On the convergence rate of the Ka\v{c}anov scheme for shear-thinning fluids

TL;DR

This paper analyzes the convergence rate of the Kaanov iteration scheme for shear-thinning fluids, providing new bounds and an a posteriori contraction factor, supported by numerical experiments.

Contribution

It introduces an improved, parameter-independent contraction factor for the Kaanov scheme applied to shear-thinning fluids, enhancing previous convergence bounds.

Findings

Energy difference contracts along the iteration sequence.

A new a posteriori computable contraction factor is proposed.

Numerical experiments confirm the analytical results.

Abstract

We explore the convergence rate of the Ka\v{c}anov iteration scheme for different models of shear-thinning fluids, including Carreau and power-law type explicit quasi-Newtonian constitutive laws. It is shown that the energy difference contracts along the sequence generated by the iteration. In addition, an a posteriori computable contraction factor is proposed, which improves, on finite-dimensional Galerkin spaces, previously derived bounds on the contraction factor in the context of the power-law model. Significantly, this factor is shown to be independent of the choice of the cut-off parameters whose use was proposed in the literature for the Ka\v{c}anov iteration applied to the power-law model. Our analytical findings are confirmed by a series of numerical experiments.