A linear edge finite element method for quad-curl problem

In this study, we explore for the application of a linear edge finite element method for the numerical solution of a quad-curl problem. We carefully construct a curl recovery operator, which serves to approximate the second order curl of the linear edge element function. The proposed scheme is creatively designed by integrating the constructed operator with the standard Ritz-Galerkin methods in a straightforward manner. We provide proof that the numerical solution derived from our proposed method converges optimally to the exact solution in both $L^2$ and $H(\text{curl},\Omega )$ norms. Our analysis uncovers several intriguing properties of the curl recovery operator. To demonstrate the optimal convergence of our scheme, we construct a series of numerical experiments. The results provide compelling evidence of the effectiveness of our approach.