We develop a high order reconstructed discontinuous approximation (RDA) method for solving a mixed formulation of the quad-curl problem in two and three dimensions. This mixed formulation is established by adding an auxiliary variable to control the divergence of the field. The approximation space for the original variables is constructed by patch reconstruction with exactly one degree of freedom per element in each dimension and the auxiliary variable is approximated by the piecewise constant space. We prove the optimal convergence rate under the energy norm and also suboptimal $L^2$ convergence using a duality approach. Numerical results are provided to verify the theoretical analysis.

翻译：本文针对二维与三维情形下的四旋度问题，提出了一种基于混合形式的高阶重构间断逼近（RDA）方法。该混合形式通过引入辅助变量来控制场的散度。原始变量的逼近空间通过单元重构技术构建，其在每个维度上每个单元仅含一个自由度；辅助变量则采用分片常数空间进行逼近。我们证明了该方法在能量范数下的最优收敛率，并利用对偶论证得到了次最优的 $L^2$ 收敛结果。数值实验验证了理论分析的正确性。