Simultaneous FETI and block FETI: Robust domain decomposition with multiple search directions

Abstract

Domain decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities. In particular, for heterogeneities along domain interfaces, the iterative techniques to solve the interface problem are lacking an efficient preconditioner. Recently, a robust approach, named finite element tearing and interconnection (FETI)-generalized eigenvalues in the overlaps (Geneo), was proposed where troublesome modes are precomputed and deflated from the interface problem. The cost of the FETI-Geneo is, however, high. We propose in this paper techniques that share similar ideas with FETI-Geneo but where no preprocessing is needed and that can be easily and efficiently implemented as an alternative to standard domain decomposition methods. In the block iterative approaches presented in this paper, the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition. Those search directions originate either from the domain-wise preconditioner (in the simultaneous FETI method) or from the block structure of the right-hand side of the interface problem (block FETI). We show on two-dimensional structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities, even when the interface is jagged or when the domains have a bad aspect ratio. The simultaneous FETI was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared with the regular FETI method.