The Schwarz alternating method can be used to solve
elliptic boundary value problems on domains which consist of two or more
overlapping subdomains. The solution is approximated by an infinite sequence of
functions which results from solving a sequence of elliptic boundary value
problems in each of the subdomains. In this paper, proofs of convergence of some Schwarz alternating methods for
nonlinear elliptic problems which are known to have solutions by the monotone
method (also known as the method of subsolutions and supersolutions) are
given. In particular, an additive Schwarz method for scalar as well as some
coupled nonlinear PDEs are shown to converge for finitely many subdomains.
These results are applicable to several models in population biology.