Suppose we want to solve the Poisson equation $\Delta u = f$ on a domain $\Omega$ with Dirichlet boundary conditions. One possible way to do is by a domain decomposition method.
There is a condition that the domain has to be decomposed into subdomains such that each subdomain touches the boundary of $\Omega$. Otherwise the scheme is supposed not to be well-posed. I do not understand that. Can you help me?