The Schwarz alternating method can be used to solveelliptic boundary value problems on domains which consist of two or moreoverlapping subdomains. The solution is approximated by an infinite sequence offunctions 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 fornonlinear elliptic problems which are known to have solutions by the monotonemethod (also known as the method of subsolutions and supersolutions) aregiven. In particular, an additive Schwarz method for scalar as well as somecoupled nonlinear PDEs are shown to converge for finitely many subdomains.These results are applicable to several models in population biology.
