a practical context where the estimator is used as a local error indicator within a self-adaptive mesh refinement process.