Abstract
We explore the convergence rate of the Kačanov 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 Kačanov iteration applied to the power-law model. Our analytical findings are confirmed by a series of numerical experiments.
Highlights
In this work, we focus on the iterative solution of nonlinear partial differential equations that arise in models of steady flows of incompressible shear-thinning fluids, including models with explicit constitutive relations of Carreau and power-law type
The derived upper bound on the contraction factor depends on the quotient m ∕M involving − and +, and may be extremely close to 1 in certain situations; interestingly, this unfavourable predicted dependence of the contraction factor on the ratio m ∕M has not been observed in numerical experiments
For the relaxed power-law model, this factor is independent of the relaxation parameters ± ; we demonstrated that it is, instead, the power-law exponent that affects the convergence rate of the iteration
Summary
We focus on the iterative solution of nonlinear partial differential equations that arise in models of steady flows of incompressible shear-thinning fluids, including models with explicit constitutive relations of Carreau and power-law type. (A3) is decreasing in the second variable, i.e., (x, t) ≤ 0 for all t ≥ 0 and all x ∈ Ω , where ′ denotes the derivative of with respect to the variable t. Page 3 of 27 4 where e( ) ∶ e( ) denotes the Frobenius inner product of e( ) and e( ) ; we refer to [2] Sect. Where 0 ∈ V is an arbitrary initial guess References concerning this iterative method include [14], where it was used to compute a stationary magnetic field in nonlinear media, and [5], where the convergence of the Kačanov iteration was investigated in the context of Galerkin methods; Fučík, Kratochvíl and Nečas point in
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