High order spline finite element method for the fourth-order parabolic equations

In this paper, we concern with the development of error estimates for high-order spline finite element approximation to a class of fourth-order parabolic equations. The L-2 estimates for semi-discrete scheme are derived by constructing two auxiliary problems (one is a steady-state problem, the other is similar to the primal problem) and by using the structure of solutions of the second-order ordinary differential equation and the similarity theory of matrices, and are optimal in terms of the regularity of the exact solution. The H-2 energy estimates for spline-element central difference approximation are established under a condition of stability (for explicit scheme), and are optimal for space variable in H-2-norm and for time variable in H-1,H-infinity- norm. Numerical examples are presented to validate the theory. (c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.