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

The quad-curl term is an essential part of the resistive magnetohydrodynamic 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 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.