3tr041211.pdf - TR2004-857 FETIDP BDDC and Block Cholesky...

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TR2004-857 FETI–DP, BDDC, and Block Cholesky Methods Jing Li * and Olof Widlund January 5, 2005 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 symmet- ric positive definite linear system of equations. Instead of introducing Lagrange multipliers to enforce the coarse level, primal continuity con- straints 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. Keywords: domain decomposition, FETI, Neumann-Neumann, BDDC, block Cholesky, primal constraints. AMS subject classification: 65F05, 65F10, 65N55. 1 Introduction The purpose of this paper is to give a simple derivation of two important domain decomposition methods namely the FETI–DP (dual-primal finite element tearing and interconnection) and the BDDC (balancing domain de- composition by constraints) algorithms. The latter, due to Clark Dohrmann, * Department of Mathematical Sciences, Kent State University, Kent, OH 44242 [email protected] , URL: li/ Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 [email protected] , URL: This work was supported in part by the US Department of Energy under Contracts DE-FG02-92ER25127 and DE-FC02-01ER25482. 1
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[13, 2, 1], represents an interesting redesign of the balancing Neumann– Neumann algorithms with the coarse, global component of a BDDC algo- rithm 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 frame- work. We believe the success of this approach has been demonstrated by several of our friends who have quickly implemented FETI–DP and BDDC algorithms on the basis of an early version of this paper. This paper is organized as follows. We first consider the case of two subdomains and two dimensions. We show that the iteration matrices of the standard Neumann–Neumann and one-level FETI methods are very closely related and that they have the same eigenvalues. A FETI–DP algorithm is then introduced in terms of a single primal constraint, which enforces the continuity of an edge average. A minimum requirement is then met that the quadratic form defined by the sum of the quadratic forms, formed with the stiffness matrices of the subdomains and subject to the primal constraints, is positive definite. In any such a case, the subdomain problems are positive
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