Abstract
In this paper we present a second order accurate, energy stable numerical scheme for the epitaxial thin film model without slope selection, with a mixed finite element approximation in space. In particular, an explicit treatment of the nonlinear term, , greatly simplifies the computational effort; only one linear equation with constant coefficients needs to be solved at each time step. Meanwhile, a second order Douglas-Dupont regularization term, , is added in the numerical scheme, so that an unconditional long time energy stability is assured. In turn, we perform an convergence analysis for the proposed scheme, with an error estimate derived. In addition, an optimal convergence analysis is provided for the nonlinear term using finite elements, which shows that the spatial convergence order can be improved to on regular rectangular mesh. A few numerical experiments are presented, which confirms the efficiency and accuracy of the proposed second order numerical scheme.