Enriched Galerkin Method for Navier-Stokes Equations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911286051209216 |
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| author | Song, Chun Feng, Minfu |
| author_facet | Song, Chun Feng, Minfu |
| contents | This paper presents an enriched Galerkin (EG) finite element method for the incompressible Navier--Stokes equations. The method augments continuous piecewise linear velocity spaces with elementwise bubble functions, yielding a locally conservative velocity approximation while retaining the efficiency of low-order continuous elements. The viscous term is discretized using a symmetric interior penalty formulation, and the divergence constraint is imposed through a stable pressure space. To enhance the robustness of the velocity approximation with respect to the pressure, a reconstruction operator is introduced in the convective and coupling terms, resulting in a pressure-robust scheme whose accuracy does not deteriorate for small viscosities. Both Picard and Newton linearizations are formulated in a fully discrete manner, and the corresponding linear systems are assembled efficiently at each iteration. Optimal a~priori error estimates are established for the velocity in the mesh-dependent energy norm and for the pressure in the $L^2$ norm. Two representative numerical experiments are presented: a smooth manufactured solution and the lid-driven cavity flow. The numerical results confirm the theoretical convergence rates, demonstrating first-order convergence of the velocity in the energy norm, second-order convergence in the $L^2$ norm, and first-order convergence of the pressure. The proposed EG scheme accurately captures characteristic flow structures, illustrating its effectiveness and robustness for incompressible flow simulation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20240 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enriched Galerkin Method for Navier-Stokes Equations Song, Chun Feng, Minfu Numerical Analysis This paper presents an enriched Galerkin (EG) finite element method for the incompressible Navier--Stokes equations. The method augments continuous piecewise linear velocity spaces with elementwise bubble functions, yielding a locally conservative velocity approximation while retaining the efficiency of low-order continuous elements. The viscous term is discretized using a symmetric interior penalty formulation, and the divergence constraint is imposed through a stable pressure space. To enhance the robustness of the velocity approximation with respect to the pressure, a reconstruction operator is introduced in the convective and coupling terms, resulting in a pressure-robust scheme whose accuracy does not deteriorate for small viscosities. Both Picard and Newton linearizations are formulated in a fully discrete manner, and the corresponding linear systems are assembled efficiently at each iteration. Optimal a~priori error estimates are established for the velocity in the mesh-dependent energy norm and for the pressure in the $L^2$ norm. Two representative numerical experiments are presented: a smooth manufactured solution and the lid-driven cavity flow. The numerical results confirm the theoretical convergence rates, demonstrating first-order convergence of the velocity in the energy norm, second-order convergence in the $L^2$ norm, and first-order convergence of the pressure. The proposed EG scheme accurately captures characteristic flow structures, illustrating its effectiveness and robustness for incompressible flow simulation. |
| title | Enriched Galerkin Method for Navier-Stokes Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2511.20240 |