Iterative residual-based vector methods to accelerate fixed point iterations

Fixed point iterations are still the most common approach to dealing with a variety of numerical problems such as coupled problems (multi-physics, domain decomposition, …) or nonlinear problems (electronic structure, heat transfer, nonlinear mechanics, …). Methods to accelerate fixed point iteration convergence or more generally sequence convergence have been extensively studied since the 1960’s. For scalar sequences, the most popular and efficient acceleration method remains the Δ2Δ2 of Aitken. Various vector acceleration algorithms are available in the literature, which often aim at being multi-dimensional generalizations of the Δ2Δ2 method.In this paper, we propose and analyse a generic residual-based formulation for accelerating vector sequences. The question of the dynamic use of this residual-based transformation during the fixed point iterations for obtaining a new accelerated fixed point method is then raised. We show that two main classes of such iterative algorithms can be derived and that this approach is generic in that various existing acceleration algorithms for vector sequences are thereby recovered.In order to illustrate the interest of such algorithms, we apply them in the field of nonlinear mechanics on a simplified “point-wise” solver used to perform mechanical behaviour unit testings. The proposed test cases clearly demonstrate that accelerated fixed point iterations based on the elastic operator (quasi-Newton method) are very useful when the mechanical behaviour does not provide the so-called consistent tangent operator. Moreover, such accelerated algorithms also prove to be competitive with respect to the standard Newton–Raphson algorithm when available.