Semi-Discrete Galerkin Finite Element Method for the Diffusive Peterlin Viscoelastic Model

In this paper, a semi-discrete Galerkin finite element method is applied to the two-dimensional diffusive Peterlin viscoelastic model which can describe the unsteady behavior of some incompressible ploymeric fluids. For the derived semi-discrete finite element spatial discretization scheme, the a priori bounds are given that does not rely on the mesh width restriction. Further, with the help of the a priori error bounds of the Stokes and Ritz projections, optimal error estimates for the velocity, the conformation tensor and the pressure are presented, respectively. Finally, in order to implement the proposed semi-discrete numerical scheme, we derive three kinds of fully discrete schemes, e.g., Newton's iterative scheme, Picard's iterative scheme and implicit-explicit time-stepping scheme. Finally, several numerical experiments are conducted to confirm our theoretical results.