We present a methodology for efficient micromagnetics simulation. The method combines an unconditionally stable, finite differences scheme with an adaptive mesh refinement technique. It enhances accuracy by covering locally special regions of the domain with a sequence of nested, progressively finer rectangular grid patches that dynamically follow sharp transitions of the magnetization field (e.g., walls and vortices). To illustrate our approach, we consider a rectangular sample of infinite thickness with strong anisotropy in the out-of-plane direction