Termination criteria for inexact fixed‐point schemes

Abstract

SummaryWe analyze inexact fixed‐point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed‐point iteration. Important applications are the Picard iteration and partitioned fluid‐structure interaction. For the analysis, the iteration is modeled as a perturbed fixed‐point iteration, and existing analysis is extended to the nested case x = F(S(x)). We prove that if the iteration converges, it converges to the exact solution irrespective of the tolerance in the inner systems, provided that a nonstandard relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property. Numerical results demonstrate the effectiveness of the approach with the nonstandard termination criterion. Copyright © 2015 John Wiley & Sons, Ltd.