GRAWITA: VLT Survey Telescope observations of the gravitational wave sources GW150914 and GW151226

We report the results of deep optical follow-up surveys of the first two gravitational-wave sources, GW150914 and GW151226, done by the GRAvitational Wave Inaf TeAm Collaboration (GRAWITA). The VLT Survey Telescope (VST) responded promptly to the gravitational-wave alerts sent by the LIGO and Virgo Collaborations, monitoring a region of $90$ deg$^2$ and $72$ deg$^2$ for GW150914 and GW151226, respectively, and repeated the observations over nearly two months. Both surveys reached an average limiting magnitude of about 21 in the $r-$band. The paper describes the VST observational strategy and two independent procedures developed to search for transient counterpart candidates in multi-epoch VST images. Several transients have been discovered but no candidates are recognized to be related to the gravitational-wave events. Interestingly, among many contaminant supernovae, we find a possible correlation between the supernova VSTJ57.77559-59.13990 and GRB150827A detected by {\it Fermi}-GBM. The detection efficiency of VST observations for different types of electromagnetic counterparts of gravitational-wave events are evaluated for the present and future follow-up surveys.