Primal and dual localized AFETI methods for the partitioned solution of heterogeneous structural systems

Partitioned methods play a crucial role in enabling scalable simulations of large-scale structural systems. While dual-based approaches such as FETI have demonstrated considerable success, they often suffer from limitations in robustness and scalability when applied to ill-conditioned heterogeneous problems. Primal methods, particularly Balancing Domain Decomposition (BDD) and its variant BDDC, have addressed some of these issues, yet the computation of their coarse problems continues to present a significant bottleneck in massively parallel settings. This paper introduces a novel class of primal domain decomposition methods derived from the algebraic FETI (AFETI) framework with localized Lagrange multipliers. Building upon our previous research on dual AFETI methods González and Park (2020) [1] , we present a localized primal variant within the same theoretical framework. The proposed primal iterative methods search for partition boundary displacements in appropriately chosen subspaces. A natural projection of these primal unknowns leads to a formulation that is cheaper than equivalent dual methods and better suited for ill-conditioned systems. Additionally, a one-level variant is introduced, incorporating additional constraints to produce a coarse problem structurally similar to BDD, but with improved robustness and lower computational overhead. Although the primal and dual forms of AFETI are theoretically equivalent, numerical experiments confirm that the primal approach offers greater generality and enhanced robustness for the solution of heterogeneous partitioned problems. These features make the proposed primal methods particularly attractive for large-scale nonlinear structural applications.