A linear HDG scheme for the diffusion type Peterlin viscoelastic problem

A linear semi-implicit hybridizable discontinuous Galerkin (HDG) scheme is proposed to solve the diffusive Peterlin viscoelastic model, allowing the diffusion coefficient $\text{ep}$ of the conformation tensor to be arbitrarily small. We investigate the well-posedness, stability, and error estimates of the scheme. In particular, we demonstrate that the $L^2$-norm error of the conformation tensor is independent of the reciprocal of $\text{ep}$. Numerical experiments are conducted to validate the theoretical convergence rates. Our numerical examples show that the HDG scheme performs better in preserving the positive definiteness of the conformation tensor compared to the ordinary finite element method (FEM).