Goodman and Kruskal's lambda: Difference between revisions

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In mathematics, '''Neumann–Neumann methods''' are domain decomposition [[preconditioner]]s named so because they solve a [[Neumann problem]] on each subdomain on both sides of the interface between the subdomains.<ref name="Klawonn-2001-FNN">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.</ref>  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.
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More specifically, consider a domain Ω, on which we wish to solve the Poisson equation
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:<math>-\Delta u = f, \qquad u|_{\partial\Omega} = 0</math>
 
for some function ''f''. Split the domain into two non-overlapping subdomains Ω<sub>1</sub> and Ω<sub>2</sub> with common boundary Γ and let ''u''<sub>1</sub> and ''u''<sub>2</sub> be the values of ''u'' in each subdomain. At the interface between the two subdomains, the two solutions must satisfy the matching conditions
 
:<math>u_1 = u_2, \qquad \partial_nu_1 = \partial_nu_2</math>
 
where ''n'' is the unit normal vector to Γ.
 
An iterative method for approximating each u<sub>i</sub> satisfying the matching conditions is to first solve the decoupled problems (i=1,2)
 
:<math>-\Delta u_i^{(k)} = f_i, \qquad u_i^{(k)}|_{\partial\Omega} = 0, \quad u^{(k)}_i|_\Gamma = \lambda^{(k)}</math>
 
for some function λ<sup>(k)</sup> on Γ. We then solve the two Neumann problems
 
:<math>-\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)}.</math>
 
We then obtain the next iterate by setting
 
:<math>\lambda^{(k+1)} = \lambda^{(k)} - \omega(\theta_1\psi_1^{(k)}|_\Gamma - \theta_2\psi_2^{(k)}|_\Gamma)</math>
 
for some parameters ω, θ<sub>1</sub> and θ<sub>2</sub>.
 
This procedure can be viewed as a [[Richardson extrapolation]] for the iterative solution of the equations arising from the [[Schur complement method]].<ref name="Quarteroni">A. Quarteroni and A. Valli, ''Domain Decomposition Methods for Partial Differential Equations'', Oxford Science Publications 1999.</ref>
 
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==
 
<references/>
 
{{Numerical PDE}}
 
{{DEFAULTSORT:Neumann-Neumann Methods}}
[[Category:Domain decomposition methods]]

Latest revision as of 08:02, 5 May 2014

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