In mathematics, Neumann–Neumann methods are domain decomposition preconditioners named so because they solve a Neumann problem on each subdomain on both sides of the interface between the subdomains.^{[1]} Just like all domain decomposition methods, so that the number of iterations does not grow with the number of subdomains, Neumann–Neumann methods require the solution of a coarse problem to provide global communication. The balancing domain decomposition is a Neumann–Neumann method with a special kind of coarse problem.
More specifically, consider a domain Ω, on which we wish to solve the Poisson equation

\Delta u = f, \qquad u_{\partial\Omega} = 0
for some function f. Split the domain into two nonoverlapping subdomains Ω_{1} and Ω_{2} with common boundary Γ and let u_{1} and u_{2} be the values of u in each subdomain. At the interface between the two subdomains, the two solutions must satisfy the matching conditions

u_1 = u_2, \qquad \partial_nu_1 = \partial_nu_2
where n is the unit normal vector to Γ.
An iterative method for approximating each u_{i} satisfying the matching conditions is to first solve the decoupled problems (i=1,2)

\Delta u_i^{(k)} = f_i, \qquad u_i^{(k)}_{\partial\Omega} = 0, \quad u^{(k)}_i_\Gamma = \lambda^{(k)}
for some function λ^{(k)} on Γ. We then solve the two Neumann problems

\Delta\psi_i^{(k)} = 0, \qquad \psi_i^{(k)}_{\partial\Omega} = 0, \quad \partial_n\psi_i^{(k)} = \partial_nu_1^{(k)}  \partial_nu_2^{(k)}.
We then obtain the next iterate by setting

\lambda^{(k+1)} = \lambda^{(k)}  \omega(\theta_1\psi_1^{(k)}_\Gamma  \theta_2\psi_2^{(k)}_\Gamma)
for some parameters ω, θ_{1} and θ_{2}.
This procedure can be viewed as a Richardson iteration for the iterative solution of the equations arising from the Schur complement method.^{[2]}
This continuous iteration can be discretized by the finite element method and then solved—in parallel—on a computer. The extension to more subdomains is straightforward, but using this method as stated as a preconditioner for the Schur complement system is not scalable with the number of subdomains; hence the need for a global coarse solve.
See also
References

^ A. Klawonn and O. B. Widlund, FETI and Neumann–Neumann iterative substructuring methods: connections and new results, Comm. Pure Appl. Math., 54 (2001), pp. 57–90.

^ A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications 1999.
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