SFEM for the unsteady Navier-Stokes Equations on a stationary surface
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| Format: | Preprint |
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2025
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| author | Elliott, Charles M. Mavrakis, Achilleas |
| author_facet | Elliott, Charles M. Mavrakis, Achilleas |
| contents | In this paper we consider a fully discrete numerical method for the unsteady Navier-Stokes equations on a smooth closed stationary surface in $\mathbb{R}^3$. We use the surface finite element method (SFEM) with a generalized Taylor-Hood finite element pair $\mathrm{\mathbf{P}}_{k_u}$-- $\mathrm{P}_{k_{pr}}$-- $\mathrm{P}_{k_λ}$, where we enforce the tangential condition of the velocity field weakly, by introducing an extra Lagrange multiplier $λ$. Depending on the richness of the finite element space involving this extra Lagrange multiplier we present a fully discrete stability and error analysis. For the velocity, we establish optimal $L^{2}(a_h)$-norm bounds ($a_h$ - an energy norm) when $k_λ=k_u$ and suboptimal with respect to the geometric approximation error when $k_λ = k_u-1$ (optimal when \emph{super-parametric finite elements} are used). For the pressure, optimal $L^2(L^2)$-norm error bounds are established when $k_λ=k_u$. Assuming further regularity assumptions for our continuous problem, we are also able to show optimal convergence (using \emph{super-parametric finite elements} again) when $k_λ=k_u-1$. Numerical simulations that confirm the established theory are provided, along with a comparative analysis against a penalty approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_09266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | SFEM for the unsteady Navier-Stokes Equations on a stationary surface Elliott, Charles M. Mavrakis, Achilleas Numerical Analysis 65N12 (Primary), 65N15, 65N30 (Secondary) In this paper we consider a fully discrete numerical method for the unsteady Navier-Stokes equations on a smooth closed stationary surface in $\mathbb{R}^3$. We use the surface finite element method (SFEM) with a generalized Taylor-Hood finite element pair $\mathrm{\mathbf{P}}_{k_u}$-- $\mathrm{P}_{k_{pr}}$-- $\mathrm{P}_{k_λ}$, where we enforce the tangential condition of the velocity field weakly, by introducing an extra Lagrange multiplier $λ$. Depending on the richness of the finite element space involving this extra Lagrange multiplier we present a fully discrete stability and error analysis. For the velocity, we establish optimal $L^{2}(a_h)$-norm bounds ($a_h$ - an energy norm) when $k_λ=k_u$ and suboptimal with respect to the geometric approximation error when $k_λ = k_u-1$ (optimal when \emph{super-parametric finite elements} are used). For the pressure, optimal $L^2(L^2)$-norm error bounds are established when $k_λ=k_u$. Assuming further regularity assumptions for our continuous problem, we are also able to show optimal convergence (using \emph{super-parametric finite elements} again) when $k_λ=k_u-1$. Numerical simulations that confirm the established theory are provided, along with a comparative analysis against a penalty approach. |
| title | SFEM for the unsteady Navier-Stokes Equations on a stationary surface |
| topic | Numerical Analysis 65N12 (Primary), 65N15, 65N30 (Secondary) |
| url | https://arxiv.org/abs/2508.09266 |