Variational Formulation for Maxwell’s Equations with Lorenz Gauge: Existence and Uniqueness of Solution

The existence and uniqueness of a vector scalar potential representation with the Lorenz gauge (Schelkunoff potential) is proven for any vector field from H(curl). This representation holds for electric and magnetic fields in the case of a piecewise smooth conductivity, permittivity and permeability, for any frequency. A regularized formulation for the magnetic field is obtained for the case when the magnetic permeability μ is constant and thus the magnetic field is divergence free. In the case of a non divergence free electric field, an equation involving scalar and vector potentials is proposed. The solution to both electric and magnetic formulations may be approximated by the nodal shape functions in the finite element method with system matrices that remain well-conditioned for low frequencies. A numerical study of a forward problem of a computation of electromagnetic fields in the diffusive electromagnetic regime shows the efficiency of the proposed method.