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

The adaptivity refines approximate solutions of ill-posed problems due to the relaxation property