A posteriori upper and lower bounds are derived for the linear finite element method (FEM) Helmholtz equation with large wavenumber. It is proved rigorously that standard residual type error estimator seriously underestimates true of FE solution mesh size h in preasymptotic regime, which first observed by Babuška et al. (Int J Numer Methods Eng 40:3443–3462, 1997) a one dimensional problem. By ...