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Overlapping technique can make each subdomain contain more nodes, and the overlapped subdomains are nonlonger disjoint, is it taking the average value of the multiple nodes as the result.

After overlapping, we will solve larger sundomins, and it will cost more time and storage, how can the benefit from taking average value of multiple nodes outperform the increse cost per iteration.

Why is it necessary and usually beneficial.

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The problem with elliptic operators is that the solution at any specific point depends on all the domain. The overlap allows to 'mix' the approximate solution between subdomains and therefore accelerates the solution of the problem. http://en.wikipedia.org/wiki/Domain_decomposition_methods

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  • $\begingroup$ Thank you. Sometimes the overlapping accelerate the convergence much, while sometimes its efficience is negigible, so, what factor affect its performance. $\endgroup$ – Z. Shen Nov 27 '14 at 9:43
  • $\begingroup$ It's not only about the overlap. In fact to design efficient overlaping DD preconditioner you have to add an additional submatrix which mixes informations between subdomains. This is done generally by using a coarse space approximation due to small eigen-vectors on each subdomain in order to avoid the 'plateau' effect in convergence. You can have a great (both theoretical and practical) overview of this in Nataf's presentation: ann.jussieu.fr/~nataf/talkCEA20130329.pdf Do not hesitate to ask if anything is unclear. Cheers $\endgroup$ – Tom Nov 27 '14 at 11:01

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