FETI-DP, BDDC, and block Cholesky methods

• Published in: International journal for numerical methods in engineering Vol. 66; no. 2; pp. 250 - 271
• Main Authors: Li, Jing, Widlund, Olof B.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 09.04.2006; Wiley
• Subjects: Algorithms; BDDC; block Cholesky; Blocking; Cholesky factorization; Computational techniques; Continuity; domain decomposition; Exact sciences and technology; FETI; Fundamental areas of phenomenology (including applications); Linear systems; Mathematical analysis; Mathematical methods in physics; Mathematical models; Neumann-Neumann; Physics; primal constraints; Spectra; Neumann problem; Neumann-Neumann; Laplace equations; Matrix factorization; Experimental study; Variable transformation; Finite element method; Lagrange multiplier; FETI; Cholesky method; Positive definite matrix; block Cholesky; Modelling; Domain decomposition; Cholesky factorization; BDDC; primal constraints
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.1553

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 formulation 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 essentially the same. Numerical experiments for a two‐dimensional Laplace's equation also confirm this result. Copyright © 2005 John Wiley & Sons, Ltd.