The aim of this work is to propose numerical strategies for obtaining iterative solution of fixed-point systems even when the conventional fixed-point iterations (also known as Picard iterations) aim to diverge. It should be noted that the construction of acceleration (or extrapolation) methods which are reliable in case of non-convergence of the fixed-point iterations, has been little investigated in the literature in the vector case. The unified vector-based acceleration formalism introduced in [Ramière & Helfer, 2025] 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: • steady state population balance of precipitation processes; • multiphysics block Gauss-Seidel coupling for nuclear fuel behaviour simulations; • contact mechanics problems with a staggered force-displacement Uzawa like solution strategy. The results highlight an interesting and rather novel trend: when the fixed-point iterations diverge in an oscillatory form, the so-called Crossed Secant method seems to give the best performances. In the end, it is the only one-step method that enables to reach the convergence of the iterative process. 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.