Convergent numerical schemes for the viscoelastic Giesekus model in two dimensions
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| Format: | Preprint |
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2025
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| _version_ | 1866915696832675840 |
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| author | Süli, Endre Trautwein, Dennis |
| author_facet | Süli, Endre Trautwein, Dennis |
| contents | 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\'ıček et al.~(Nonlinearity, 2022). Finally, we verify the practicality of the proposed method through numerical experiments, including convergence studies and typical benchmark problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22831 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergent numerical schemes for the viscoelastic Giesekus model in two dimensions Süli, Endre Trautwein, Dennis Numerical Analysis Analysis of PDEs 76M10, 76A10, 65M12, 65M22 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\'ıček et al.~(Nonlinearity, 2022). Finally, we verify the practicality of the proposed method through numerical experiments, including convergence studies and typical benchmark problems. |
| title | Convergent numerical schemes for the viscoelastic Giesekus model in two dimensions |
| topic | Numerical Analysis Analysis of PDEs 76M10, 76A10, 65M12, 65M22 |
| url | https://arxiv.org/abs/2512.22831 |