Convergence of the Dirichlet-Neumann method for semilinear elliptic equations

Convergence of the Dirichlet-Neumann method for semilinear elliptic equations

The Dirichlet-Neumann method is a common domain decomposition method for nonoverlapping domain decomposition and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial...

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Veröffentlicht in:arXiv.org 2024-10
1. Verfasser: Engström, Emil
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Convergence
Domain decomposition methods
Elliptic functions
Hilbert space
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description The Dirichlet-Neumann method is a common domain decomposition method for nonoverlapping domain decomposition and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet-Neumann method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov-Poincaré formulation of the Dirichlet-Neumann method.
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subjects Convergence
Domain decomposition methods
Elliptic functions
Hilbert space
title Convergence of the Dirichlet-Neumann method for semilinear elliptic equations
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