Volume 49, pp. 1-27, 2018.

Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems

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 transformation-of-basis approach is used to improve the convergence by combining the local assembly with a change of basis. Suitable basis vectors can be constructed by the recently introduced adaptive coarse space approaches. The resulting FETI-DP and BDDC methods fulfill a condition number bound independent of heterogeneities in the problem. The adaptive method with a transformation of basis presented here builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses a coarse space constructed from solving small, local eigenvalue problems on closed faces and on a small number of edges. In contrast to our earlier work on adaptive FETI-DP, the coarse space correction is not implemented by using balancing (or deflation), which requires the use of an exact coarse space solver, but by using local transformations. This will make it simpler to extend the method to a large number of subdomains and large supercomputers. The recently established theory of a generalized transformation-of-basis approach yields a condition number estimate for the preconditioned operator that is independent of jumps of the coefficients across and inside subdomains when using the local adaptive constraints. It is shown that all results are also valid for BDDC. Numerical results are presented in three dimensions for FETI-DP and BDDC. We also provide a comparison of different scalings, i.e., deluxe, rho, stiffness, and multiplicity for our adaptive coarse space in 3D.

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Key words

domain decomposition, FETI-DP, BDDC, coarse space, adaptive, eigenvalue problem, elliptic partial differential equations

AMS subject classifications


Links to the cited ETNA articles

[9]Vol. 45 (2016), pp. 524-544 Juan G. Calvo and Olof B. Widlund: An adaptive choice of primal constraints for BDDC domain decomposition algorithms
[34]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
[53]Vol. 46 (2017), pp. 273-336 Clemens Pechstein and Clark R. Dohrmann: A unified framework for adaptive BDDC

ETNA articles which cite this article

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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