An A-$\Phi$ Formulation Solver in Electromagnetics Based on Discrete Exterior Calculus

An efficient numerical solver for the A - $\Phi$ formulation in electromagnetics based on discrete exterior calculus (DEC) is proposed in this paper. The A - $\Phi$ formulation is immune to low-frequency breakdown and ideal for broadband and multi-scale analysis. The generalized Lorenz gauge is used in this paper, which decouples the A equation and the $\Phi$ equation. The A - $\Phi$ formulation is discretized by using the DEC, which is the discretized version of exterior calculus in differential geometry. In general, DEC can be viewed as a generalized version of the finite difference method, where Stokes' theorem and Gauss's theorem are naturally preserved. Furthermore, compared with finite difference method, where rectangular grids are applied, DEC can be implemented with unstructured mesh schemes, such as tetrahedral meshes. Thus, the proposed DEC A - $\Phi$ solver is inherently stable, free of spurious solutions and can capture highly complex structures efficiently. In this paper, the background knowledge about the A - $\Phi$ formulation and DEC is introduced, as well as technical details in implementing the DEC A - $\Phi$ solver with different boundary conditions. Numerical examples are provided for validation purposes as well.