2 Charbel Farhat And

We present a formulation of the FETI substructuring method as a matrix preconditioning algorithm and report on theoretical results and practical experience in large scale parallel implementations. The method has quasi-optimal convergence properties: the condition number can be proved to be independent of the number of subdomains, and to grow at most polylogarithmically with the number of elements per subdomain, for both solid and plate elements, and for static as well as dynamic problems. In practical tests, the method has performed very well even for irregular domains and shell models with junctures. The FETI (Finite Element Tearing and Interconnecting) method is a non-overlapping domain decomposition algorithm for the iterative solution of systems of equations arising from the nite element discretization of self-adjoint elliptic partial diierential equations. It is based on using direct solvers in subdomains and enforcing continuity at subdomain interfaces by Lagrange multipliers. The dual problem for the Lagrange multipliers is solved by a preconditioned conjugate gradient (PCG) algorithm. The FETI method was developed in Far91, FR91, FR92], and discussed in detail in the monograph FR94]. Unlike other related domain decomposition methods using Lagrange multipliers as unknowns GW88, Rou90], the FETI method uses the null spaces of the subdomain stiiness matrices (rigid body modes) to construct a small \coarse" problem that is solved in each PCG iteration. It was recognized in FMR94] and proved mathematically in MT96] that solving this coarse problem accomplishes a global exchange of information between the subdomains and results in a method which, for elasticity problems, has a condition number that grows only polylogarithmically DD9 Proceedings