A linearizing-decoupling finite element method with stabilization for the Peterlin viscoelastic model

We propose a linearizing-decoupling finite element method for the nonstationary diffusive Peterlin viscoelastic system with shear-dependent viscosity modeling the incompressible polymeric fluid flow, where the equation of the conformation tensor \({\varvec{C}}\) contains a diffusion term with a tiny diffusion coefficient \(\epsilon\). By using the stabilizing terms \(\alpha _1^{-1} \Delta ({\varvec{u}}^{n+1} - {\varvec{u}}^{n})\) and \(\alpha _2^{-1} \Delta ({\varvec{C}}^{n+1} - {\varvec{C}}^{n})\), at every time level, the velocity \({\varvec{u}}\) and each component \(C_{ij}\) of the conformation tensor \({\varvec{C}}\) can be computed in parallel by our scheme. We obtain the error estimate \(C(\tau + h^2)\) for the P2/P1/P2 element, where the constant C depends on the norm of the solution but is not explicitly related to the reciprocal of \(\epsilon\). We conduct several numerical simulations and compute the experimental convergence rates to compare with the theoretical results.