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Coupled Domain-Boundary Variational Formulations for Hodge-Helmholtz Operators

by E. Schulz and R. Hiptmair

(Report number 2020-21)

Abstract
We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded three dimensional Lipschitz domain with the first-kind boundary integral equation arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderon projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Garding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a “rough” surface. The low-frequency robustness of the potential formulation of Maxwell’s equations makes this model a promising starting point for Galerkin discretization.

Keywords: Maxwell, electromagnetism, scattering, Hodge-laplace, Hodge-Helmholtz, Hodge decomposition, Helmholtz decomposition, Calderon projector, symmetric coupling, T-coercivity

@Techreport{SH20_894,
  author = {E. Schulz and R. Hiptmair},
  title = {Coupled Domain-Boundary Variational Formulations for Hodge-Helmholtz Operators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-21},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-21.pdf },
  year = {2020}
}

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