The aim of this work is to propose numerical strategies for obtaining convergence of fixed-point systems even when the conventional fixed-point iterations (also known as Picard iterations) diverge. It should be noted that acceleration (or extrapolation) methods [1] reliable in case of non-convergences of the fixed-point iterations have been little investigated in the literature in the vector case. The unified vector-based acceleration formalism introduced in [2] is used. Several vector acceleration approaches (such that Anderson, n-scalar Steffensen, Irons & Tucks, Secant,...) are then challenged against industrial test cases of interest using a fixed-point based solution where the standard fixed-point iterations may diverge: ˆ- steady state population balance of precipitation processes [3]; ˆ- multiphysics block Gauss-Seidel coupling for nuclear fuel behaviour simulations [4]; ˆ- contact mechanics problems with a staggered force-displacement solution strategy. The results highlight an interesting and rather novel trend: when the fixed-point iterations diverge in an oscillatory form, the Crossed Secant method [2] seems to give the best performances. In the end, it is the only one-step method that enables to reach the fixed-point convergence. It can effectively compete with the n-scalar Steffensen and the Irons & Tuck approaches, both two-step methods originally derived from Aitken’s methodology. Anderson-like sequences fail to achieve convergence in these cases. REFERENCES [1] C. Brezinski. Convergence acceleration during the 20th century. Journal of Comput. Appl. Math., 122:1–21, 2000. [2] I. Ramière and T. Helfer. Iterative residual-based vector methods to accelerate fixed point iterations. Computers and Mathematics with Applications, 70(9):2210 – 2226, 2015. [3] C. C. Ruiz Vasquez, N. Lebaz, I. Ramière, S. Lalleman, D. Mangin, and M. Bertrand. Fixed point convergence and acceleration for steady state population balance modelling of precipitation processes: Application to neodymium oxalate. Chemical Engineering Research and Design, 177:767–777, 2022. [4] S. Bernaud, I. Ramière, G. Latu, and B. Michel. PLEIADES: A numerical framework dedicated to the multiphysics and multiscale nuclear fuel behavior simulation. Annals of Nuclear Energy, 205:110577, 2024.