Nonconforming finite element approximations and the analysis of Nitsche's method for a singularly perturbed quad-curl problem in three dimensions

TL;DR

This paper develops a robust nonconforming finite element method for a 3D singularly perturbed quad-curl problem, demonstrating uniform convergence and improved boundary treatment via Nitsche's method, supported by numerical experiments.

Contribution

It introduces a new nonconforming finite element approach for a complex 3D quad-curl problem, analyzing robustness and boundary treatment effects.

Findings

Method is robust with respect to the perturbation parameter psilon.

Numerical solutions converge uniformly with order h^{1/2}.

Weak boundary treatment via Nitsche's method yields sharper error estimates.

Abstract

We introduce and analyze a robust nonconforming finite element method for a three dimensional singularly perturbed quad-curl model problem. For the solution of the model problem, we derive proper a priori bounds, based on which, we prove that the proposed finite element method is robust with respect to the singular perturbation parameter $\varepsilon$ and the numerical solution is uniformly convergent with order $h^{1/2}$. In addition, we investigate the effect of treating the second boundary condition weakly by Nitsche's method. We show that such a treatment leads to sharper error estimates than imposing the boundary condition strongly when the parameter $\varepsilon< h$. Finally, numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.