Dynamic subscales and unconditional semi-discrete stability for non-residual VMS approximations of generalised Newtonian flows

Abstract

We analyse a non-residual variational multiscale finite element formulation with dynamic subscales for incompressible generalised Newtonian Navier--Stokes flows. The apparent viscosity is assumed to be bounded and Lipschitz continuous with respect to the shear rate, covering several regularised rheological laws. The method separates, through orthogonal projections, the unresolved pressure-divergence, convective, and pressure-gradient contributions. For the linearised semi-discrete problem, we prove well-posedness and an unconditional stability estimate in an anisotropic VMS norm. The estimate controls viscous dissipation, subscale energies, discrete divergence, and a coupled acceleration-convection-pressure balance. For the non-linear formulation, a fixed-point argument yields existence and optimal-order a priori bounds under suitable regularity and smallness assumptions. The analysis also provides weak-in-time control of the discrete pressure-convection balance. A key point is that the dynamic pressure-related subscale provides the time-derivative contribution needed to control the projected pressure-gradient error, a mechanism unavailable in the corresponding quasi-static setting.