A Quadrilateral Edge Element Method Of H(curl) For Maxwell Equations

In the finite element method of Maxwell equations,including many methods,e.g.the nodal-based method,the discontinuous method,the edge element method,etc.Among others,the edge element method is the most popular method.This type of method is usually refereed to as the Nedelec element method.A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral version of version of Raviart-Thomas edge element,which was designed in the year 2003.first,establishing the interpolation error estimates for the Duan-Liang quadrilateral version of version of Raviart-Thomas edge element when the interpolated function has a low regularity,The solution having a low regularity is rather common in electromagnetism.Second,proposing two finite element methods with the use of Duan-Liang quadrilateral version of version of Raviart-Thomas edge element.One method is symmetric positive definite,but it is based on a continuous variational problem with a noncompact resolvent.The other method is a mixed method,based on a continuous mixed variational problem with a compact resolvent.Third,establishing the kernel-coercivity,a type of Poincare inequality,and the Inf-Sup condition for the well-posedess of the proposed finite element methods.Final,proving the posed two finite element methods when applied to the eigenvalue problem,spectral-correct and spurious-free approximations are obtained.Stability and error estimates are obtained,including the interpolation error estimates and the error estimates between the finite element solution and the exact solution.The method is suitable singular solution as well smooth solution,and consequently,the method is valid for nonconvex domains which may have a number of reentrant corners.Of course,the method is suitable for arbitrary quadrilaterals(under shape-regular condition).