Selecting Constraints in Dual-Primal FETI Methods for Elasticity in Three Dimensions
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 Farhat et al.  and then extended to three dimensional elasticity problems by Farhat et al. . 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 Klawonn et al. [2002a,b]. 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.
KeywordsDomain Decomposition Method Primal Vertex Primal Constraint Edge Average Rigid Body Mode
Unable to display preview. Download preview PDF.
- P. G. Ciarlet. Mathematical Elasticity Volume I: Three-Dimensional Elasticity. North-Holland, 1988.Google Scholar
- A. Klawonn, O. Rheinbach, and O. B. Widlund. Some computational results for Dual-Primal FETI methods for three dimensional elliptic problems. In R. Kornhuber, R. Hoppe, D. Keyes, J. Périaux, O. Pironneau, and J. Xu, editors, Domain Decomposition Methods. Springer-Verlag, Lecture Notes in Computational Science and Engineering, 2003. Proceedings of the 15th International Conference on Domain Decomposition Methods, Berlin, July 21–25, 2003.Google Scholar
- A. Klawonn and O. Widlund. Dual-Primal FETI methods for linear elasticity. Technical report, Courant Institute of Mathematical Sciences, Department of Computer Science, 2004. In preparation.Google Scholar
- A. Klawonn, O. B. Widlund, and M. Dryja. Dual-Primal FETI methods with face constraints. In L. F. Pavarino and A. Toselli, editors, Recent developments in domain decomposition methods, pages 27–40. Springer-Verlag, Lecture Notes in Computational Science and Engineering, Volume 23, 2002b.Google Scholar