Finite element method for epitaxial growth with thermodynamic boundary conditions
- Bänsch, Eberhard
- Haußer, Frank
- Voigt, Axel
2010 Mathematics Subject Classification
- 35Q99 35R35 65N30 65Z05 74S05
- epitaxial growth, island dynamics, free or moving boundary problem, adatom diffusion, surface diffusion, mean curvature flow, Gibbs-Thomson, finite elements, adaptivity, front tracking
We develop an adaptive finite element method for island dynamics in epitaxial growth. We study a step-flow model, which consists of an adatom (adsorbed atom) diffusion equation on terraces of different height, thermodynamic boundary conditions on terrace boundaries including anisotropic line tension, and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional (possibly anisotropic) ``surface'' diffusion of edge-adatoms along the step-edges. 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. A penalty method is used in order to incorporate the boundary conditions. The evolution of the terrace boundaries includes both the weighted/anisotropic mean curvature flow and the weighted/anisotropic surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements.