A linear HDG scheme for the diffusion type Peterlin viscoelastic problem

TL;DR

The paper addresses the diffusive Peterlin viscoelastic system in the convection-dominated regime by developing a linear semi-implicit HDG scheme that remains stable and accurate as the diffusion coefficient $\epsilon$ tends to zero. It establishes unique solvability, mass conservation, and a stability framework that is uniform in $\epsilon^{-1}$, and proves error estimates with constants independent of $\epsilon^{-1}$ through Ritz projections and standard inf-sup arguments. Theoretical results show optimal convergence in natural norms with $\|oldsymbol{u}-oldsymbol{u}_h\|$, $\|{\rm tr}\boldsymbol{C}-{\rm tr}\boldsymbol{C}_h\|$, and $\|p-p_h\|$ behaving like $O(h^k)$ or $O(\tau)$ depending on the quantity, plus $O(h^{2k})$ components in certain endpoint measures. Numerical experiments corroborate the theory across $\epsilon=1$, $10^{-3}$, and $0$, and demonstrate superior preservation of the positive definiteness of the conformation tensor and stability relative to standard FEM in the small-$\epsilon$ regime, highlighting the HDG method’s practical impact for diffusion-dominated and convection-dominated viscoelastic flows.

Abstract

A linear semi-implicit hybridizable discontinuous Galerkin (HDG) scheme is proposed to solve the diffusive Peterlin viscoelastic model, allowing the diffusion coefficient $\ep$ 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 $\ep$. 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).

Figures (6)

• Figure 1: $\epsilon=1$: The numerical solution $(\bm{u}_h,p_h,\bm{C}_h)$ at $t=0.2$
• Figure 2: The numerical solution $(\bm{u}_h,p_h,\bm{C}_h)$ of the HDG scheme at $t=1$ with $(h=2^{-6},\tau=\frac{1}{100})$
• Figure 3: The numerical solution $(\bm{u}_h,p_h,\bm{C}_h)$ of the ordinary FEM scheme at $t=0.55$ with $(h=2^{-6},\tau=\frac{1}{100})$
• Figure 4: The numerical solution $(\bm{u}_h,p_h,\bm{C}_h)$ of the ordinary FEM scheme at $t=0.45$ with $(h=2^{-7},\tau=\frac{1}{100})$
• Figure 5: The numerical solution $(\bm{u}_h,p_h,\bm{C}_h)$ of the ordinary FEM scheme at $t=1$ with $(h=2^{-6},\tau=\frac{1}{120})$

Theorems & Definitions (14)

• Remark 2.1
• Lemma 3.1: Hana2015
• Lemma 3.2: Norm equivalence Hana2015
• Lemma 3.3: Sander2017
• Lemma 3.4: Aycil2017
• Lemma 3.5: Aycil2017
• Proposition 3.1
• proof
• Remark 3.1
• Theorem 3.1