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.

翻译：本文研究针对泊松-玻尔兹曼模型的区域分解法的收敛性，该模型可表述为内-外传输问题。为分析其收敛性，我们引入了一个内-外常数，该常数可为任意内部调和函数的$L^2$范数提供上界，并通过建立相关狄利克雷-诺伊曼算子的谱等价性来估计内-外迭代算子的谱。由于外部子域的无界性，该分析与经典非重叠有界子域的施瓦茨交替法分析有本质区别，使得分析具有非平凡性。研究证明，在实际线性泊松-玻尔兹曼溶剂模型中，当松弛参数取值在0到2之间时，内-外迭代的收敛性得以保证。该收敛结果解释了文献[SIAM Journal on Scientific Computing, 41 (2019), pp. B320-B350]中发展的ddLPB方法（其松弛参数设为1）表现良好的内在机理。数值模拟验证了收敛性分析结论，并探究了内-外迭代的最优松弛参数。