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.

翻译：本文设计并分析了一种适用于非牛顿不可压缩流体稳态运动的虚拟单元离散格式。引入了一种专门定制的稳定化方法，以模拟连续算子的单调性和有界性性质，并对其进行了理论探究。所提方法具有若干显著特点，包括精确满足无散度条件以及能够使用完全通用的多边形网格。在非线性规律的温和假设下（涵盖诸如Carreau-Yasuda模型等常见示例），本文给出了所提方法的完整适定性及收敛性分析。数值实验验证了理论界，并展示了所提公式的实际应用能力。