A simple third order compact finite element method for 1D BVP

TL;DR

This paper introduces a straightforward third order compact finite element method for 1D Sturm-Liouville boundary value problems, achieving high accuracy with simple modifications and validated by numerical tests.

Contribution

It presents a novel, simple third order finite element approach for 1D BVPs based on interpolation error estimates and modified basis functions.

Findings

Achieves third order accuracy in L^2 norm

Attains second order accuracy in H^1 norm

Numerical examples confirm theoretical error estimates

Abstract

A simple third order compact finite element method is proposed for one-dimensional Sturm-Liouville boundary value problems. The key idea is based on the interpolation error estimate, which can be related to the source term. Thus, a simple posterior error analysis or a modified basis functions based on original piecewise linear basis function will lead to a third order accurate solution in the $L^2$ norm, and second order in the $H^1$ or the energy norm. Numerical examples have confirmed our analysis.