  In this paper, we partially answer open questions about the convergence of
overlapping Schwarz methods. We prove that overlapping Schwarz methods with
Dirichlet transmission conditions for semilinear elliptic and parabolic
equations always converge. While overlapping Schwarz methods with Robin
transmission conditions only converge for semilinear parabolic equations, but
not for semilinear elliptic ones. We then provide some conditions so that
overlapping Schwarz methods with Robin transmission conditions converge for
semilinear elliptic equations. Our new techniques can also be potentially
applied to others kinds of partial differential equations.
