| Abstract: |
| In this talk, we review a class of energy-stable numerical methods for the viscoelastic Oldroyd-B and Giesekus models. 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 models transport and nonlinear relaxation effects and is usually stated in terms of the elastic deformation gradient, the conformation tensor, or the log-conformation approach. 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.
The core of this presentation introduces a novel convergence result. We prove the (subsequence) convergence of a proposed numerical method to a large-data global weak solution of the Giesekus model in two dimensions. Crucially, this result is achieved without the use of artificial cut-offs or regularization in the limit system, providing a constructive alternative to the existence proof by Bul\`{i}\v{c}ek et al.~(Nonlinearity, 2022). Finally, we demonstrate the robustness of the method through numerical experiments, including convergence rate studies and typical benchmark problems. |
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