A Finite Element Method by Patch Reconstruction for the Quad-Curl Problem Using Mixed Formulations (Advances in Applied Mathematics and Mechanics, 2024)
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 tocontrol the divergence of the field. The approximation space for the original variableis constructed by patch reconstruction with exactly one degree of freedom per elementin each dimension and the auxiliary variable is approximated by the piecewise constant space. We prove the optimal convergence rate under the energy norm and alsosuboptimal $L^2$ convergence using a duality approach. Numerical results are providedto verify the theoretical analysis.