A mixed finite element method for thin film epitaxy

We present a mixed finite element method for the thin film epitaxy problem. Comparing to the primal formulation which requires $C^2$ elements in the discretization, the mixed formulation only needs to use $C^1$ elements, by introducing proper dual variables. The dual variable in our method is defined naturally from the nonlinear term in the equation, and its accurate approximation will be essential for understanding the long-time effect of the nonlinear term. For time-discretization, we use a backward-Euler semi-implicit scheme, which involves a convex–concave decomposition of the nonlinear term. The scheme is proved to be unconditionally stable and its convergence rate is analyzed.