Keywords: domain decomposition, total FETI, hybrid FETI, scalable algorithm.

The original FETI method, also called the FETI-1 method, was originally introduced for the numerical solution of the large linear systems arising in linearized engineering problems by Farhat and Roux [1]. In both FETI methods a body is decomposed into several non-overlapping subdomains and the continuity between the subdomains is enforced by Lagrange multipliers. Using the theory of duality, a smaller and relatively well conditioned dual problem can be derived and efficiently solved using a suitable variant of the conjugate gradient algorithm.

The original FETI algorithm, where only the favorable distribution of the spectrum of the dual Schur complement matrix was considered, was efficient only for a small number of subdomains. So it was later extended by introducing a natural coarse problem, whose solution was implemented by auxiliary projectors so that the resulting algorithm became in a sense optimal.

In total FETI method also the Dirichlet boundary conditions are enforced using Lagrange multipliers. Hence all subdomain stiffness matrices are singular with a-priori known kernels which is a great advantage in the numerical solution. With known kernel basis we can regularize effectively the stiffness matrix and use any standard Cholesky type decomposition method for nonsingular matrices.

Even if there are several efficient coarse problem parallelization strategies there are still some size limitations of the coarse problem. So several hybrid (multilevel) methods were proposed. The key idea is to aggregate small number of neighbouring subdomains into the clusters, which naturally results into the smaller coarse problem. In our hybrid total FETI, the aggregation of subdomains into the clusters is enforced again by Lagrange multipliers. Thus the total FETI method is used on both cluster and subdomain levels. This approach simplifies implementation of the hybrid FETI methods and enables to the parallelization of the original problem to be extended up to tens of thousands of cores as a result of the lesser memory requirements. This has the positive effect of reducing the coarse space. The negative one has a worse convergence rate compared with the original TFETI. To improve it the transformation of basis originally introduced by Klawonn and Widlund [2], Klawonn and Rheinbach [3] and Li and Widlund [4] is applied to the derived hybrid algorithm.

The proposed hybrid total FETI algorithm was implemented in C++ and the results provided in the full paper demonstrate the ability of the algorithm to overcome the memory bottleneck of the original total FETI algorithm.

1

C. Farhat, F-X. Roux, "An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems", SIAM J. Sci. Stat. Comput. 13: 379-396, 1992.

2

A. Klawonn and O. B. Widlund, "Dual-primal FETI methods for linear elasticity", Communications on Pure and Applied Mathematics, 59(11):1523-1572, 2006

3

A. Klawonn and O. Rheinbach, "A parallel implementation of Dual-Primal FETI methods for three dimensional linear elasticity using a transformation of basis", SIAM J. Sci. Comput., 28(5):1886-1906, 2006

4

J. Li and O. B. Widlund, "FETI-DP, BDDC, and block Cholesky methods", International Journal for Numerical Methods in Engineering, 66:250-271, 2006.

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