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.

翻译：我们引入并分析三维异常扰动的二次曲线模型问题的强力不兼容的有限要素方法。 为了解决模型问题,我们得出了适当的先验界限,在此基础上,我们证明拟议的有限要素方法对于单振动参数$\varepsilon$和数字解决方案是稳健的。此外,我们还调查了用尼采的方法对第二个边界条件进行微弱处理的效果。我们表明,这种处理比当参数$\varepsilon < h$时强加边界条件的误差估计数更明显。最后,提供了数字实验,以说明该方法的良好性能并证实我们的理论预测。