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In this work, we develop and analyse a novel Hybrid High-Order discretisationof the Brinkman problem. The method hinges on hybrid discrete velocity unknownsat faces and elements and on discontinuous pressures. Based on the discreteunknowns, we reconstruct inside each element a Stokes velocity one degreehigher than face unknowns, and a Darcy velocity in theRaviart-Thomas-N\'ed\'elec space. These reconstructed velocities arerespectively used to formulate the discrete versions of the Stokes and Darcyterms in the momentum equation, along with suitably designed penaltycontributions. The proposed construction is tailored to yield optimal errorestimates that are robust throughout the entire spectrum of local (Stokes- orDarcy-dominated) regimes, as identified by a dimensionless number which can beinterpreted as a friction coefficient. The singular limit corresponding to theDarcy equation is also fully supported by the method. Numerical examplescorroborate the theoretical results. This paper also contains two contributionswhose interest goes beyond the specific method and application treated in thiswork: an investigation of the dependence of the constant in the second Korninequality on star-shaped domains and its application to the study of theapproximation properties of the strain projector in general Sobolev seminorms.

A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits