  A nonoverlapping domain decomposition method is studied for the linearized
Poisson--Boltzmann equation, which is essentially an interior-exterior
transmission problem with bounded interior and unbounded exterior. This problem
is different from the classical Schwarz alternating method for bounded
nonoverlapping subdomains well studied by Lions in 1990, and is challenging due
to the existence of unbounded subdomain. To obtain the convergence, a new
concept of interior-exterior Sobolev constant is introduced and a spectral
equivalence of related Dirichlet-to-Neumann operators is established
afterwards. We prove rigorously that the spectral equivalence results in the
convergence of interior-exterior iteration. Some numerical simulations are
provided to investigate the optimal stepping parameter of iteration and to
verify our convergence analysis.
