Surface Stokes Without Inf-Sup Condition
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908493531840512 |
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| author | Nochetto, Ricardo H. Shakipov, Mansur |
| author_facet | Nochetto, Ricardo H. Shakipov, Mansur |
| contents | For a $d$-dimensional hypersurface of class $C^3$ without boundary, we reformulate the surface Stokes equations as a nonsymmetric indefinite elliptic problem governed by two Laplacians. We then use this elliptic reformulation as a basis for a numerical method based on lifted parametric FEM. Assuming no geometric error for simplicity, we prove its well-posedness, quasi-best approximation in a robust mesh-dependent $H^1$-norm for any polynomial degree, as well as an optimal $L^2$ error estimate for both velocity and pressure. This entails a sufficiently small mesh size that solely depends on the Weingarten map and circumvents the usual discrete inf-sup condition. We present numerical experiments for velocity-pressure pairs with equal and disparate polynomial degrees, demonstrating that the proposed method is both accurate and practical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13342 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Surface Stokes Without Inf-Sup Condition Nochetto, Ricardo H. Shakipov, Mansur Numerical Analysis Analysis of PDEs 65N30, 58J05, 35J47, 35J50 G.1.8 For a $d$-dimensional hypersurface of class $C^3$ without boundary, we reformulate the surface Stokes equations as a nonsymmetric indefinite elliptic problem governed by two Laplacians. We then use this elliptic reformulation as a basis for a numerical method based on lifted parametric FEM. Assuming no geometric error for simplicity, we prove its well-posedness, quasi-best approximation in a robust mesh-dependent $H^1$-norm for any polynomial degree, as well as an optimal $L^2$ error estimate for both velocity and pressure. This entails a sufficiently small mesh size that solely depends on the Weingarten map and circumvents the usual discrete inf-sup condition. We present numerical experiments for velocity-pressure pairs with equal and disparate polynomial degrees, demonstrating that the proposed method is both accurate and practical. |
| title | Surface Stokes Without Inf-Sup Condition |
| topic | Numerical Analysis Analysis of PDEs 65N30, 58J05, 35J47, 35J50 G.1.8 |
| url | https://arxiv.org/abs/2508.13342 |