The Balancing Domain Decomposition by Constraints (BDDC) method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory.
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Brenner SC, He Q (2003) Lower bounds for three-dimensional nonoverlapping domain decomposition algorithms. Numer Math 93: 445–470
Brenner SC, Sung L-Y (2000) Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions. Math Comp 69: 1319–1339
Brenner SC, Sung L-Y (2007) BDDC and FETI-DP without matrices or vectors. Comput Methods Appl Mech Eng 196: 1429–1435
Dohrmann CR (2003) A preconditioner for substructuring based on constrained energy minimization. SIAM J Sci Comput 25: 246–258
Dryja M, Widlund OB (1995) Schwarz methods of Neumann–Neumann type for three-dimensional elliptic finite element problems. Comm Pure Appl Math 48: 121–155
Farhat C, Lesoinne M, Le Tallec P, Pierson K, Rixen D (2001) FETI-DP: a dual-primal unified FETI method. I. A faster alternative to the two-level FETI method. Int J Numer Methods Eng 50: 1523–1544
Farhat C, Lesoinne M, Pierson K (2000) A scalable dual-primal domain decomposition method. Numer linear algebra appl, vol 7, pp 687–714. Preconditioning techniques for large sparse matrix problems in industrial applications (Minneapolis, MN, 1999)
Farhat C, Roux F-X (1991) A method of finite element tearing and interconnecting and its parallel solution algorithm. Int J Numer Methods Eng 32: 1205–1227
Klawonn A, Widlund OB (2006) Dual-primal FETI methods for linear elasticity. Comm Pure Appl Math 59: 1523–1572
Li J, Widlund OB (2006) FETI-DP, BDDC, and block Cholesky methods. Int J Numer Methods Eng 66: 250–271
Li J, Widlund OB (2007) On the use of inexact subdomain solvers for BDDC algorithms. Comput Methods Appl Mech Eng 196: 1415–1428
Mandel J (1993) Balancing domain decomposition. Comm Numer Methods Eng 9: 233–241
Mandel J, Dohrmann CR (2003) Convergence of a balancing domain decomposition by constraints and energy minimization. Numer Linear Algebra Appl 10: 639–659
Mandel J, Dohrmann CR, Tezaur R (2005) An algebraic theory for primal and dual substructuring methods by constraints. Appl Numer Math 54: 167–193
Mandel J, Sousedík B (2007) Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods. Comput Methods Appl Mech Eng 196: 1389–1399
Mandel J, Sousedík B, Dohrmann CR (2007) On multilevel BDDC. Lecture notes in computational science and engineering, vol 60, pp 287–294. Domain Decomposition Methods in Science and Engineering XVII
Smith BF, Bjørstad PE, Gropp WD (1996) Domain decomposition. Cambridge University Press, Cambridge (Parallel multilevel methods for elliptic partial differential equations)
Toselli A, Widlund O (2005) Domain decomposition methods—algorithms and theory. Springer series in computational mathematics, vol 34. Springer, Berlin
Tu X (2007) Three-level BDDC in three dimensions. SIAM J Sci Comput 29: 1759–1780
Tu X (2007) Three-level BDDC in two dimensions. Int J Numer Methods Eng 69: 33–59
J. Mandel and B. Sousedík were supported in part by the National Science Foundation under grants CNS-0325314, DMS-0713876, and CNS-0719641. B. Sousedík was supported by the program of the Information society of the Academy of Sciences of the Czech Republic 1ET400760509 and by the Grant Agency of the Czech Republic GA ČR 106/08/0403. C. R. Dohrmann was supported by Sandia, which is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94-AL85000.
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Mandel, J., Sousedík, B. & Dohrmann, C.R. Multispace and multilevel BDDC. Computing 83, 55–85 (2008). https://doi.org/10.1007/s00607-008-0014-7
- Iterative substructuring
- Additive Schwarz method
- Balancing domain decomposition
- Multispace BDDC
- Multilevel BDDC
Mathematics Subject Classification (2000)