Abstract: Firstly, we used the cubic B-spline finite element method to solve a class of fourth-order semi-linear parabolic equation with the variable coefficient, and proved the stability and convergence of the semi-discrete scheme. Secondly, by using the Crank-Nicolson method to discretize the time variable, we obtained the fully discrete scheme and discussed the stability and convergence of the fully discrete scheme. Finally, in the numerical example, we used the Picard iteration method to handle the nonlinear term and obtained the convergence order of the finite element method according to L² norm and H² norm.

Key words: fourth-order semi-linear parabolic equation, variable coefficient, cubic B-spline finite element method, stability, convergence