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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?
This result applies only to the most naive one-level Neumann-Neumann domain decomposition methods. If a subdomain does not touch a Dirichlet boundary, it will be "floating", thus has a null space. Classical Neumann-Neumann and original FETI methods identify the null space to control the sudomain problems and converge with general partitions. The early two-level methods use this null space characterization to build coarse spaces. Those methods tend to be fragile and have fallen out of favor since FETI-DP and BDDC (essentially equivalent dual and primal Neumann-Neumann methods) were developed. These methods use simpler and more general coarse spaces to control the subdomain null spaces.
