Optimized transmission conditions in domain decomposition methods have been the focus of intensive research efforts over the past decade. Traditionally, transmission conditions are optimized for two subdomain model configurations, and then used in practice for many subdomains. We optimize here transmission conditions for the first time directly for many subdomains for a class of complex diffusion problems. Our asymptotic analysis leads to closed form optimized transmission conditions for many subdomains, and shows that the asymptotic best choice in the mesh size only differs from the two subdomain best choice in the constants, for which we derive the dependence on the number of subdomains explicitly, including the limiting case of an infinite number of subdomains, leading to new insight into scalability. Our results include both Robin and Ventcell transmission conditions, and we also optimize for the first time a two-sided Ventcell condition. We illustrate our results with numerical experiments, both for situations covered by our analysis and situations that go beyond.

翻译：在过去十年中,对域分解方法的最佳传输条件一直是集中研究的重点。传统上,对两个子域模型配置进行优化,然后对许多子域进行实际应用。我们首次对许多子域的传输条件进行优化,以便处理一系列复杂的扩散问题。我们的无症状分析导致许多子域的封闭形式最佳传输条件,并表明网状尺寸的无症状最佳选择仅不同于常数中两个子域的最佳选择,为此,我们明确依赖子域的数目,包括无限数量子域的有限案例,从而导致对可缩放性的新认识。我们的结果包括罗宾和文特赛尔的传输条件,我们还首次优化了许多子域的双向通风槽条件。我们用数字实验来说明我们的结果,包括分析所涉情况和超出范围的情况。