Analysis of a mixed finite element method for the quad-curl problem

Quad-curl term plays an essential role in the numerical analysis of the resistive magnetohydrodynamics (MHD) and the forth order inverse electromagnetic scattering problem. It is desirable to develop simple and efficient numerical methods for the quad-curl problem. In this paper, we firstly give a regularity result for the quad-curl problem on Lipschitz polyhedron domains, which is {\em new} in literatures. Then, we propose a mixed finite element method for the quad-curl problem. With {\em novel} discrete Sobolev embedding inequalities for the piecewise polynomials, we obtain stability results and derive {\em optimal} error estimates relying on a low regularity assumption of the exact solution. To the best of our knowledge, this low regularity assumption is {\em lower} than the regularity requirements in existing works.

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