TY - JOUR T1 - A Mixed Finite Element Scheme for Quad-Curl Source and Eigenvalue Problems AU - Chen , Huangxin AU - Li , Jingzhi AU - Qiu , Weifeng AU - Wang , Chao JO - Communications in Computational Physics VL - 4 SP - 1125 EP - 1151 PY - 2021 DA - 2021/02 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0108 UR - https://global-sci.org/intro/article_detail/cicp/18649.html KW - Quad-curl problem, mixed finite element scheme, error estimates, eigenvalue problem. AB -

The quad-curl problem arises in the resistive magnetohydrodynamics (MHD) and the electromagnetic interior transmission problem. In this paper we study a new mixed finite element scheme using Nédélec's edge elements to approximate both 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 for Maxwell's equations. Via a discrete energy norm stability due to element-wise stabilization, we prove optimal convergence under a low regularity condition. We also extend the mixed finite element scheme to the quad-curl eigenvalue problem and provide corresponding convergence analysis based on that of source problem. Numerical examples are provided to show the viability and accuracy of the proposed method for quad-curl source problem.