An algebraic theory for primal and dual substructuring methods by constraints

Some Recent Tools and a BDDC Algorithm for 3D Problems in H(curl)

“…From Theorem 25 of [10], the condition number of the BDDC preconditioned operator is bounded above by…”

“…Background information and related theory for BDDC can be found in several references including [2,9,10,8,1]. Let u i and u denote vectors of finite element coefficients associated with Γ i and Γ .…”

Lower Bounds in Domain Decomposition

“…But the generalization of the results of Sections 3 and 4 to three dimensions is straightforward, and the results in Section 5 have been generalized [5] to three dimensions (wire-basket algorithm [9]) and Neumann-Neumann algorithms [12]. Since the balancing domain decomposition by constraint (BDDC) method has the same condition number as the FETI-DP method [17,15], the sharpness of the condition number estimate for BDDC [16] also follows from Theorem 4.…”

A BDDC Preconditioner for Saddle Point Problems

“…[10,Section 6.2] where references to earlier work can also be found. It has also been established that the preconditioned operators of a pair of BDDC and FETI-DP algorithms, with the same primal constraints, have the same nonzero eigenvalues for positive definite elliptic problems; see [9,3,7].…”

“…Introduced by Dohrmann [1] and analyzed in the elliptic case by him, Mandel, and Tezaur [9], these methods represent an important advance over the balancing Neumann-Neumann methods that have been used extensively in the past to solve large finite element problems; cf. [10,Section 6.2] where references to earlier work can also be found.…”