Fully H(gradcurl)-nonconforming Finite Element Method for The Singularly Perturbed Quad-curl Problem on Cubical Meshes

TL;DR

This paper introduces two new fully nonconforming finite elements on cubical meshes for the singularly perturbed quad-curl problem, achieving robust and optimal convergence with numerical verification.

Contribution

It develops novel H(grad curl)-nonconforming finite elements on cubical meshes that are compatible with the Stokes complex and provide robust solutions for the singularly perturbed quad-curl problem.

Findings

Optimal convergence order O(h) for fixed epsilon

Uniform convergence order O(h^{1/2}) for all epsilon

Numerical examples confirm theoretical results

Abstract

In this paper, we develop two fully nonconforming (both H(grad curl)-nonconforming and H(curl)-nonconforming) finite elements on cubical meshes which can fit into the Stokes complex. The newly proposed elements have 24 and 36 degrees of freedom, respectively. Different from the fully H(grad curl)-nonconforming tetrahedral finite elements in [9], the elements in this paper lead to a robust finite element method to solve the singularly perturbed quad-curl problem. To confirm this, we prove the optimal convergence of order $O(h)$ for a fixed parameter $\epsilon$ and the uniform convergence of order $O(h^{1/2})$ for any value of $\epsilon$. Some numerical examples are used to verify the correctness of the theoretical analysis.