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Selecting constraints in dual-primal FETI methods for elasticity in three dimensions. (English) Zbl 1152.74386
Kornhuber, Ralf (ed.) et al., Domain decomposition methods in science and engineering. Selected papers of the 15th international conference on domain decomposition, Berlin, Germany, July 21–25, 2003. Berlin: Springer (ISBN 3-540-22523-4/pbk). Lecture Notes in Computational Science and Engineering 40, 67-81 (2005).
Summary: Iterative substructuring methods with Lagrange multipliers for the elliptic system of linear elasticity are considered. The algorithms belong to the family of dual-primal FETI methods which was introduced for linear elasticity problems in the plane by C. Farhat, M. Lesoinne, P. LeTallec, K. Pierson and D. Rixen [Int. J. Numer. Methods Eng. 50, No. 7, 1523–1544 (2001; Zbl 1008.74076)] and then extended to three dimensional elasticity problems by C. Farhat, M. Lesoinne and K. Pierson [Numer. Linear Algebra Appl. 7, No. 7–8, 687–714 (2000; Zbl 1051.65119)]. In dual-primal FETI methods, some continuity constraints on primal displacement variables are required to hold throughout the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by the use of dual Lagrange multipliers, as in the older one-level FETI algorithms. The primal constraints should be chosen so that the local problems become invertible. They also provide a coarse problem and they should be chosen so that the iterative method converges rapidly.
Recently, the family of algorithms for scalar elliptic problems in three dimensions was extended and a theory was provided in A. Klawonn, O. B. Widlund and M. Dryja [SIAM J. Numer. Anal. 40, No. 1, 159–179 (2002; Zbl 1032.65031); Lect. Notes in Comput. Scii. Eng. 23, 27–40 (2002; Zbl 1009.65069)]. It was shown that the condition number of the dual-primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds can otherwise be made independent of the number of subdomains, the mesh size, and jumps in the coefficients.
In the case of the elliptic system of partial differential equations arising from linear elasticity, essential changes in the selection of the primal constraints have to be made in order to obtain the same quality bounds for elasticity problems as in the scalar case. Special emphasis is given to developing robust condition number estimates with bounds which are independent of arbitrarily large jumps of the material coefficients. For benign coefficients, without large jumps, selecting an appropriate set of edge averages as primal constraints are sufficient to obtain good bounds, whereas for arbitrary coefficient distributions, additional primal first order moments are also required.
For the entire collection see [Zbl 1049.65003].

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
74B05 Classical linear elasticity
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