FETI-DP, BDDC, and block Cholesky methods

Abstract

Two popular non-overlapping domain decomposition methods, the FETI-DP and BDDC algorithms, are reformulated using Block Cholesky factorizations, an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations. Instead of introducing Lagrange multipliers to enforce the coarse level, primal continuity constraints in these algorithms, a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom. With the new formulations of these algorithms, a simplified proof is provided that the spectra of a pair of FETI-DP and BDDC algorithms, with the same set of primal constraints, are the same. Results of numerical experiments also confirm this result.

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.…”

A BDDC Preconditioner for Saddle Point Problems

“…These basis functions correspond to the primal interface velocity continuity constraints. We will always assume that the basis has been changed so that each primal basis function corresponds to an explicit degree of freedom which is shared by the neighboring subdomains; see [7], [5,Section 6], and [4] for more details of the change of basis. The complimentary space W ∆ is the direct sum of the subdomain dual interface velocity spaces W (i) ∆ , which correspond to the remaining interface velocity degrees of freedom and are spanned by basis functions which vanish at the primal degrees of freedom.…”

“…To compute the product of S −1 and a vector, a coarse level saddle point problem, for the primal variables, and subdomain Neumann problems, each with a few primal constraints, need to be solved; cf. [7,8].…”

“…[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].…”