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

$u_1 = u_2, \qquad \partial_nu_1 = \partial_nu_2$

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)

$-\Delta u_i^{(k)} = f_i, \qquad u_i^{(k)}|_{\partial\Omega} = 0, \quad u^{(k)}_i|_\Gamma = \lambda^{(k)}$

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

$-\Delta\psi_i^{(k)} = 0, \qquad \psi_i^{(k)}|_{\partial\Omega} = 0, \quad \partial_n\psi_i^{(k)} = \partial_nu_1^{(k)} - \partial_nu_2^{(k)}.$

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.