  A posteriori upper and lower bounds are derived for the linear finite element
method (FEM) for the Helmholtz equation with large wave number. It is proved
rigorously that the standard residual type error estimator seriously
underestimates the true error of the FE solution for the mesh size $h$ in the
preasymptotic regime, which is first observed by Babu\v{s}ka, et al. for an one
dimensional problem. By establishing an equivalence relationship between the
error estimators for the FE solution and the corresponding elliptic projection
of the exact solution, an adaptive algorithm is proposed and its convergence
and quasi-optimality are proved under condition that $k^3h_0^{1+\al}$ is
sufficiently small, where $h_0$ is the initial mesh size and $\frac12<\al\le 1$
is a regularity constant depending on the maximum reentrant angle of the
domain. Numerical tests are given to verify the theoretical findings and to
show that the adaptive continuous interior penalty finite element method
(CIP-FEM) with appropriately selected penalty parameters can greatly reduce the
pollution error and hence the residual type error estimator for this CIP-FEM is
reliable and efficient even in the preasymptotic regime.
