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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?

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  • $\begingroup$ 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. $\endgroup$ – Shuhao Cao Mar 2 '12 at 3:16
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    $\begingroup$ Is the title supposed to be "Compatibility conditions in domain decomposition methods"? $\endgroup$ – Geoff Oxberry Mar 2 '12 at 7:04
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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.

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  • $\begingroup$ Or he can use the Total FETI method which makes all subdomains float. In this case the kernel is the same. $\endgroup$ – Zoltán Csáti Nov 15 '16 at 15:11

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