| dc.description | In this paper we introduce and analyze a new mixed finite element method for the 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 ≥ 0 for the pseudostress, and continuous
piecewise polynomials of degree k + 1 for the Lagrange multiplier. We also derive a reliable and ef-
ficient 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 Cl´ement 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 of
the method, confirming the theoretical properties of the estimator, and showing the behaviour of
the associated adaptive algorithm, are provided. | |
