On Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs
Department of Mathematics, Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong. (email@example.com)
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.
Mathematics Subject Classification: 65N55 / 65J15
Key words: Domain decomposition / nonlinear elliptic PDE / Schwarz alternating method / monotone methods / subsolution / supersolution.
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