Volume 45, pp. 75-106, 2016.

A comparison of adaptive coarse spaces for iterative substructuring in two dimensions

Axel Klawonn, Patrick Radtke, and Oliver Rheinbach


The convergence rate of iterative substructuring methods generally deteriorates when large discontinuities occur in the coefficients of the partial differential equations to be solved. In dual-primal Finite Element Tearing and Interconnecting (FETI-DP) and Balancing Domain Decomposition by Constraints (BDDC) methods, sophisticated scalings, e.g., deluxe scaling, can improve the convergence rate when large coefficient jumps occur along or across the interface. For more general cases, additional information has to be added to the coarse space. One possibility is to enhance the coarse space by local eigenvectors associated with subsets of the interface, e.g., edges. At the center of the condition number estimates for FETI-DP and BDDC methods is an estimate related to the $P_D$-operator which is defined by the product of the transpose of the scaled jump operator $B_D^T$ and the jump operator $B$ of the FETI-DP algorithm. Some enhanced algorithms immediately bring the $P_D$-operator into focus using related local eigenvalue problems, and some replace a local extension theorem and local Poincaré inequalities by appropriate local eigenvalue problems. Three different strategies, suggested by different authors, are discussed for adapting the coarse space together with suitable scalings. Proofs and numerical results comparing the methods are provided.

Full Text (PDF) [877 KB], BibTeX

Key words

FETI-DP, BDDC, eigenvalue problem, coarse space, domain decomposition, multiscale

AMS subject classifications

65F10, 65N30, 65N55

ETNA articles which cite this article

Vol. 45 (2016), pp. 524-544 Juan G. Calvo and Olof B. Widlund: An adaptive choice of primal constraints for BDDC domain decomposition algorithms
Vol. 46 (2017), pp. 273-336 Clemens Pechstein and Clark R. Dohrmann: A unified framework for adaptive BDDC
Vol. 49 (2018), pp. 1-27 Axel Klawonn, Martin Kühn, and Oliver Rheinbach: Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems
Vol. 49 (2018), pp. 28-40 Leszek Marcinkowski and Talal Rahman: Additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems
Vol. 49 (2018), pp. 64-80 Hyea Hyun Kim, Eric Chung, and Junxian Wang: BDDC and FETI-DP algorithms with a change of basis formulation on adaptive primal constraints
Vol. 48 (2018), pp. 156-182 Alexander Heinlein, Axel Klawonn, Jascha Knepper, and Oliver Rheinbach: Multiscale coarse spaces for overlapping Schwarz methods based on the ACMS space in 2D
Vol. 52 (2020), pp. 43-76 Axel Klawonn, Martin Kühn, and Oliver Rheinbach: Coarse spaces for FETI-DP and BDDC Methods for heterogeneous problems: connections of deflation and a generalized transformation-of-basis approach
Vol. 53 (2020), pp. 562-591 Alexander Heinlein, Axel Klawonn, Martin Lanser, and Janine Weber: A frugal FETI-DP and BDDC coarse space for heterogeneous problems

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