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

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1373-A1396, June 2024.
Abstract. 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 [math] and [math] only, where [math] and [math] are the typical diameters of an element and a subdomain, respectively, and [math] measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings.