A Domain Decomposition Based Two-Level Newton Scheme for Nonlinear Problems

One can refer to the paper by Keyes [1992] and the book by Smith et al. [1996] for in-depth reviews of domain decomposition (DD) methods. Of particular interest here are non-overlapping schemes such as iterative substructuring (Bjørstad et al. [2001]) and FETI methods (Farhat et al. [2001]). When solving non-linear boundary value problems (BVPs) via domain decomposition it is common to use Newton type algorithms and then to apply existing DD approaches to the ensuing linearized problems (Knoll and Keyes [2002]). The NK-Schwarz scheme (Keyes [1995]) as the name suggests, uses a Krylov scheme equipped with a Schwarz preconditioner to solve this linear update equation. If the non-linear effects are unbalanced, i.e., the nonlinearity has a significant spatial variation, then the Jacobian becomes ill-conditioned and hence the NK-Schwarz scheme is not effective cf. Cai and Keyes [2002]. A scheme that proves effective for problems with unbalanced nonlinearities is the multi level Newton Schwarz (MLN-Schwarz) scheme that was originally introduced to solve multi-physics problems (Bächtold et al. [1995], Aluru and White [1999]). Kim et al. [2003] implemented a serial version of the MLN-Schwarz to solve fluid-structure interaction problems which had the flavor of a multiplicative Schwarz approach. The scheme employs a global consistency equation in