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

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 ($\mathbf{A}$-$Φ$) 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

💡 Research Summary

The paper presents a novel hybrid computational framework that combines Discrete Exterior Calculus (DEC) for the interior heterogeneous region with Surface Integral Equations (SIE) for the exterior unbounded region, all formulated in the magnetic vector potential–electric scalar potential (A‑Φ) representation under the Lorenz gauge. By working with potentials rather than fields, the method inherently avoids low‑frequency breakdown, a notorious problem for traditional differential‑equation solvers.

In the interior, the authors employ DEC, a coordinate‑free discretization that treats electromagnetic quantities as differential k‑forms evaluated on a simplicial complex. Unlike conventional DEC implementations that model the magnetic vector potential A as a 1‑form (edge‑based), the paper proposes to represent each Cartesian component of A (Ax, Ay, Az) as independent 0‑forms (node‑based). This unconventional choice aligns the DEC representation with the scalar‑component formulation of the SIE, dramatically simplifying the coupling at the interface.

For the exterior, the authors derive a scalar‑component SIE that reduces the usual fourteen integral operators (involving tangential, normal, divergence, and curl terms) to just two classical operators: the single‑layer potential S and the double‑layer potential D. The surface unknowns are the Cartesian components of A and the scalar potential Φ, together with their normal derivatives. The resulting jump conditions (Eqs. 17‑20) involve only S and D, eliminating the need for complex operator families and reducing computational cost and memory footprint.

A key geometric device is the introduction of a homogeneous buffer region surrounding the heterogeneous interior. This ensures that the DEC–SIE interface Γ lies entirely in free space, allowing the potentials to be continuous across Γ without artificial absorbing boundary conditions (ABC) or perfectly matched layers (PML). Consequently, the radiation condition is satisfied exactly by the SIE, while the DEC domain remains free of truncation artifacts.

The continuous formulation is first expressed in exterior calculus, where the differential operators d (exterior derivative) and ⋆ (Hodge star) replace vector calculus operators. The A‑Φ system becomes a block matrix involving the operators VEC = d⋆d + k₀²⋆ε and LSC = d⋆ε d + k₀²⋆ε². After discretization, the DEC matrices are sparse, well‑conditioned, and naturally respect topological invariants. The scalar‑component SIE contributes only two dense blocks (S and D), which are coupled to the DEC system through simple node‑based continuity constraints.

Numerical experiments on three‑dimensional objects with multiple dielectric inclusions demonstrate the method’s accuracy, convergence, and robustness. Compared with traditional FEM‑BEM (or FEM‑SIE) hybrids, the proposed approach reduces the total number of unknowns by roughly 30 % and the memory usage by about 40 %, while achieving comparable or superior error levels. Low‑frequency tests confirm that the potential‑based formulation remains well‑conditioned, and iterative solvers converge rapidly without the stagnation typical of field‑based formulations.

Overall, the paper contributes four major advances: (1) a low‑frequency‑stable potential formulation, (2) a structure‑preserving DEC discretization that leverages 0‑form representation of vector‑potential components, (3) a dramatically simplified SIE requiring only single‑ and double‑layer operators, and (4) an exact radiation treatment via a free‑space interface. The authors argue that the framework is readily extensible to multiphysics coupling (e.g., electro‑thermal, magneto‑mechanical) and to more complex material models (anisotropic permeability, nonlinear conductivities). Future work will explore time‑domain extensions, higher‑order DEC basis, and integration with fast multipole or hierarchical matrix accelerators for large‑scale scattering problems.