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We focus here on a class of fourth-order parabolic equations that can bewritten as a system of second-order equations by introducing an auxiliaryvariable. We design a novel second-order fully discrete mixed finite elementmethod to approximate these equations. In our approach, we propose newtechniques using the second-order backward differentiation formula for the timederivative and a special technique for the approximation of nonlinear terms.The use of the proposed technique for nonlinear terms makes the developednumerical scheme efficient in terms of computational cost since the proposedmethod only deals with a linear system at each time step and no iterativeresolution is needed. A numerical convergence study is performed using themethod of manufactured and analytical solutions of the system where weinvestigate different boundary conditions. With respect to the spatialdiscretization, convergence rates are found to at least match a priori errorestimates available for linear problems. The convergence analysis is completedwith an investigation of the temporal discretization where we numericallydemonstrate the second-order time-accuracy of the proposed scheme using themethod of reference solution. We present a series of numerical tests todemonstrate the efficiency and robustness of the proposed scheme.

Efficient second-order semi-implicit finite element method for fourth-order nonlinear diffusion equations