Abstract: We used  finite element method to numerically solve the viscous Cahn-Hilliard equation. Firstly, the equivalent form of the viscous Cahn-Hilliard equation was obtained by introducing the Lagrange multiplier r of the auxiliary variable. Secondly, the second order linear finite element numerical scheme for the viscous Cahn-Hilliard equation was given by using the mixed finite element approximation  in space and the implicit backward differentiation formula (BDF)  for discretization in time, and the unconditional stability in energy and error estimation of the given scheme were analyzed in detail. Finally, a series of numerical examples were used to verify the accuracy and effectiveness of the given scheme. The results show that the proposed numerical scheme is ideal and has the characteristics of simultaneously satisfying linear, unconditional stability in energy and second order accuracy.

Key words: viscous Cahn-Hilliard equation, Lagrange multiplier, backward differentiation formula (BDF), unconditional stability in energy