A scalar and vector potential formulation for finite element solutions to Maxwell's equations

Finite-element discretization of Maxwell's equations using conventional nodal-based elements has been plagued by spurious modes. A method for deterministic problems based on Helmholtz's equation and specific boundary conditions has previously been shown to eliminate these unwanted modes. In the present work, using a scalar and vector potential formulation and a particular choice of gauge, this Helmoltz-based method is derived in greater generality from first principles. Under certain conditions, the scalar potential can be eliminated completely and an equation for the electric field alone results which is free from spurious modes in nonresonant cases. Consideration of the uniqueness of the scalar potential leads to the derived boundary conditions. A second gauge condition leads to a slightly different formulation, more amenable to finite-element discretization but coupling the vector and scalar potentials in general. In the presence of physical resonances, boundary conditions alone are not sufficient to discriminate against spurious modes. Additional enforcement of the gauge condition, however, leads to the exact cancellation of the spurious modes in the vector potential by the gradient of the scalar potential.< >