Nonlinearly preconditioned FETI method

International audienceWe consider the Finite Element approximation of the solution to nonlinear elliptic partial differential equations such as the ones encountered in (quasi)-static mechanics, in transient mechanics with implicit time integration, or in thermal diffusion. Non-overlapping domain decomposition methods (DDM) offer an interesting framework for the distribution of the resolution. We focus on methods allowing independent nonlinearcomputations on the subdomains, sometimes called “nonlinear relocalization techniques”. Nonlinear counterparts to classical non-overlapping DDM have been proposed: non-linear primal (Dirichlet) and mixed (Robin) approach [Philippe Cresta, Olivier Allix, Christian Rey, and Stéphane Guinard. Nonlinear localization strategies for domain decomposition methods: Application to post-buckling analyses. Computer Methods in Applied Mechanics and Engineering, 196(8):1436–1446, 2007], dual approach [Julien Pebrel, Christian Rey, and Pierre Gosselet. A nonlinear dual-domain decomposition method: Application to structural problems with damage. International Journal for Multiscale Computational Engineering, 6(3), 2008], and nonlinear FETIDP and BDDC [Axel Klawonn, Martin Lanser, and Oliver Rheinbach. Nonlinear FETI-DP and BDDC Methods. SIAM Journal on Scientific Computing, 36(2):A737–A765, 2014]. The latter methods were improved and assessed at a large scale in [Axel Klawonn, Martin Lanser, Oliver Rheinbach, and Matthias Uran. Nonlinear FETI-DP and BDDC methods: A unified framework and parallel results. SIAM J. Sci. Comput., 39(6):C417–C451, 2017]. A global framework for primal/dual/mixed approaches was also proposed [Camille Negrello, Pierre Gosselet, Christian Rey, and Julien Pebrel. Substructured formulations of nonlinear structure problems–influence of the interface condition. International Journal for Numerical Methods in Engineering, 2016] and the impedance of the mixed approach was improved [Camille Negrello, Pierre Gosselet, and Christian Rey. A new impedance accounting for short and long range effects in mixed substructured formulations of nonlinear problems. International Journal for Numerical Methods in Engineering, 2017].Our objective is to double the intensity of the local independent nonlinear computations by modifying the condensed problem to be solved. The method can be interpreted as proposing a nonlinear preconditioner [Peter R Brune, Matthew G Knepley, Barry F Smith, and Xuemin Tu. Composing scalable nonlinear algebraic solvers. SIAM Review, 57(4):535–565, 2015] to the nonlinear DDM. It appears that this idea applies particularly easily to the dual approach, under an hypothesis equivalent to infinitesimal strain in mechanics. When applying a Newton algorithm to this nonlinear preconditioned condensed system, one alternates a sequence of two independent nonlinear local solves (one Neumann problem and one Dirichlet problem separated by one all-neighbor communication) and an interface tangent solve which exactly has the structure of a linear preconditioned FETI problem. Academic assessments show that the sequence of two local nonlinear solves can reduce the need of global Newton iterations and thus the number of calls to the communication-demanding Krylov solver