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# 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

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

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

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

where n is the unit normal vector to Γ.

An iterative method for approximating each ui satisfying the matching conditions is to first solve the decoupled problems (i=1,2)

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

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

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

We then obtain the next iterate by setting

${\displaystyle \lambda ^{(k+1)}=\lambda ^{(k)}-\omega (\theta _{1}\psi _{1}^{(k)}|_{\Gamma }-\theta _{2}\psi _{2}^{(k)}|_{\Gamma })}$

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.

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• ✪ MIT Numerical Methods for PDE Lecture 7: von Neumann stability analysis
• ✪ ch11 5. Laplace equation with Neumann boundary condition. Wen Shen
• ✪ Kyoto U. "An Introduction to Subfactors in Mathematics and Physics" Prof. Vaughan F. R. Jones, L.1