# A Family of Dual-Primal FETI Domain Decomposition Methods in 3D

## Axel Klawonn

**Oct. 7, 2003, 3:30 p.m. P 215**

In this talk, iterative substructuring methods with Lagrange multipliers for stationary diffusion equations and for the elliptic system of linear elasticity are considered. The algorithms belong to the family of dual-primal FETI methods which were introduced for elliptic problems in the plane by Farhat, Lesoinne, Le Tallec, Pierson, and Rixen and were algorithmically extended to three dimensional problems by Farhat, Lesoinne, and Pierson.

In dual-primal FETI methods, some continuity constraints on primal displacement variables are forced to hold throughout iterations, as in primal methods, while other constraints are enforced by the use of dual Lagrange multiplier variables, as in standard one-level FETI. The primal constraints have to be chosen such that the local problems become invertible but they also provide a coarse problem.

Recently, the family of algorithms for scalar elliptic problems in three dimensions was further extended and new theoretical estimates were provided. 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 are otherwise independent of the number of subdomains, the mesh size, and jumps in the diffusion coefficients. Numerical results support the theoretical estimates.

In the case of the elliptic system of partial differential equations arising from linear elasticity, essential changes to the convergence theory have to be made. In this talk, recent results on the convergence theory for dual-primal FETI methods for linear elasticity are presented which are again only polylogarithmically dependent on the dimension of the individual subregion problems and otherwise independent of the number of subdomains, the mesh size, and jumps in the material coefficients, namely Young's modulus. The presented results are joint work with Olof Widlund, New York and in parts with Maksymilian Dryja, Warsaw and Oliver Rheinbach, Essen.