A diffuse-interface approximation for solving partial differential equations on evolving surfaces was described. The model of interest was a fourth-order geometrical evolution equation, for a growing surface, with an additional diffusive adatom density on the surface. Such models arose in the description of epitaxial growth, where the surface of interest was a solid/vapour interface. The model permitted the handling of complex geometries in an implicit manner by considering an evolution equation for a phase-field variable that described the surface, and an evolution equation for an extended adatom concentration on a time-independent domain. Matched asymptotic analysis revealed a formal convergence towards the sharp interface model and numerical results, based upon adaptive finite elements, demonstrated the value of the approach.

A Diffuse-Interface Approximation for Surface Diffusion Including Adatoms. A.Rätz, A.Voigt: Nonlinearity, 2007, 20[1], 177-92