  Adaptive Finite Element Method (adaptivity) is known to be an effective
numerical tool for some ill-posed problems. The key advantage of the adaptivity
is the image improvement with local mesh refinements. A rigorous proof of this
property is the central part of this paper. In terms of Coefficient Inverse
Problems with single measurement data, the authors consider the adaptivity as
the second stage of a two-stage numerical procedure. The first stage delivers a
good approximation of the exact coefficient without an advanced knowledge of a
small neighborhood of that coefficient. This is a necessary element for the
adaptivity to start iterations from. Numerical results for the two-stage
procedure are presented for both computationally simulated and experimental
data.
