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

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