A velocity-vorticity-pressure formulation for the steady Navier--Stokes--Brinkman--Forchheimer problem
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911225113214976 |
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| author | Badia, Santiago Carstensen, Carsten Martin, Alberto F. Ruiz-Baier, Ricardo Villa-Fuentes, Segundo |
| author_facet | Badia, Santiago Carstensen, Carsten Martin, Alberto F. Ruiz-Baier, Ricardo Villa-Fuentes, Segundo |
| contents | The flow of incompressible fluid in highly permeable porous media in vorticity - velocity - Bernoulli pressure form leads to a double saddle-point problem in the Navier--Stokes--Brinkman--Forchheimer equations. The paper establishes, for small sources, the existence of solutions on the continuous and discrete level of lowest-order piecewise divergence-free Crouzeix--Raviart finite elements. The vorticity employs a vector version of the pressure space with normal and tangential velocity jump penalisation terms. A simple Raviart--Thomas interpolant leads to pressure-robust a priori error estimates. An explicit residual-based a posteriori error estimate allows for efficient and reliable a posteriori error control. The efficiency for the Forchheimer nonlinearity requires a novel discrete inequality of independent interest. The implementation is based upon a light-weight forest-of-trees data structure handled by a highly parallel set of adaptive mesh refining algorithms. Numerical simulations reveal robustness of the a posteriori error estimates and improved convergence rates by adaptive mesh-refining. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10533 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A velocity-vorticity-pressure formulation for the steady Navier--Stokes--Brinkman--Forchheimer problem Badia, Santiago Carstensen, Carsten Martin, Alberto F. Ruiz-Baier, Ricardo Villa-Fuentes, Segundo Numerical Analysis The flow of incompressible fluid in highly permeable porous media in vorticity - velocity - Bernoulli pressure form leads to a double saddle-point problem in the Navier--Stokes--Brinkman--Forchheimer equations. The paper establishes, for small sources, the existence of solutions on the continuous and discrete level of lowest-order piecewise divergence-free Crouzeix--Raviart finite elements. The vorticity employs a vector version of the pressure space with normal and tangential velocity jump penalisation terms. A simple Raviart--Thomas interpolant leads to pressure-robust a priori error estimates. An explicit residual-based a posteriori error estimate allows for efficient and reliable a posteriori error control. The efficiency for the Forchheimer nonlinearity requires a novel discrete inequality of independent interest. The implementation is based upon a light-weight forest-of-trees data structure handled by a highly parallel set of adaptive mesh refining algorithms. Numerical simulations reveal robustness of the a posteriori error estimates and improved convergence rates by adaptive mesh-refining. |
| title | A velocity-vorticity-pressure formulation for the steady Navier--Stokes--Brinkman--Forchheimer problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.10533 |