Local Ultraconvergence of Quadratic Rectangular Element

Abstract

A state of the art technology is employed to investigate the local ultraconvergence properties of a quadratic rectangular element for the Poisson equation. The proposed method combine advantages of a novel interpolation postprocessing operator $\overline{P}^6_{6h,m} R^∗_h ,$ the Richardson extrapolation technique, and properties of a discrete Green’s function. The local ultraconvergence of the post-processed gradient of the finite element solution is derived with the order $\mathcal{O}(h^6 |{\rm ln}h|^2).$ A numerical example shows a good agreement with the theoretical findings.