Error analysis for the finite element approximation of the Darcy–Brinkman–Forchheimer model for porous media with mixed boundary conditions

This paper deals with the finite element approximation of the Darcy–Brinkman- Forchheimer equation, involving a porous media with spatially-varying porosity, with mixed boundary condition such as inhomogeneous Dirichlet and traction boundary conditions. We first prove that the considered problem has a unique solution if the source terms are small enough. The convergence of a Taylor–Hood finite element approximation using a finite element interpolation of the porosity is then proved under similar smallness assumptions. Some optimal error estimates are obtained if the solution to the Darcy–Brinkman–Forchheimer model is smooth enough. We end this paper by providing a fixed-point method to solve the discrete non-linear problems and with some numerical experiments to make more precise the smallness assumptions on the source terms and to illustrate the theoretical convergence results.