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Mathematics of Computation

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BDDC Algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields


Authors: Duk-Soon Oh, Olof B. Widlund, Stefano Zampini and Clark R. Dohrmann
Journal: Math. Comp.
MSC (2010): Primary 65F08, 65F10, 65N30, 65N55
DOI: https://doi.org/10.1090/mcom/3254
Published electronically: June 21, 2017
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Abstract: A BDDC domain decomposition preconditioner is defined by a coarse component, expressed in terms of primal constraints, a weighted average across the interface between the subdomains, and local components given in terms of solvers of local subdomain problems. BDDC methods for vector field problems discretized with Raviart-Thomas finite elements are introduced. The methods are based on a deluxe type of weighted average and an adaptive selection of primal constraints developed to deal with coefficients with high contrast even inside individual subdomains. For problems with very many subdomains, a third level of the preconditioner is introduced.

Under the assumption that the subdomains are all built from elements of a coarse triangulation of the given domain, that the meshes of each subdomain are quasi uniform and that the material parameters are constant in each subdomain, a bound is obtained for the condition number of the preconditioned linear system which is independent of the values and the jumps of these parameters across the interface between the subdomains as well as the number of subdomains. Numerical experiments, using the PETSc library, are also presented which support the theory and show the effectiveness of the algorithms even for problems not covered by the theory. Included are also experiments with Brezzi-Douglas-Marini finite element approximations.


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Additional Information

Duk-Soon Oh
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
Email: duksoon@math.rutgers.edu

Olof B. Widlund
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012
Email: widlund@cims.nyu.edu

Stefano Zampini
Affiliation: Computer, Electrical and Mathematical Sciences and Engineering Division, Extreme Computing Research Center, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
Email: stefano.zampini@kaust.edu.sa

Clark R. Dohrmann
Affiliation: Computational Solid Mechanics and Structural Dynamics, Sandia National Laboratories, Albuquerque, New Mexico, 87185
Email: crdohrm@sandia.gov

DOI: https://doi.org/10.1090/mcom/3254
Keywords: Domain decomposition, BDDC preconditioner, Raviart-Thomas finite elements, multilevel preconditioners, adaptive selection of coarse spaces
Received by editor(s): January 18, 2016
Received by editor(s) in revised form: October 7, 2016
Published electronically: June 21, 2017