  This paper proposes a finite element method for solving the periodic
steady-state problem for the scalar-valued and vector-valued Poisson equations,
a simple reduction model of the Maxwell equations under the Coulomb gauge.
Introducing a new potential variable, we reformulate two systems composed of
the scalar-valued and vector-valued Poisson problems to a single Hodge-Laplace
problem for the 1-form in $\mathbb{R}^4$ using the standard de Rham complex.
Consequently, we can directly apply the Finite Element Exterior Calculus (FEEC)
theory in $\mathbb{R}^4$ to deduce the well-posedness, stability, and
convergence. Numerical examples using the cubical element are reported to
validate the theoretical results.
