Sharp convergence rates for the homogenization of the Stokes equations in a perforated domain

Loïc Balazi (1, 2) , Grégoire Allaire (2) , Pascal Omnes (1, 3)

1 SGLS - Service de Génie Logiciel pour la Simulation
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 LAGA - Laboratoire Analyse, Géométrie et Applications

This paper is concerned with the homogenization of the Stokes equations in a periodic perforated domain. The homogenized model is known to be Darcy's law in the full domain. We establish a sharp convergence rate $O(\sqrt{\varepsilon})$ for the energy norm of the difference of the velocities, where $\varepsilon$ represents the size of the solid obstacles. This is achieved by using a two-scale asymptotic expansion of the Stokes equations and a new construction of a cut-off function which avoids the introduction of boundary layers. The main novelty is that our analysis applies for the physically relevant case of a porous medium where each of the fluid and solid parts is a connected subdomain.

Domaines: Mathématiques [math]

Loïc Balazi, Grégoire Allaire, Pascal Omnes. Sharp convergence rates for the homogenization of the Stokes equations in a perforated domain. 2024. ⟨hal-04541828⟩