A Virtual Element method for non-Newtonian fluid flows
P. F. Antonietti, L. Beirao da Veiga, M. Botti, G. Vacca, M. Verani
TL;DR
This work advances the numerical analysis of non-Newtonian incompressible flows by developing a divergence-free Virtual Element Method on general polygonal meshes for the steady Carreau–Yasuda model. The method combines a carefully designed nonlinear stabilization with VEM spaces that yield exactly divergence-free discrete velocities, and it proves strong monotonicity and discrete inf-sup stability leading to well-posedness. An a priori error theory tailored to the degenerate case $\delta=0$ provides convergence rates that depend on the polynomial degree and model exponent, with $\|\boldsymbol u-\boldsymbol u_h\|_{W^{1,r}}$ scaling like $h^{kr/2}$ under full regularity and $\|p-p_h\|_{L^{r'}}$ like $h^{k(r-1)}$. Numerical experiments on multiple polygonal meshes validate the theory and demonstrate practical robustness, including a two-step fixed-point strategy for the nonlinear solve and observed convergence trends under varying $r$ and $\delta$.
Abstract
In this paper, we design and analyze a Virtual Element discretization for the steady motion of non-Newtonian, incompressible fluids. A specific stabilization, tailored to mimic the monotonicity and boundedness properties of the continuous operator, is introduced and theoretically investigated. The proposed method has several appealing features, including the exact enforcement of the divergence free condition and the possibility of making use of fully general polygonal meshes. A complete well-posedness and convergence analysis of the proposed method is presented under mild assumptions on the non-linear laws, encompassing common examples such as the Carreau--Yasuda model. Numerical experiments validating the theoretical bounds as well as demonstrating the practical capabilities of the proposed formulation are presented.