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
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
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)
for some function λ^{(k)} on Γ. We then solve the two Neumann problems
We then obtain the next iterate by setting
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.
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