Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems

In this paper, third-order compact and fourth-order finite element methods (FEMs) based on simple modifications of traditional FEMs are proposed for solving one-dimensional Sturm-Liouville boundary value problems (BVPs). The key idea is based on interpolation error estimates. A simple posterior error analysis of the original piecewise linear finite element space leads to a third-order accurate solution in the L2 norm, second-order in the H1, and the energy norm. The novel idea is also applied to obtain a fourth-order FEM based on the quadratic finite element space. The basis functions in the new fourth-order FEM are more compact compared with that of the classic cubic basis functions. Numerical examples presented in this paper have confirmed the convergence order and analysis. A generalization to a class of nonlinear two-point BVPs is also discussed and tested.