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Coupled Domain-Boundary Variational Formulations for Hodge-Helmholtz Operators.

Erick Schulz1, Ralf Hiptmair1

  • 1ETH Zürich, SAM, HG G 58.3, 8092 Zürich, Switzerland.

Integral Equations and Operator Theory
|February 28, 2022
PubMed
Summary
This summary is machine-generated.

This study couples variational problems with boundary integral equations for electromagnetic wave scattering. The developed model ensures stability, offering a robust starting point for numerical simulations of complex physical phenomena.

Keywords:
Calderón projectorHodge decompositionHodge–Helmholtz equationHodge–Laplace equationSymmetric couplingT-coercivity

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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.