# Compability conditions in domain decomposition methods

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?

• Stiffness matrix for floating subdomain is generally singular, you could think it being like a pure Neumann problem, where the existence and uniqueness depends on whether the $f$ is in the range of the operator. – Shuhao Cao Mar 2 '12 at 3:16
• Is the title supposed to be "Compatibility conditions in domain decomposition methods"? – Geoff Oxberry Mar 2 '12 at 7:04