Preprint 2025-26
Alonso J. Bustos, Sergio Caucao:
A Banach space mixed formulation for the unsteady Brinkman problem with spatially varying porosity
Abstract:
We propose and analyze a new mixed formulation for the Brinkman equations with spatially varying porosity, modeling the time-dependent flow of an incompressible fluid through heterogeneous porous media. The formulation is developed within a Banach space framework and introduces the stress and vorticity tensors as additional unknowns. This approach eliminates the pressure, which can be recovered via post-processing, yielding a stress-velocity-vorticity system. The well-posedness of the continuous problem is proved under an appropriate small-porosity assumption, by employing monotone operator techniques together with recent advances on the solvability of perturbed saddle-point problems in Banach spaces. At the discrete level, we first introduce a semidiscrete continuous-in-time scheme employing finite element spaces stable for elasticity, such as the PEERS and Arnold–Falk–Winther elements. We prove the well-posedness of this scheme and derive the corresponding a priori error estimates. Subsequently, a fully discrete method is obtained by applying the backward Euler scheme for the time discretization, for which we also establish well-posedness and derive optimal convergence rates with respect to the spatial and temporal discretization parameters. Under this setting, momentum is conserved provided that the porosity, the permeability tensor, and the external forces are piecewise constant. Finally, several two- and three-dimensional numerical experiments, involving both manufactured and non-manufactured solutions, are presented, which confirm the theoretical convergence rates and highlight the capability of the proposed method to handle challenging geometries featuring strong contrasts in physical parameters such as permeability and porosity.
