© 2020, Springer-Verlag GmbH Germany, part of Springer Nature. 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 recent paper (Kopteva in Numer Math 137:607¿642, 2017) is efficient on partially structured anisotropic meshes.