Numerical investigation of the reliability of a posteriori error estimation for advection-diffusion equations

A numerical investigation of the reliability of a posteriori error estimation for advection-diffusion equations is presented. The estimator used is based on the solution of local problems subjected to Neumann boundary conditions. The estimated errors are calculated in a weighted energy norm, a stability norm and an approximate fractional order norm in order to study the effect of the error norm on both the effectivity index of the estimated errors and the mesh adaptivity process. The reported numerical results are in general better than what is available in the literature. The results reveal that the reliability of the estimated errors depends on the relation between the mesh size and the size of local features in the solution. The stability norm is found to have some advantages over the weighted energy norm in terms of producing effectivity indices closer to the optimal unit value, especially for problems with internal sharp layers. Meshes adapted by the element residual method measured in the stability norm conform to the sharp layers and are shown to be less dependent on the wind direction.