An all-frequency stable integral system for Maxwell’s equations in 3-D penetrable media: continuous and discrete model analysis

• Published in: Advances in computational mathematics Vol. 51; no. 1
• Main Authors: Ganesh, Mahadevan, Hawkins, Stuart C., Volkov, Darko
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.02.2025
• Subjects: Algorithms; Constraints; Integral equations; Linear operators; Linear systems; Mathematics; Maxwell's equations; Numerical methods; Operators (mathematics); Well posed problems
• DOI: 10.1007/s10444-024-10218-4
• ISSN: 1019-7168

We introduce a new system of surface integral equations for Maxwell’s transmission problem in three dimensions (3-D). This system has two remarkable features, both of which we prove. First, it is well-posed at all frequencies. Second, the underlying linear operator has a uniformly bounded inverse as the frequency approaches zero, ensuring that there is no low-frequency breakdown. The system is derived from a formulation we introduced in our previous work, which required additional integral constraints to ensure well-posedness across all frequencies. In this study, we eliminate those constraints and demonstrate that our new self-adjoint, constraints-free linear system—expressed in the desirable form of an identity plus a compact weakly-singular operator—is stable for all frequencies. Furthermore, we propose and analyze a fully discrete numerical method for these systems and provide a proof of spectrally accurate convergence for the computational method. We also computationally demonstrate the high-order accuracy of the algorithm using benchmark scatterers with curved surfaces.