The Local $L^2$ Projected $C^0$ Finite Element Method for Maxwell Problem

Abstract

An element-local $L^2$-projected $C^0$ finite element method is presented to approximate the nonsmooth solution being not in $H^1$ of the Maxwell problem on a nonconvex Lipschitz polyhedron with reentrant corners and edges. The key idea lies in that element-local $L^2$ projectors are applied to both curl and div operators. The $C^0$ linear finite element (enriched with certain higher degree bubble functions) is employed to approximate the nonsmooth solution. The coercivity in $L^2$ norm is established uniform in the mesh-size, and the condition number ${\cal O}(h^{-2})$ of the resulting linear system is proven. For the solution and its curl in $H^r$ with $r<1$ we obtain an error bound ${\cal O}(h^r)$ in an energy norm. Numerical experiments confirm the theoretical error bound.