Additive Schwarz Methods for Semilinear Elliptic Problems with Convex Energy Functionals: Convergence Rate Independent of Nonlinearity

.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 \(H/δ\) only, where \(h\) and \(H\) are the typical diameters of an element and a subdomain, respectively, and \(δ\) measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings.