Superconvergence of a nonconforming brick element for the quad-curl problem

TL;DR

This paper demonstrates the superconvergence properties of a recently introduced nonconforming brick element for the quad-curl problem, achieving higher accuracy through modifications and postprocessing, supported by numerical verification.

Contribution

It introduces modifications to the interpolation and formulation of a nonconforming brick element, achieving superconvergence for the quad-curl problem.

Findings

Achieves $O(h^2)$ superclose order in $H(grad curl)$ norm.

Proposes a postprocessing method for global superconvergence.

Numerical results confirm theoretical predictions.

Abstract

This short note shows the superconvergence of an $H(\mathrm{grad}\,\mathrm{curl})$-nonconforming brick element very recently introduced in [17] for the quad-curl problem. The supercloseness is based on proper modifications for both the interpolation and the discrete formulation, leading to an $O(h^2)$ superclose order in the discrete $H(\mathrm{grad}\,\mathrm{curl})$ norm. Moreover, we propose a suitable postprocessing method to ensure the global superconvergence. Numerical results verify our theory.