Robust Augmented Mixed Finite Element Methods for Stoke Interface Problems with Discontinuous Viscosity in Multiple Subdomains
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912494026489856 |
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| author | Liang, Yuxiang Zhang, Shun |
| author_facet | Liang, Yuxiang Zhang, Shun |
| contents | A stationary Stokes problem with a piecewise constant viscosity coefficient in multiple subdomains is considered in the paper. For standard finite element pairs, a robust inf-sup condition is required to show the robustness of the discretization error with respect to the discontinuous viscosity, which has only been proven for the two-subdomain case in the paper [Numer. Math. (2006) 103: 129--149]. To avoid the robust inf-sup condition of a discrete finite element pair for multiple subdomains, we propose an ultra-weak augmented mixed finite element formulation. By adopting a Galerkin-least-squares method, the augmented mixed formulation can achieve stability without relying on the inf-sup condition in both continuous and discrete settings. The key step to having a robust priori error estimate is to use two norms, one energy norm and one full norm, in robust continuity. The robust coercivity is proved for the energy norm. A robust a priori error estimate in the energy norm is then derived with the best approximation property in the full norm for the case of multiple subdomains. Additionally, the paper introduces a singular Kellogg-type example with exact solutions for the first time. Extensive numerical tests are conducted to validate the robust error estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20655 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Robust Augmented Mixed Finite Element Methods for Stoke Interface Problems with Discontinuous Viscosity in Multiple Subdomains Liang, Yuxiang Zhang, Shun Numerical Analysis A stationary Stokes problem with a piecewise constant viscosity coefficient in multiple subdomains is considered in the paper. For standard finite element pairs, a robust inf-sup condition is required to show the robustness of the discretization error with respect to the discontinuous viscosity, which has only been proven for the two-subdomain case in the paper [Numer. Math. (2006) 103: 129--149]. To avoid the robust inf-sup condition of a discrete finite element pair for multiple subdomains, we propose an ultra-weak augmented mixed finite element formulation. By adopting a Galerkin-least-squares method, the augmented mixed formulation can achieve stability without relying on the inf-sup condition in both continuous and discrete settings. The key step to having a robust priori error estimate is to use two norms, one energy norm and one full norm, in robust continuity. The robust coercivity is proved for the energy norm. A robust a priori error estimate in the energy norm is then derived with the best approximation property in the full norm for the case of multiple subdomains. Additionally, the paper introduces a singular Kellogg-type example with exact solutions for the first time. Extensive numerical tests are conducted to validate the robust error estimate. |
| title | Robust Augmented Mixed Finite Element Methods for Stoke Interface Problems with Discontinuous Viscosity in Multiple Subdomains |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.20655 |