Convergent numerical schemes for the viscoelastic Giesekus model in two dimensions

TL;DR

This paper introduces stable, convergent numerical schemes for the 2D viscoelastic Giesekus model, proving convergence to global weak solutions and validating through numerical experiments.

Contribution

It develops and rigorously proves convergence of numerical methods for the Giesekus model without regularization, providing an alternative proof of existence results.

Findings

Proven convergence of the numerical scheme to large-data weak solutions.

Numerical experiments confirm the scheme's stability and accuracy.

Validated the method on benchmark problems.

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 Bul\'{\i}\v{c}ek et al.~(Nonlinearity, 2022). Finally, we verify the practicality of the proposed method through numerical experiments, including convergence studies and typical benchmark problems.