AN ANALYSIS OF EQUIVALENT OPERATOR PRECONDITIONING FOR EQUATION-FREE NEWTON-KRYLOV METHODS

We consider the computation of a fixed point of a time-stepper using Newton-Krylov methods, and propose and analyze equivalent operator preconditioning for the resulting linear systems. For a linear, scalar advection-reaction-diffusion equation, we investigate in detail how the convergence rate depends upon the choice of preconditioner parameters and upon the time discretization. The results are especially valuable when computing fixed points of a coarse time-stepper in the equation-free multiscale framework, in which one simulates an unavailable coarse-scale model by wrapping a set of computational routines around appropriately initialized fine-scale simulations. Both analytical results and numerical experiments are presented, showing that one can speed up the convergence of iterative methods significantly for a wide range of parameter values in the preconditioner.