Virtual Element methods for non-Newtonian shear-thickening fluid flow problems
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In [1], we present a comprehensive theoretical analysis for Virtual Element discretizations of incompressible non-Newtonian flows governed by the (r,delta)-Carreau--Yasuda constitutive law, in the shear-thickening regime (r > 2) including both degenerate (delta = 0) and non-degenerate (delta > 0) cases. The proposed Virtual Element method features two distinguishing advantages: the construction of an exactly divergence-free discrete velocity field and compatibility with general polygonal meshes. The analysis presented in this work extends the results of [2], where only shear-thinning behavior (1 < r < 2) was considered. Indeed, the theoretical analysis of the shear-thickening setting requires several novel analytical tools, including: an inf–sup stability analysis of the discrete velocity-pressure coupling in non-Hilbertian norms, a stabilization term specifically designed to address the nonlinear structure as the exponent r > 2; and the introduction of a suitable discrete norm tailored to the underlying nonlinear constitutive relation. We prove that if r > 2, and the underlying solution is sufficiently smooth, the order of convergence of the velocity and the pressure is k/(r-1). In the non-degenerate case, i.e. delta > 0, we show that the order of convergence is 2k/r. Numerical results demonstrate the practical performance of the proposed formulation. [1] Antonietti P.F., Beirao da Veiga L., Botti M., Harnist A., Vacca G., Verani M., Virtual Element methods for non-Newtonian shear-thickening fluid flow problems, preprint, 2026. [2] Antonietti P.F., Beirao da Veiga L., Botti M., Vacca G., Verani M., A Virtual Element method for non-Newtonian pseudoplastic Stokes flows, Journal of Computer Methods in Applied Mechanics and Engineering, Vol. 428, pp. 117079, Year 2024.
