3. Dual and Dual-Primal FETI Methods for Elliptic Problems with Discontinuous Coefficients in Three Dimensions

The Finite Element Tearing and Interconnecting (FETI) methods were first introduced by Farhat and Roux [FMR94]. An important advance, making the rate of convergence of the iteration less sensitive to the number of unknowns of the local problems, was made by Farhat, Mandel, and Roux a few years later [FMR94]. For a detailed introduction, see [FR94] and we also refer to our own papers for many additional references. Our own work, cf. [KW01, KW00b], owes a great deal to the pioneering theoretical work by Mandel and Tezaur [MT96, MT00]. The principal purpose of this paper is to survey some recent results developed by the authors. We introduce new one-parameter families of one-level FETI as well as of dual–primal FETI preconditioners which have a rate of convergence which is bounded independently of possible jumps of the coefficients of an elliptic model problem often considered in the theory of Neumann–Neumann and other iterative substructuring algorithms; see, e.g., [DW95, DSW94, MB96] and the references therein. Our new results become possible because of special scalings. One of them, for the preconditioner, is closely related to an important algorithmic idea used in the best of the Neumann–Neumann methods. The other scaling affects the choice of the projection which is used in each step of the one–level FETI iteration, whether preconditioned or not. For a certain choice of the two scalings, our preconditioner for the one–level FETI methods results in a method that is identical to one recently tested successfully for very difficult and large problems by Bhardwaj et al. [BDF00]. The scaling used in the preconditioner was originally introduced by Rixen and Farhat; see [RF99]. We note that, by now, many variants of the FETI algorithms have been designed and that a number of them have been tested extensively; see in particular [RFTM99]. Some of our results have also already been extended to Maxwell’s equation in two dimensions by Toselli and Klawonn [TK99]. Recently, Farhat et al. [FLLT99] introduced a dual–primal FETI algorithm suitable for second order elliptic problems in the plane and for plate problems. A convergence analysis in the case of benign coefficients is given by Mandel and Tezaur [MT00]. Numerical experiments show a poor performance for this algorithm in three