Neumann–Neumann methods

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

−Δu=f,u|∂Ω=0

for some function f. Split the domain into two non-overlapping subdomains Ω1 and Ω2 with common boundary Γ and let u1 and u2 be the values of u in each subdomain. At the interface between the two subdomains, the two solutions must satisfy the matching conditions

u1=u2,∂n1u1=∂n2u2

where ni is the unit normal vector to Γ in each subdomain.

An iterative method with iterations k = 0, 1, ... for the approximation of each ui (i = 1, 2) that satisfies the matching conditions is to first solve the Dirichlet problems

−Δui(k)=fiinΩi,ui(k)|∂Ω=0,ui(k)|Γ=λ(k)

for some function λ(k) on Γ, where λ(0) is any inexpensive initial guess. We then solve the two Neumann problems

−Δψi(k)=0inΩi,ψi(k)|∂Ω=0,∂niψi(k)|Γ=ω(∂n1u1(k)+∂n2u2(k)).

We then obtain the next iterate by setting

λ(k+1)=λ(k)−ω(θ1ψ1(k)+θ2ψ2(k))onΓ

for some parameters ω, θ1 and θ2.

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

* Neumann–Dirichlet method

## References

1. ↑ 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.
2. ↑ A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications 1999.

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Numerical partial differential equations by method

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* Forward-time central-space (FTCS)
* Crank–Nicolson

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Hyperbolic|

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* Alternating direction-implicit (ADI)
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* Godunov
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* hp-FEM
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Meshless/Meshfree|

* Smoothed-particle hydrodynamics (SPH)
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* Material point method (MPM)
* Particle-in-cell (PIC)

Domain decomposition|

* Schur complement
* Fictitious domain
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* additive
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* Neumann–Dirichlet
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* Poincaré–Steklov operator
* Balancing (BDD)
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Others|

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