A new Galerkin spectral element method for fourth-order boundary value problems

Abstract

In this paper, we propose a new Galerkin spectral element method for one-dimensional fourth-order boundary value problems. We first introduce some quasi-orthogonal approximations in one dimension, and establish a series of results on these approximations, which serve as powerful tools in the spectral element method. By applying these results to the fourth-order boundary value problems, we establish sharp and error bounds of the Galerkin spectral element method. The efficient algorithm is implemented in detail. Numerical results demonstrate its high accuracy, and confirm the theoretical analysis well.