  We design an adaptive virtual element method (AVEM) of lowest order over
triangular meshes with hanging nodes in 2d, which are treated as polygons. AVEM
hinges on the stabilization-free a posteriori error estimators recently derived
in [8]. The crucial property, that also plays a central role in this paper, is
that the stabilization term can be made arbitrarily small relative to the a
posteriori error estimators upon increasing the stabilization parameter. Our
AVEM concatenates two modules, GALERKIN and DATA. The former deals with
piecewise constant data and is shown in [8] to be a contraction between
consecutive iterates. The latter approximates general data by piecewise
constants to a desired accuracy. AVEM is shown to be convergent and
quasi-optimal, in terms of error decay versus degrees of freedom, for solutions
and data belonging to appropriate approximation classes. Numerical experiments
illustrate the interplay between these two modules and provide computational
evidence of optimality.
