  The present paper extends the theory of Adaptive Virtual Element Methods
(AVEMs) to the three-dimensional meshes showing the possibility to bound the
stabilization term by the residual-type error estimator. This new bound enables
a stabilization-free a posteriori control for the energy error. Following the
recent studies for the bi-dimensional case, we investigate the case of
tetrahedral elements with aligned edges and faces. We believe that the AVEMs
can be an efficient strategy to address the mesh conforming requirements of
standard three-dimensional Adaptive Finite Element Methods (AFEMs), which
typically extend the refinement procedure to non-marked mesh cells. Indeed,
numerical tests on the Fichera corner shape domain show that this method can
reduce the number of three-dimensional cells generated in the refinement
process by about 30% with compared to standard AFEMs, for a given error
threshold.
