An extended Lagrange FEM for the Maxwell eigenvalue problem

We construct an extended Lagrange FE space to solve the Maxwell equation and its eigenvalue problem in $\mathbb R^d$ $(d=2,3)$, which is the sum of the vectorial $p-$order Lagrange FE space ($p\ge1$) and the gradient of the $p+1-$order Lagrange FE space. The two lowest-order methods in 3D adopt slightly less degrees of freedom than the second family of the same order edge element methods in 3D. We construct a Cl\'{e}ment interpolant operator to prove the discrete compactness of the FE space and the convergence of the new methods for both Maxwell equation and its eigenvalue problem. For the extended linear Lagrange element method, an average-type curl recovery approach is designed to obtain numerical solution of super-convergence. In the numerical part, we verify the optimal convergence order for the two lowest-order methods, discuss the upper bound property of numerical eigenvalues and investigate the lower bound property by the average-type curl recovery approach.