Solving the problem of fast eruptive events in magnetically dominated astrophysical plasmas requires the use of particularly well adapted numerical tools. Indeed, the central mechanism based on magnetic reconnection is determined by a complex behavior with quasi-singular forming current layers enriched by their associated small-scale magnetic islands called plasmoids. A new code is thus presented for the solution of two-dimensional dissipative magnetohydrodynamics (MHD) equations in cartesian geometry specifically developed to this end. A current–vorticity formulation representative of an incompressible model is chosen in order to follow the formation of the current sheets and the ensuing magnetic reconnection process. A finite-element discretization using triangles with quadratic basis functions on an unstructured grid is employed, and implemented via a highly adaptive characteristic-Galerkin scheme. The adaptivity of the code is illustrated on simplified test equations and finally for magnetic reconnection associated with the nonlinear development of the tilt instability between two repelling current channels. Varying the Lundquist number S has allowed us to study the transition between the steady-state Sweet–Parker reconnection regime (for S ≲ 104) and the plasmoid-dominated reconnection regime (for S ≳ 105). The implications for the understanding of the mechanism explaining the fast conversion of free magnetic energy in astrophysical environments such as the solar corona are briefly discussed.