Convergent numerical schemes for the viscoelastic Giesekus model in two dimensions

TL;DR

This work develops stable, convergent fully discrete schemes for the viscoelastic Giesekus model in two dimensions by coupling Navier–Stokes with a deformation gradient evolution. It proves subsequence convergence of the spatially discrete solutions to a large-data weak solution in 2D, without cut-offs or extra regularization, and demonstrates energy stability through a discrete energy inequality with a time-step restriction Δt<λ/μ. The analysis introduces multiple convective-derivative discretizations, including a Lambda-based discrete chain rule, and shows strong compactness of the deformation gradient in 2D via differential inequalities and renormalization; positivity of det(F) is ensured in the limit under suitable initial data. Numerical experiments validate convergence rates and benchmark performance, including a 4:1 planar contraction, while illustrating the effects of stabilization on energy dissipation and stress localization. Overall, the paper provides a rigorous numerical foundation for simulating 2D viscoelastic flows modeled by the Giesekus system, with practical implications for stable, accurate simulations at high Weissenberg numbers without ad-hoc regularization.

Abstract

In this work, we develop a class of stable and convergent numerical methods for the approximate solution of the viscoelastic Giesekus model in two space dimensions. The model couples the incompressible Navier--Stokes equations with an evolution equation for an additional stress tensor accounting for elastic effects. This coupled evolution equation is stated here in terms of the elastic deformation gradient and models transport and nonlinear relaxation effects. In the existing literature, numerical schemes for such models often suffer from accuracy limitations and convergence problems, usually due to the lack of rigorous existence results or inherent limitations of the discretization. Therefore, our main goal is to prove the (subsequence) convergence of the proposed numerical method to a large-data global weak solution in two dimensions, without relying on cut-offs or additional regularization. This also provides an alternative proof of the recent existence result by Bulíček et al.~(Nonlinearity, 2022). Finally, we verify the practicality of the proposed method through numerical experiments, including convergence studies and typical benchmark problems.

Figures (8)

• Figure 1: Studying the temporal convergence of the $L^2(\Omega_T)$ errors for the velocity $\mathbf{v}$ (left), the pressure $p$ (center), and the deformation gradient $\mathbb{F}$ (right).
• Figure 2: Studying the spatial convergence of the $L^2(\Omega_T)$ errors for the velocity $\mathbf{v}$ (left), the pressure $p$ (center), and the deformation gradient $\mathbb{F}$ (right).
• Figure 3: Schematic sketch of the $4{:}1$ planar contraction geometry.
• Figure 4: Zoomed view of the computational mesh near the contraction plane $x_1=0$. The triangulation shows the unstructured base mesh ($h=1/20$) with five levels of local refinement applied near the re-entrant corners $(0, \pm L)^\top$ to resolve the stress singularity.
• Figure 5: Evolution of the energy $\int_\Omega \left( \frac{\rho}{2} |\mathbf{v}|^2 + \frac{\mu}{2} |\mathbb{F}|^2 \right)$ for different Weissenberg numbers $\textit{Wi} \in \{0, 0.1, 1, 3, 5, 8\}$. (a) The standard scheme \ref{['eq:system_FE']}. (b) The standard scheme with a refined time step size $\Delta t = 0.0025$. (c) The scheme with stress diffusion scaled by $(\Delta t)^2$ instead of $\Delta t$. (d) The scheme without stress diffusion. In case (a), the stabilization term results in strong dissipation and decaying total energy in the long run. Reducing the time step size in (b) does not fully eliminate this dissipation. However, in case (c), the additional dissipation appears negligible, and the curves coincide with the non-stabilized case (d). All results agree with the stability result \ref{['eq:stability_FE']}.

Theorems & Definitions (14)

• Theorem
• Definition 2.1
• Theorem 2.2
• Remark 3.1
• Lemma 3.2
• Remark 3.3
• Lemma 3.4
• Theorem 3.5
• Theorem 3.6
• Theorem 4.1