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

$-\Delta u=f,\qquad u|_{{\partial \Omega }}=0$

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

$u_{1}=u_{2},\qquad \partial _{n}u_{1}=\partial _{n}u_{2}$

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)

$-\Delta u_{i}^{{(k)}}=f_{i},\qquad u_{i}^{{(k)}}|_{{\partial \Omega }}=0,\quad u_{i}^{{(k)}}|_{\Gamma }=\lambda ^{{(k)}}$

for some function λ^(k) on Γ. We then solve the two Neumann problems

$-\Delta \psi _{i}^{{(k)}}=0,\qquad \psi _{i}^{{(k)}}|_{{\partial \Omega }}=0,\quad \partial _{n}\psi _{i}^{{(k)}}=\partial _{n}u_{1}^{{(k)}}-\partial _{n}u_{2}^{{(k)}}.$

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, 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.