Coupled Domain-Boundary Variational Formulations for Hodge-Helmholtz Operators

Area of Science:

• Computational electromagnetics
• Mathematical physics
• Numerical analysis

Background:

• The generalized Hodge-Helmholtz and Hodge-Laplace equations are fundamental in physics.
• Boundary integral equations are crucial for solving problems in unbounded domains.
• Coupling these formulations presents significant mathematical challenges.

Purpose of the Study:

• To develop a stable and robust numerical method for analyzing electromagnetic wave scattering.
• To couple variational formulations with boundary integral equations for 3D Lipschitz domains.
• To provide a foundation for Galerkin discretization of Maxwell's equations.

Main Methods:

• Coupling of mixed variational problems with first-kind boundary integral equations.
• Utilizing recently developed Calderón projectors for symmetric coupling.
• Proving stability via a generalized Gårding inequality (T-coercivity).

Main Results:

• A stable coupled problem formulation is established away from resonant frequencies.
• The system accurately describes monochromatic electromagnetic wave scattering.
• Demonstrated low-frequency robustness for potential formulations.

Conclusions:

• The coupled model provides a stable framework for electromagnetic wave scattering problems.
• The approach is suitable for bodies with complex geometries and inhomogeneous properties.
• This work is a promising starting point for Galerkin discretization methods.