Analysis of an interior penalty DG method for the quad-curl problem

Abstract: The quad-curl term is an essential part of the resistive magnetohydrodynamic (MHD) equation and the fourth order inverse electromagnetic scattering problem, which are both of great significance in science and engineering. It is desirable to develop efficient and practical numerical methods for the quad-curl problem. In this paper, we first present some new regularity results for the quad-curl problem on Lipschitz polyhedron domains and then propose a mixed finite element method for solving the quad-curl problem. With a {\em novel} discrete Sobolev imbedding inequality for the piecewise polynomials, we obtain stability results and derive error estimates based on a relatively low regularity assumption of the exact solution.