A Virtual Element method for non-Newtonian fluid flows

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.