An unconditionally stable fully discrete method for Navier–Stokes-type viscoelastic fluid models with tempered kernel

This paper develops a fully discrete finite element method with first-order temporal accuracy and second-order spatial accuracy for a class of Navier–Stokes-type viscoelastic fluid equations with a tempered power-law memory kernel. The finite element method is employed for spatial discretization, while a first-order convolution quadrature is utilized to approximate the nonlocal memory term, and a semi-implicit scheme is adopted to treat the nonlinear convection term. The proposed method is rigorously analyzed in terms of stability, consistency, and convergence. In particular, stability estimates, truncation error bounds, and optimal L2-norm error estimates for the velocity are established. Extensive numerical experiments are conducted to verify the theoretical convergence rates for both manufactured solutions and benchmark problems without available analytical solutions. Furthermore, a two-dimensional 4:1 planar contraction channel is considered to investigate the impact of viscoelastic effects on complex flow dynamics. The numerical results reveal that the elastic stress coefficient and the tempering rate parameter play a crucial role in regulating the evolution of corner vortices and recirculation structures. In particular, the competition between viscous dissipation and elastic stress effects leads to non-monotonic vortex evolution, including vortex suppression and subsequent enlargement with increasing elasticity. These observations demonstrate the capability of the proposed method to capture the characteristic flow behaviors of viscoelastic fluids, including elasticity-induced instability and drag-reduction phenomena.