We investigate additive Schwarz methods for semilinear elliptic problems with convex energy functionals, which have wide scientific applications. A key observation is that the convergence rates of both one- and two-level additive Schwarz methods have bounds independent of the nonlinear term in the problem. That is, the convergence rates do not deteriorate by the presence of nonlinearity, so that solving a semilinear problem requires no more iterations than a linear problem. Moreover, the two-level method is scalable in the sense that the convergence rate of the method depends on $H/h$ and $H/\delta$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $\delta$ measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings.

翻译：我们研究具有凸能量泛函的半线性椭圆问题的加性Schwarz方法，该方法具有广泛的科学应用。关键发现是，单层和双层加性Schwarz方法的收敛率均存在与问题中非线性项无关的界。即收敛率不会因非线性的存在而退化，因此求解半线性问题所需的迭代次数不超过线性问题。此外，双层方法具有可扩展性，其收敛率仅依赖于$H/h$和$H/\delta$，其中$h$和$H$分别表示单元和子域的典型直径，$\delta$衡量子域间的重叠程度。数值实验结果支持了我们的理论发现。