A domain decomposition method that converges in two steps
for three subdomains

Sébastien Loisel
2 4, rue du Lièvre, Case postale 64
1211 Genève 4 (Suisse)
loisel@math.unige.ch

In Schwarz-like domain decomposition methods, a domain $ \Omega$ is broken into two or more subdomains and Dirichlet, Neumann, Robin or pseudo-differential problems are iteratively solved on each subdomain. For certain problems, it is well-known that the Dirichlet-Neumann iteration for two subdomains will converge in two steps. Let $ \Omega$ be an open domain and $ \Omega_{1},\Omega_{2},\Omega_{3}$ a domain decomposition of $ \Omega$ such that each pair of subdomains shares an interface (for instance, $ \Omega=\{ z\in\mathbb{C}\;\vert\; \vert z\vert<1\}$ and $ \Omega_{j}=\{ re^{i\theta} \;\vert\; 0<r<1$ and $ \theta\in(2j\pi/3,2(j+1)\pi/3)\}$, $ j=1,2,3$). We will show a new Schwarz-like domain decomposition method that converges in two iterations in this situation.