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</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. 989–1015.
</ref>. In its original formulation, BDD performs well only for 2nd order problems, such [[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
</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 mutiplicative 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--659.
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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 belongs to class of [[Neumann
==References==
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