A Nitsche method for incompressible fluids with general dynamic boundary conditions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866929714226003968 |
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| author | Gazca-Orozco, Pablo Alexei Gmeineder, Franz Kokavcová, Erika Maringová Tscherpel, Tabea |
| author_facet | Gazca-Orozco, Pablo Alexei Gmeineder, Franz Kokavcová, Erika Maringová Tscherpel, Tabea |
| contents | Both Newtonian and non-Newtonian fluids may exhibit complex slip behaviour at the boundary. We examine a broad class of slip boundary conditions that generalises the commonly used Navier slip, perfect slip, stick-slip and Tresca friction boundary conditions. In particular, set-valued, non-monotone, noncoercive and dynamic relations may occur. For a unifying framework of such relations, we present a fully discrete numerical scheme for the time-dependent Navier-Stokes equations subject to impermeability and general slip type boundary conditions on polyhedral domains. Based on compactness arguments, we prove convergence of subsequences, finally ensuring the existence of a weak solution. The numerical scheme uses a general inf-sup stable pair of finite element spaces for the velocity and pressure, a regularisation approach for the implicit slip boundary condition and, most importantly, a Nitsche method to impose the impermeability and a backward Euler time stepping. One of the key tools in the convergence proof is an inhomogeneous Korn inequality that includes a normal trace term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09550 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Nitsche method for incompressible fluids with general dynamic boundary conditions Gazca-Orozco, Pablo Alexei Gmeineder, Franz Kokavcová, Erika Maringová Tscherpel, Tabea Numerical Analysis 65N30, 76D07, 76M10 Both Newtonian and non-Newtonian fluids may exhibit complex slip behaviour at the boundary. We examine a broad class of slip boundary conditions that generalises the commonly used Navier slip, perfect slip, stick-slip and Tresca friction boundary conditions. In particular, set-valued, non-monotone, noncoercive and dynamic relations may occur. For a unifying framework of such relations, we present a fully discrete numerical scheme for the time-dependent Navier-Stokes equations subject to impermeability and general slip type boundary conditions on polyhedral domains. Based on compactness arguments, we prove convergence of subsequences, finally ensuring the existence of a weak solution. The numerical scheme uses a general inf-sup stable pair of finite element spaces for the velocity and pressure, a regularisation approach for the implicit slip boundary condition and, most importantly, a Nitsche method to impose the impermeability and a backward Euler time stepping. One of the key tools in the convergence proof is an inhomogeneous Korn inequality that includes a normal trace term. |
| title | A Nitsche method for incompressible fluids with general dynamic boundary conditions |
| topic | Numerical Analysis 65N30, 76D07, 76M10 |
| url | https://arxiv.org/abs/2502.09550 |