Coupled domain-boundary variational formulations for Hodge–Helmholtz operators

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 first-kind boundary integral equations arising when constant coefficients are assumed in the unbounded complement. Using recently developed Calderón projectors for symmetric coupling, we prove stability away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a rough surface.