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/\delta only, where h and H are the typical diameters of an element and a sub domain, respectively, and \delta measures the overlap among the sub domains. Numerical results are provided to support our theoretical findings.