Structure-preserving parametric finite element methods for two-phase Stokes flow based on Lagrange multiplier approaches
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916904523792384 |
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| author | Garcke, Harald Trautwein, Dennis Zhang, Ganghui |
| author_facet | Garcke, Harald Trautwein, Dennis Zhang, Ganghui |
| contents | We present a novel formulation for parametric finite element methods to approximate two-phase Stokes flow. The new formulation is based on the classical Stokes equation in the bulk and a novel choice of interface conditions with additional Lagrange multipliers. This new Lagrange multiplier approach ensures that the numerical methods exactly preserve two physical structures of two-phase Stokes flow at the fully discrete level: (i) the energy-decaying and (ii) the volume-preserving properties. Moreover, different types of higher-order time discretization methods are employed, including the Crank--Nicolson method and the second-order backward differentiation formula approach. The resulting schemes are nonlinear and can be efficiently solved by using the Newton method with a decoupling technique. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy while preserving the two physical structures of the two-phase Stokes system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12326 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure-preserving parametric finite element methods for two-phase Stokes flow based on Lagrange multiplier approaches Garcke, Harald Trautwein, Dennis Zhang, Ganghui Numerical Analysis We present a novel formulation for parametric finite element methods to approximate two-phase Stokes flow. The new formulation is based on the classical Stokes equation in the bulk and a novel choice of interface conditions with additional Lagrange multipliers. This new Lagrange multiplier approach ensures that the numerical methods exactly preserve two physical structures of two-phase Stokes flow at the fully discrete level: (i) the energy-decaying and (ii) the volume-preserving properties. Moreover, different types of higher-order time discretization methods are employed, including the Crank--Nicolson method and the second-order backward differentiation formula approach. The resulting schemes are nonlinear and can be efficiently solved by using the Newton method with a decoupling technique. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy while preserving the two physical structures of the two-phase Stokes system. |
| title | Structure-preserving parametric finite element methods for two-phase Stokes flow based on Lagrange multiplier approaches |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2508.12326 |