Volume 52, pp. 43-76, 2020.

Coarse spaces for FETI-DP and BDDC Methods for heterogeneous problems: connections of deflation and a generalized transformation-of-basis approach

Axel Klawonn, Martin Kühn, and Oliver Rheinbach


In FETI-DP (Finite Element Tearing and Interconnecting) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods, the convergence behavior of the iterative scheme can be improved by implementing a coarse space using a transformation of basis and local assembly. This is an alternative to coarse spaces implemented by deflation or balancing. The transformation-of-basis approaches are more robust with respect to inexact solvers than deflation and therefore more suitable for multilevel extensions. In this paper, we show a correspondence of FETI-DP or BDDC methods using a generalized transformation-of-basis approach and of FETI-DP methods using deflation or balancing, where the deflation vectors are obtained from the transformation of basis. These methods then have essentially the same eigenvalues. As opposed to existing theory, this result also applies to general scalings and highly heterogeneous problems. We note that the new methods differ slightly from the classic FETI-DP and BDDC methods using a transformation of basis and that the classic theory has to be replaced. An important application for the theory presented in this paper are FETI-DP and BDDC methods with adaptive coarse spaces, i.e., where deflation vectors are obtained from approximating local eigenvectors. These methods have recently gained considerable interest.

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Key words

domain decomposition, FETI-DP, BDDC, coarse space, deflation, transformation of basis, change of variables, elliptic partial differential equations

AMS subject classifications


Links to the cited ETNA articles

[7]Vol. 45 (2016), pp. 524-544 Juan G. Calvo and Olof B. Widlund: An adaptive choice of primal constraints for BDDC domain decomposition algorithms
[33]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
[38]Vol. 45 (2016), pp. 75-106 Axel Klawonn, Patrick Radtke, and Oliver Rheinbach: A comparison of adaptive coarse spaces for iterative substructuring in two dimensions
[55]Vol. 46 (2017), pp. 273-336 Clemens Pechstein and Clark R. Dohrmann: A unified framework for adaptive BDDC

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