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

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