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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.21392 |
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| _version_ | 1866908470606823424 |
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| author | Zheng, Nan Guo, Xu Pei, Wenlong Zhao, Wenju |
| author_facet | Zheng, Nan Guo, Xu Pei, Wenlong Zhao, Wenju |
| contents | This paper is concerned with mixed finite element method (FEM) for solving the two-dimensional, nonlinear fourth-order active fluid equations. By introducing an auxiliary variable $w=-Δu$, the original fourth problem is transformed into a system of second-order equations, which relaxes the regularity requirements of standard $H^2$-conforming finite spaces. To further enhance the robustness and efficiency of the algorithm, an additional auxiliary variable $ϕ$, treated analogously to the pressure, is introduced, leading to a divergence-free preserving mixed finite element scheme. A fully discrete scheme is then constructed by coupling the spatial mixed FEM with the variable-step Dahlquist-Liniger-Nevanlinna (DLN) time integrator. The boundedness of the scheme and corresponding error estimates can be rigorously proven under appropriate assumptions due to unconditional non-linear stability and second-order accuracy of the DLN method. To enhance computational efficiency in practice, we develop an adaptive time-stepping strategy based on a minimum-dissipation criterion. Several numerical experiments are displayed to fully validate the theoretical results and demonstrate the accuracy and efficiency of the scheme for complex active fluid simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21392 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Divergence-free Preserving Mix Finite Element Methods for Fourth-order Active Fluid Model Zheng, Nan Guo, Xu Pei, Wenlong Zhao, Wenju Numerical Analysis This paper is concerned with mixed finite element method (FEM) for solving the two-dimensional, nonlinear fourth-order active fluid equations. By introducing an auxiliary variable $w=-Δu$, the original fourth problem is transformed into a system of second-order equations, which relaxes the regularity requirements of standard $H^2$-conforming finite spaces. To further enhance the robustness and efficiency of the algorithm, an additional auxiliary variable $ϕ$, treated analogously to the pressure, is introduced, leading to a divergence-free preserving mixed finite element scheme. A fully discrete scheme is then constructed by coupling the spatial mixed FEM with the variable-step Dahlquist-Liniger-Nevanlinna (DLN) time integrator. The boundedness of the scheme and corresponding error estimates can be rigorously proven under appropriate assumptions due to unconditional non-linear stability and second-order accuracy of the DLN method. To enhance computational efficiency in practice, we develop an adaptive time-stepping strategy based on a minimum-dissipation criterion. Several numerical experiments are displayed to fully validate the theoretical results and demonstrate the accuracy and efficiency of the scheme for complex active fluid simulations. |
| title | Divergence-free Preserving Mix Finite Element Methods for Fourth-order Active Fluid Model |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.21392 |