First-order long-time convergence of fully discrete methods for viscoelastic flow with tempered power-law kernel

This work presents a rigorous analysis of fully discrete numerical schemes for a viscoelastic flow model with memory effects governed by a tempered power-law kernel. The finite element method is employed for spatial discretization, and a backward Euler method is adopted for time discretization. Two discretization strategies are considered: a fully implicit scheme and a semi-implicit scheme, each incorporating a convolution quadrature rule tailored to the tempered fractional integral. Regularity results for the spatially semi-discrete solutions are established. Optimal error estimates with respect to both time and space are derived, with particular attention to their long-time behavior. Numerical experiments are conducted to validate the theoretical convergence results of the schemes and to explore the influence of the viscoelastic parameter ρ on the numerical solutions.