Convergence of a balancing domain decomposition by constraints and energy minimization

Abstract

Dedicated to Professor Ivo Marek on the occasion of his 70th birthday. AbstractA convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts. The preconditioner is formulated as an Additive Schwarz method and analyzed by building on existing results for Balancing Domain Decomposition. The main result is a bound on the condition number based on inequalities involving the matrices of the preconditioner. Estimates of the usual form C(1 + log 2 (H/h)) are obtained under the standard assumptions of substructuring theory. Computational results demonstrating the performance of method are included.

FETI-DP, BDDC, and block Cholesky methods

“…The BDDC algorithm, first introduced by Dohrmann, [13,2,1], is a variant of the two-level Neumann-Neumann type preconditioner for solving the interface problem (27). In the BDDC preconditioner, the coarse-level problem is assembled from a special set of coarse basis functions, which are the minimum energy extension on the subdomains subject to sets of primal constraints; these coarse-level basis functions, in fact, correspond to the matrix Φ in the block Cholesky elimination in Equation (3).…”

“…The latter, due to Clark Dohrmann, [13,2,1], represents an interesting redesign of the balancing NeumannNeumann algorithms with the coarse, global component of a BDDC algorithm expressed in terms of a set of primal constraints, just as in the FETI-DP algorithms [4,5]. Throughout this paper, we will employ the language of block Cholesky elimination and our discussion can therefore also be seen as a guide to the design of domain decomposition methods using such a framework.…”

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

“…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.…”