A Mixed Finite Element Scheme for Quad-Curl Source and Eigenvalue Problems (Communications in Computational Physics, 2021)
The quad-curl problem arises in the resistive magnetohydrodynamics(MHD) and the electromagnetic interior transmission problem. In this paper we studya new mixed finite element scheme using Nédélec's edge elements to approximateboth the solution and its curl for quad-curl problem on Lipschitz polyhedral domains.We impose element-wise stabilization instead of stabilization along mesh interfaces.Thus our scheme can be implemented as easy as standard Nédélec's methods forMaxwell's equations. Via a discrete energy norm stability due to element-wise stabilization, we prove optimal convergence under a low regularity condition. We alsoextend the mixed finite element scheme to the quad-curl eigenvalue problem and provide corresponding convergence analysis based on that of source problem. Numericalexamples are provided to show the viability and accuracy of the proposed method forquad-curl source problem.