  The process of fast magnetic reconnection supported by the formation of
plasmoid chains in the high Lundquist number ($S$) regime is investigated using
a recently developed adaptive finite-element magnetohydrodynamic (MHD) code. We
employ a two-dimensional incompressible model with a set of reduced
visco-resistive MHD equations. The tilt instability setup is chosen to provide
a three-step mechanism, where two curved current sheets initially form on an
Alfv\'enic time scale followed by a second phase of super-Alfv\'enic growth of
plasmoid chains for $S \ge S_c$ (Baty 2020). A third phase is reached where an
ensuing stochastic time-dependent reconnection regime with a fast time-averaged
rate independent of $S$ is obtained. We reveal the multi-scale current
structures during magnetic reconnection, where merging events of plasmoids give
rise to monster plasmoids with shocks bounding the outflow regions. At high
enough $S$ values (typically for $S \sim 100 S_c$), a dynamical Petschek-type
reconnection is achieved with pairs of slow-mode shocks emanating from a small
central region containing a few plasmoids. Finally, we briefly discuss the
relevance of our results to explain the flaring activity in solar corona and
internal disruptions in tokamaks.
