A mixed finite element method for a time-fractional fourth-order partial differential equation

摘要

In this paper, a numerical theory based on the mixed finite element method for a time-fractional fourth-order partial differential equation (PDE) is presented and analyzed. An auxiliary variable σ=Δu is introduced, then the fourth-order equation can be split into the coupled system of two second-order equations. The time Caputo-fractional derivative is discretized by a finite difference method and the spatial direction is approximated by the mixed finite element method. The stabilities based on a priori analysis for two variables are discussed and some a priori error estimates in L2-norm for the scalar unknown u and the variable σ=Δu, are derived, respectively. Moreover, an a priori error result in H1-norm for the scalar unknown u also is proved. For verifying the theoretical analysis, a numerical test is made by using Matlab procedure.