A stabilization-free mixed virtual element approximation for unsteady non-Newtonian pseudoplastic Stokes flows

In this work, we propose and analyze a mixed virtual element method within a Banach space framework to numerically investigate the unsteady motion of non-Newtonian pseudoplastic Stokes flows. Motivated by the growing interest in non-Newtonian fluid problems where the stress response plays a pivotal role, our formulation introduces additional unknowns, including the rate of strain and the symmetric stress tensor. This leads to a mixed variational formulation that incorporates the velocity, strain rate, and stress within a Banach space setting. We establish the well-posedness of the weak solution and derive stability estimates using classical results from the theory of nonlinear monotone operators. The discretization in both space and time is performed using an [Formula: see text]-conforming virtual element method and the implicit Euler scheme, respectively. Specifically, the pseudostress is approximated using a virtual element subspace of [Formula: see text], while piecewise polynomial subspaces of degree [Formula: see text] are employed for approximating the velocity and the rate of strain tensor. The nonlinear term is treated implicitly in the time discretization, and the resulting scheme is shown to be well-posed and unconditionally stable. A rigorous convergence analysis is provided for all unknowns in their natural norms, establishing optimal convergence rates with respect to both the spatial mesh size and the time step. Finally, a series of numerical experiments is presented to validate the accuracy and effectiveness of the proposed method.