Badia, SantiagoCarstensen, CarstenMartín, Alberto F.Ruiz-Baier, RicardoVilla-Fuentes, Segundo2025-12-162025-12-160045-7825Bibtex:BADIA2025118343https://hdl.handle.net/1885/733795228The 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.This work has been supported by Monash Mathematics through a Gordon Preston Sabbatical Fellowship (C. Carstensen); by the Australian Research Council through the Future Fellowship grant FT220100496 (R. Ruiz-Baier) and Discovery Project grant DP22010316 (S. Badia and R. Ruiz-Baier); and by the National Research and Development Agency (ANID) of the Ministry of Science, Technology, Knowledge and Innovation of Chile through the postdoctoral program Becas Chile grant 74220026 (S. Villa-Fuentes). Computational resources were provided by the Australian Government through NCI under the National Computational Merit Allocation Scheme (NCMAS) and the ANU Merit Allocation Scheme (ANUMAS) (A.F. Martín).27en© 2025 The Author(s)Navier–Stokes–Brinkman–Forchheimer equationsPressure robustnessNonconforming finite elementsBanach fixed-point theoryA priori and a posteriori error estimates2025-12-0110.1016/j.cma.2025.118343105014945957