Balancing ___domain decomposition method: Difference between revisions

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</ref> In its original formulation, BDD performs well only for 2nd order problems, such [[Elasticity (physics)|elasticity]] in 2D and 3D. For 4th order problems, such as [[plate bending]], it needs to be modified by adding to the coarse problem special basis functions that enforce continuity of the solution at subdomain corners,<ref name="LeTallec-1998-NND">P. Le Tallec, J. Mandel, and M. Vidrascu, ''A Neumann&ndash;Neumann ___domain decomposition algorithm for solving plate and shell problems'', SIAM Journal on Numerical Analysis, 35 (1998), pp. 836&ndash;867. {{doi|10.1137/S0036142995291019}}

</ref> which makes it however more expensive. The [[BDDC]] method uses the same corner basis functions as,<ref name="LeTallec-1998-NND"/> but in an additive rather than multiplicative fashion.<ref name="Mandel-2003-CBD">J. Mandel and C. R. Dohrmann, ''Convergence of a balancing ___domain decomposition by constraints and energy minimization'', Numer. Linear Algebra Appl., 10 (2003), pp. 639--659639–659. {{doi|10.1002/nla.341}}

</ref> The dual counterpart to BDD is [[FETI]], which enforces the equality of the solution between the subdomain by Lagrange multipliers. The base versions of BDD and FETI are not mathematically equivalent, though a special version of FETI designed to be robust for hard problems <ref name="Bhardwaj-2000-AFM">M. Bhardwaj, D. Day, C. Farhat, M. Lesoinne, K. Pierson, and D. Rixen, ''Application of the FETI method to ASCI problems --– scalability results on 1000 processors and discussion of highly heterogeneous problems'', International Journal for Numerical Methods in Engineering, 47 (2000), pp. 513&ndash;535. {{doi|10.1002/(SICI)1097-0207(20000110/30)47:1/3<513::AID-NME782>3.0.CO;2-V}}

</ref> has the same [[eigenvalue]]s and thus essentially the same performance as BDD.<ref name="Fragakis-2007-FDD">Y. Fragakis, ''Force and displacement duality in Domain Decomposition Methods for Solid and Structural Mechanics''. To appear in Comput. Methods Appl. Mech. Engrg., 2007.

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</ref><ref name="Sousedik-2008-EPD">B. Sousedík and J. Mandel, ''On the equivalence of primal and dual substructuring preconditioners''. arXiv:math/0802.4328, 2008.</ref>

The operator of the system solved by BDD is the same as obtained by eliminating the unknowns in the interiors of the subdomain, thus reducing the problem to the [[Schur complement]] on the subdomain interface. Since the BDD preconditioner involves the solution of [[Neumann problem]]s on all subdomain, it belongsis toa classmember of the [[Neumann&ndash;NeumannNeumann–Neumann methods|Neumann–Neumann class of methods]], named so named because they solve a Neumann problem on both sides of the interface between subdomains.

In the simplest case, the [[coarse problem|coarse space]] of BDD consists of functions constant on each subdomain and averaged on the interfaces. More generally, on each subdomain, the coarse space needs to only contain the [[nullspace]] of the problem as a subspace.

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==External links==

* [https://web.archive.org/web/20080228054053/http://www.mgnet.org/mgnet/Codes/jmandel/ BDD reference implementation at mgnet.org]

* [http://www.___domain-decomposition.com Domain Decomposition – Theory, publications, methods, algorithms.] {{Webarchive|url=https://web.archive.org/web/20110710143436/http://www.___domain-decomposition.com/ |date=2011-07-10 }}

{{Numerical PDE}}