Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems
DOI:
https://doi.org/10.37256/rrcs.2120232118Keywords:
high-order compact finite element method, posterior analysis, modified basis functions, Sturm-Liouville boundary value problemAbstract
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.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Baiying Dong, Zhilin Li, Juan Ruiz-Álvarez
This work is licensed under a Creative Commons Attribution 4.0 International License.