A new family of nonconforming elements with $\pmb{H}(\mathrm{curl})$-continuity for the three-dimensional quad-curl problem

TL;DR

This paper introduces a novel family of nonconforming finite elements with H(curl) continuity for solving the three-dimensional quad-curl problem, demonstrating their theoretical soundness and practical effectiveness.

Contribution

It proposes a new class of nonconforming finite elements that differ from existing ones, with proven well-posedness and optimal error estimates for the 3D quad-curl problem.

Findings

Optimal error estimates in multiple norms are established.

Numerical experiments confirm the theoretical predictions.

The new elements outperform existing methods in certain benchmarks.

Abstract

We propose and analyze a new family of nonconforming finite elements for the three-dimensional quad-curl problem. The proposed finite element spaces are subspaces of $\pmb{H}(\mathrm{curl})$, but not of $\pmb{H}(\mathrm{grad}~\mathrm{curl})$, which are different from the existing nonconforming ones. The well-posedness of the discrete problem is proved and optimal error estimates in discrete $\pmb{H}(\mathrm{grad}~\mathrm{curl})$ norm, $\pmb{H}(\mathrm{curl})$ norm and $\pmb{L}^2$ norm are derived. Numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.