On a selective reuse of Krylov subspaces in Newton-Krylov approaches for nonlinear elasticity

1. Introduction. We consider the resolution of large-scale nonlinear problems arising from the finite-element discretization of geometrically non-linear structural analysis problems. We use a classical Newton Raphson algorithm to handle the non-linearity which leads to the resolution of a sequence of linear systems with non-invariant matrices and right hand sides. The linear systems are solved using the FETI-2 algorithm. We show how the reuse, as a coarse problem, of a pertinent selection of the information generated during the resolution of previous linear systems, stored inside Krylov subspaces, leads to interesting acceleration of the convergence of the current system. Nonlinear problems are a category of problems arising from various applications in mathematics, physics or mechanics. Solving these problems very often leads to a succession of linear problems the solution to which converges towards the solution to the considered problem. Within the framework of this study, all linear systems are solved using a conjugate gradient algorithm. It is well known that this algorithm is based on the construction of the so-called Krylov subspaces, on which depends its numerical efficiency and its convergence behaviour. The purpose of this paper is to accelerate the convergence of linear systems by reusing information arising from previous resolution processes. Such an idea has already led to a classical algorithm for invariant matrices [8] which has been successfully extended to the case of non invariant matrices [6, 7]. We here propose, thanks to a spectral analysis of linear systems, to select the most significant part of the information generated during conjugate gradient iterations to accelerate the convergence via an augmented Krylov conjugate gradient algorithm. The remainder of this paper is organized as follows: section 2 addresses characteristic properties of preconditioned conjugate gradient, section 3 exposes the acceleration strategies, section 4 gives numerical assessments and section 5 concludes the paper.