A multiple-dynamics-preserving splitting mixed finite element method for 2D time-fractional molecular beam epitaxy model with slope selection

Abstract

This paper deals with a new numerical method and its discrete dynamical analysis for 2D time-fractional molecular beam epitaxy model with slope selection. A multiple-dynamics-preserving splitting mixed finite element method is proposed to solve the model. A unique solvability criterion of the method is given. Under no stepsize constraint, the method is proved to be energy-stability-preserving, variational-energy-dissipation-law-preserving and $L^2$-norm-stability-preserving in the discrete sense. Moreover, an error estimate of the method is derived under the suitable condition, which shows that the method can arrive at convergence order $2-\alpha$ (resp. $m+1$) in time (resp. in space), where $\alpha$ and $m+1$ denote the order of fractional derivatives in the model and the dimension of the used finite element space, respectively. By performing a series of numerical experiments, the error estimate and discrete dynamical properties of the method are further confirmed.