A Hybrid DEC-SIE Framework for Potential-Based Electromagnetic Analysis of Heterogeneous Media

Abstract

Analyzing electromagnetic fields in complex, multi-material environments presents substantial computational challenges. To address these, we propose a hybrid numerical method that couples discrete exterior calculus (DEC) with surface integral equations (SIE) in the potential-based formulation of Maxwell's equations. The method employs the magnetic vector and electric scalar potentials ($\mathbf{A}$-$\Phi$) under the Lorenz gauge, offering natural compatibility with multi-physics couplings and inherent immunity to low-frequency breakdown. To effectively handle both bounded and unbounded regions, we divide the computational domain: the inhomogeneous interior is discretized using DEC, a coordinate-free framework that preserves topological invariants and enables structure-preserving discretization on unstructured meshes, while the homogeneous exterior is treated using SIEs, which inherently satisfy the radiation condition and eliminate the need for artificial domain truncation. A key contribution of this work is a scalar reformulation of the SIEs, which reduces the number of surface integral operators from fourteen to two by expressing the problem in terms of the Cartesian components of the vector potential and their normal derivatives. This simplification motivates a corresponding adaptation in the DEC domain: each vector potential component is represented as a discrete 0-form, in contrast to the conventional 1-form representation. This novel treatment improves compatibility at the interface and significantly enhances numerical performance. The proposed hybrid method thus offers a unified, efficient, and physically consistent framework for solving electromagnetic scattering and radiation problems in complex geometries and heterogeneous materials