SFEM for the Lagrangian formulation of the surface Stokes problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915368168062976 |
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| author | Elliott, Charles M. Mavrakis, Achilleas |
| author_facet | Elliott, Charles M. Mavrakis, Achilleas |
| contents | We consider the surface Stokes equation with Lagrange multiplier and approach it numerically. Using a Taylor-Hood surface finite element method, along with an appropriate estimate for the additional Lagrange multiplier, we derive a new inf-sup condition to help with the stability and convergence results. We establish optimal velocity convergence both in energy and tangential $L^2$ norms, along with optimal $L^2$ norm convergence for the two pressures, in the case of super-parametric finite elements. Furthermore, if the approximation order of the velocities matches that of the extra Lagrange multiplier, we achieve optimal order convergence even in the standard iso-parametric case. In this case, we also establish some new estimates for the normal $L^2$ velocity norm. In addition, we provide numerical simulations that confirm the established error bounds and also perform a comparative analysis against the penalty approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_19470 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | SFEM for the Lagrangian formulation of the surface Stokes problem Elliott, Charles M. Mavrakis, Achilleas Numerical Analysis 65N12 (Primary) 65N15, 65N30 (Secondary) We consider the surface Stokes equation with Lagrange multiplier and approach it numerically. Using a Taylor-Hood surface finite element method, along with an appropriate estimate for the additional Lagrange multiplier, we derive a new inf-sup condition to help with the stability and convergence results. We establish optimal velocity convergence both in energy and tangential $L^2$ norms, along with optimal $L^2$ norm convergence for the two pressures, in the case of super-parametric finite elements. Furthermore, if the approximation order of the velocities matches that of the extra Lagrange multiplier, we achieve optimal order convergence even in the standard iso-parametric case. In this case, we also establish some new estimates for the normal $L^2$ velocity norm. In addition, we provide numerical simulations that confirm the established error bounds and also perform a comparative analysis against the penalty approach. |
| title | SFEM for the Lagrangian formulation of the surface Stokes problem |
| topic | Numerical Analysis 65N12 (Primary) 65N15, 65N30 (Secondary) |
| url | https://arxiv.org/abs/2410.19470 |