# On monotone and Schwarz alternating methods for nonlinear elliptic PDEs

• Volume: 35, Issue: 1, page 1-15
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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Lui, Shiu-Hong. "On monotone and Schwarz alternating methods for nonlinear elliptic PDEs." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.1 (2001): 1-15. <http://eudml.org/doc/194043>.

@article{Lui2001,
abstract = {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.},
author = {Lui, Shiu-Hong},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {domain decomposition; nonlinear elliptic PDE; Schwarz alternating method; monotone methods; subsolution; supersolution; nonlinear elliptic PDEs; convergence; difference methods; population biology},
language = {eng},
number = {1},
pages = {1-15},
publisher = {EDP-Sciences},
title = {On monotone and Schwarz alternating methods for nonlinear elliptic PDEs},
url = {http://eudml.org/doc/194043},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Lui, Shiu-Hong
TI - On monotone and Schwarz alternating methods for nonlinear elliptic PDEs
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 1
SP - 1
EP - 15
AB - 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.
LA - eng
KW - domain decomposition; nonlinear elliptic PDE; Schwarz alternating method; monotone methods; subsolution; supersolution; nonlinear elliptic PDEs; convergence; difference methods; population biology
UR - http://eudml.org/doc/194043
ER -

## References

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