  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.
