A Legendre spectral-element method for the one-dimensional fourth-order equations

A Legendre-Galerkin spectral-element method is proposed to solve the one-dimensional fourth-order equations. C¹-continuity between the elemental-faces is imposed by constructing appropriate basis functions. The method leads to linear systems with sparse matrices for the discrete variational formulations. Rigorous error analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results.