A finite element method based on the first order system LL* (FOSLL*) approach is derived for time harmonic Maxwell's equations in three dimensional domains. The finite element solution is a potential for the original field in a sense that the original field U is given by U = L*u. The Maxwellian boundary data appears as natural boundary condition. Homogeneous Dirichlet boundary conditions for the potential must be imposed, and they are circumvented with weak enforcement of boundary conditions and it is proved that the sesquilinear form of the finite element system is elliptic in the space where the Dirichlet boundary conditions are satisfied weakly.
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