Optimal rate convergence analysis of a second order scheme for a thin film model with slope selection

Abstract In this paper, an energy stable, second-order mixed finite element scheme is proposed and analyzed for the thin film epitaxial growth model with slope selection. We employ second-order backward differentiation formula (BDF) scheme with a second-order stabilized term, which guarantees the long time energy stability to approximate the continuous model. In terms of the convergence analysis, the key difficulty to derive an optimal rate spatial estimate is associated with the appearance of the gradient operator in the nonlinear terms, which may lead to a loss of optimal accuracy order. To overcome this well-known difficulty, we make use of some auxiliary techniques over triangular elements, and obtain an optimal convergence rate O ( h q + 1 + Δ t 2 ) , in comparison with O ( h q + Δ t 2 ) rate from a standard projection estimate. Furthermore, we use an efficient preconditioned steepest descent (PSD) solver for the numerical implementation. A few numerical examples are presented to validate the stability and convergence.