We present an adaptive finite element method for epitaxial island growth. The problem consists of an adatom diffusion equation on terraces of different height; boundary conditions on steps including the kinetic asymmetry in the adatom attachment and detachment; and the normal velocity law for the motion of the steps determined by a two-sided flux, together with diffusion of edge-adatoms along the steps. The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the boundary evolution. Some applications demonstrating the influence of the Ehrlich-Schwoebel and inverse Ehrlich-Schwoebel barrier and the anisotropy are shown. (C) 2004 Elsevier B.V. All rights reserved.