World Library  
Flag as Inappropriate
Email this Article

Neumann–Neumann methods

Article Id: WHEBN0016252918
Reproduction Date:

Title: Neumann–Neumann methods  
Author: World Heritage Encyclopedia
Language: English
Subject: Domain decomposition methods, Abstract additive Schwarz method, FETI-DP, Neumann–Dirichlet method, Fictitious domain method
Collection: Domain Decomposition Methods
Publisher: World Heritage Encyclopedia

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 ui 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


  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.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.