Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes

Summary. Both for the $H^1$- and $L^2$-norms, we prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes. We also show that, with a correct scaling, edge residuals yield a robust error estimator for a singularly perturbed reaction-diffusion equation.