Multispace and multilevel BDDC

Abstract

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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Correspondence to Jan Mandel.

Additional information

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

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Keywords

  • Iterative substructuring
  • Additive Schwarz method
  • Balancing domain decomposition
  • BDD
  • BDDC
  • Multispace BDDC
  • Multilevel BDDC

Mathematics Subject Classification (2000)

  • 65N55
  • 65M55
  • 65Y05