Nonconforming Finite Element Methods For Approximating Maxwell's Equations

Maxwell's equations are very important equations and the fundamental laws governing electromagnetic fields. Theory analysis and numerical solution of the Maxwell's equations is a hot topic in both numerical mathematics and engineering communities. FEM is a popular one for solving such problems. So far, almost all of previous analysis only concentrated on conforming finite element methods. A class of nonconforming finite element approximations to Maxwell's equations are discussed in this paper. Firstly, we apply two Crouzeix-Raviart type nonconforming finite elements to Maxwell's equations, the convergence analysis of them are studied on a mixed finite element scheme and a finite element scheme, respectively. Based on the two elements' special properties and some novel approaches, the optimal error estimates for one under anisotropic meshes and the other with regular meshes are obtained, which are as same as those in the previous literature for conforming elements under regular meshes. Secondly, a nonconforming arbitrary quadrilateral element, so-called Quasi-Wilson element, is applied to Maxwell's equations on the finite element scheme. The convergence analysis of its finite element scheme is derived under arbitrary quadrilateral meshes.