BDDC

In numerical analysis, BDDC (balancing domain decomposition by constraints) is a domain decomposition method for solving large symmetric, positive definite systems of linear equations that arise from the finite element method. BDDC is used as a preconditioner to the conjugate gradient method. A specific version of BDDC is characterized by the choice of coarse degrees of freedom, which can be values at the corners of the subdomains, or averages over the edges or the faces of the interface between the subdomains. One application of the BDDC preconditioner then combines the solution of local problems on each subdomains with the solution of a global coarse problem with the coarse degrees of freedom as the unknowns. The local problems on different subdomains are completely independent of each other, so the method is suitable for parallel computing. With a proper choice of the coarse degrees of freedom (corners in 2D, corners plus edges or corners plus faces in 3D) and with regular subdomain shapes, the condition number of the method is bounded when increasing the number of subdomains, and it grows only very slowly with the number of elements per subdomain. Thus the number of iterations is bounded in the same way, and the method scales well with the problem size and the number of subdomains.

BDDC was introduced by Dohrmann as a simpler primal alternative to the FETI-DP domain decomposition method by Farhat et al. The name of the method was coined by Mandel and Dohrmann, because it can be understood as further development of the BDD (balancing domain decomposition) method. The same method was also proposed independently by Fragakis and Papadrakakis under the name P-FETI-DP, and by Cros, which, however, was not recognized for some time. See for a proof that these are all actually the same method as BDDC. Mandel, Dohrmann, and Tezaur proved that the eigenvalues of BDDC and FETI-DP are identical, except for the eigenvalue equal to one, which may be present in BDDC but not for FETI-DP, and thus their number of iterations is practically the same. Much simpler proofs of this fact were obtained later by Li and Widlund and by Brenner and Sung.

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