Finite Element Methods for One Dimensional Fourth Order Semilinear Partial Differential Equation

In this paper, we consider one dimensional fourth order semilinear partial differential equation. Some a priori bounds using Lyapunov functional are derived and existence and uniqueness results for the weak solution are proved. We discuss the finite element Galerkin methods and establish optimal error estimates for the semidiscrete case. Crank–Nicolson scheme is used in the temporal direction and optimal error estimates are derived. Finally, we discuss some numerical experiments and validate with the theoretical results.