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/\) only, where \ (h\) and \ (H\) are the typical diameters of an element and a subdomain, respectively, and \ (\) measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings. Keywordsadditive Schwarz methodssemilinear elliptic problemsconvex optimizationconvergence analysisdomain decomposition methodsMSC codes65N5565J1535J6190C25