Balancing ___domain decomposition method: Difference between revisions

Revision as of 17:01, 24 December 2010

Latest revision as of 10:38, 23 September 2023

(10 intermediate revisions by 10 users not shown)

Line 1:

In [[numerical analysis]], the '''balancing ___domain decomposition method (BDD)''' is an [[iterative method]] to find the solution of a [[symmetric]] [[positive definite]] system of [[linear]] [[algebraic equation]]s arising from the [[finite element method]] .<ref name="Mandel-1993-BDD">J. Mandel, ''Balancing ___domain decomposition'', Comm. Numer. Methods Engrg., 9 (1993), pp. 233&ndash;241. {{doi|10.1002/cnm.1640090307}}

</ref>. In each iteration, it combines the solution of local problems on non-overlapping subdomains with a coarse problem created from the subdomain [[null space|nullspaces]]. BDD requires only solution of subdomain problems rather than access to the matrices of those problems, so it is applicable to situations where only the solution operators are available, such as in [[oil reservoir]] [[simulation]] by [[mixed finite elements]] .<ref name="Cowsar-1995-BDD">L. C. Cowsar, J. Mandel, and M. F. Wheeler, ''Balancing ___domain decomposition for mixed finite elements'', Math. Comp., 64 (1995), pp. 989&ndash;1015. {{doi|10.1090/S0025-5718-1995-1297465-9}}

</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 mutiplicativemultiplicative 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.

Line 13:

</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.

Line 22:

==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}}