Convergence analysis of a solver for the linear Poisson--Boltzmann model

This work investigates the convergence of a domain decomposition method for the Poisson-Boltzmann model that can be formulated as an interior-exterior transmission problem. To study its convergence, we introduce an interior-exterior constant providing an upper bound of the $L^2$ norm of any harmonic function in the interior, and establish a spectral equivalence for related Dirichlet-to-Neumann operators to estimate the spectrum of interior-exterior iteration operator. This analysis is nontrivial due to the unboundedness of the exterior subdomain, which distinguishes it from the classical analysis of the Schwarz alternating method with nonoverlapping bounded subdomains. It is proved that for the linear Poisson-Boltzmann solvent model in reality, the convergence of interior-exterior iteration is ensured when the relaxation parameter lies between 0 and 2. This convergence result interprets the good performance of ddLPB method developed in [SIAM Journal on Scientific Computing, 41 (2019), pp. B320-B350] where the relaxation parameter is set to 1. Numerical simulations are conducted to verify our convergence analysis and to investigate the optimal relaxation parameter for the interior-exterior iteration.