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, \(\frac{\nabla u}{1+|\nabla u|^2}\), 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, \(A\tau \varDelta ^2 ( u^{n+1} - u^n)\), is added in the numerical scheme, so that an unconditional long time energy stability is assured. In turn, we perform an \(\ell ^\infty (0,T; L^2)\) convergence analysis for the proposed scheme, with an \(O (\tau ^2 + h^q)\) error estimate derived. In addition, an optimal convergence analysis is provided for the nonlinear term using \(Q_q\) finite elements, which shows that the spatial convergence order can be improved to \(q+1\) on regular rectangular mesh. A few numerical experiments are presented, which confirms the efficiency and accuracy of the proposed second order numerical scheme.