Neumann–Neumann methods explained

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, u| _{\partial\Omega} = 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

u _{1} = u _{2,} \partial _{n _{1}} u _{1} = \partial _{n _{2}} u _{2}

where $n\_$ is the unit normal vector to Γ in each subdomain.

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

- \Delta u ^{(k)} _{i} = f _{i} in \Omega _{i} , u ^{(k)} _{i} | _{\partial\Omega} = 0 , u ^{(k)} _{i|} _{\Gamma} = λ ^{(k)}

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

- \Delta\psi ^{(k)} _{i} = 0 in \Omega _{i} , \psi ^{(k)} _{i} | _{\partial\Omega} = 0 , \partial _{n _{i}} \psi ^{(k)} _{i} | _{\Gamma} = \omega(\partial _{n _{1}} u ^{(k)} _{1} + \partial _{n _{2}} u ^{(k)} _{2} ).

We then obtain the next iterate by setting

λ ^{(k+1)} = λ ^{(k)} - \omega(\theta _{1\psi} ^{(k)} ^{1} + \theta _{2\psi} ^{(k)} ^{2} ) on \Gamma

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.