Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow

Abstract

In this paper we introduce and analyze a new mixed finite element method for the two-dimensional Brinkman model of porous media flow with mixed boundary conditions. We use a dual-mixed formulation in which the main unknown is given by the pseudostress. The original velocity and pressure unknowns are easily recovered through a simple postprocessing. In addition, since the Neumann boundary condition becomes essential, we impose it in a weak sense, which yields the introduction of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange multiplier. We apply the Babuska---Brezzi theory to establish sufficient conditions for the well-posedness of the resulting continuous and discrete formulations. In particular, a feasible choice of finite element subspaces is given by Raviart---Thomas elements of order $$k \ge 0$$ k ? 0 for the pseudostress, and continuous piecewise polynomials of degree $$k + 1$$ k + 1 for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator for this problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Raviart---Thomas and Clement interpolation operators are the main tools for proving the reliability. Then, Helmholtz's decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, several numerical results illustrating the performance and the robustness of the method, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are provided.