  We prove residual-type a posteriori error estimates in the maximum norm for a
linear scalar elliptic convection-diffusion problem that may be singularly
perturbed. Similar error analysis in the energy norm by Verf\"{u}rth indicates
that a dual norm of the {convective derivative of the} error must be added to
the natural energy norm in order for the natural residual estimator to be
reliable and efficient. We show that the situation is similar for the maximum
norm. In particular, we define a mesh-dependent weighted seminorm of the
convective error which functions as a maximum-norm counterpart to the dual norm
used in the energy norm setting. The total error is then defined as the sum of
this seminorm, the maximum norm of the error, and data oscillation. The natural
maximum norm residual error estimator is shown to be equivalent to this total
error notion, with constant independent of singular perturbation parameters.
These estimates are proved under the assumption that certain natural estimates
hold for the Green's function for the problem at hand. Numerical experiments
confirm that our estimators effectively capture the maximum-norm error behavior
for singularly perturbed problems, and can effectively drive adaptive
refinement in order to capture layer phenomena.
