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

Abstract : 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 next obtained when assuming the solution to the Darcy-Brinkman-Forchheimer model are smooth enough. We end this paper by providing a fixed-point method to solve iteratively 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.
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Submitted on : Sunday, May 3, 2020 - 6:52:58 AM
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Pierre-Henri Cocquet, Michaël Rakotobe, Delphine Ramalingom, Alain Bastide. Error analysis for the finite element approximation of the Brinkmann-Darcy-Forchheimer model for porous media with mixed boundary conditions. Journal of Computational and Applied Mathematics, Elsevier, 2020, ⟨10.1016/j.cam.2020.113008⟩. ⟨hal-02561058⟩

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