A Second Order Energy Stable Linear Scheme for a Thin Film Model Without Slope Selection.

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, ∇u1+|∇u|2<inline-graphic></inline-graphic>, 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, AτΔ2(un+1-un)<inline-graphic></inline-graphic>, is added in the numerical scheme, so that an unconditional long time energy stability is assured. In turn, we perform an ℓ∞(0,T;L2)<inline-graphic></inline-graphic> convergence analysis for the proposed scheme, with an O(τ2+hq)<inline-graphic></inline-graphic> error estimate derived. In addition, an optimal convergence analysis is provided for the nonlinear term using Qq<inline-graphic></inline-graphic> finite elements, which shows that the spatial convergence order can be improved to q+1<inline-graphic></inline-graphic> on regular rectangular mesh. A few numerical experiments are presented, which confirms the efficiency and accuracy of the proposed second order numerical scheme. [ABSTRACT FROM AUTHOR]