Lower a posteriori error estimates on anisotropic meshes

Abstract: Lower a posteriori error bounds obtained using the standard bubble function approach are reviewed in the context of anisotropic meshes. A numerical example is given that clearly demonstrates that the short-edge jump residual terms in such bounds are not sharp. Hence, for linear finite element approximations of the Laplace equation in polygonal domains, a new approach is employed to obtain essentially sharper lower a posteriori error bounds and thus to show that the upper error estimator in the paper [N. Kopteva, Numer. Math., 137 (2017), 607-642] is efficient on certain anisotropic meshes.