  We propose and analyze a mixed formulation for the Brinkman-Forchheimer
equations for unsteady flows. Our approach is based on the introduction of a
pseudostress tensor related to the velocity gradient, leading to a mixed
formulation where the pseudostress tensor and the velocity are the main
unknowns of the system. We establish existence and uniqueness of a solution to
the weak formulation in a Banach space setting, employing classical results on
nonlinear monotone operators and a regularization technique. We then present
well-posedness and error analysis for semidiscrete continuous-in-time and fully
discrete finite element approximations on simplicial grids with spatial
discretization based on the Raviart-Thomas spaces of degree $k$ for the
pseudostress tensor and discontinuous piecewise polynomial elements of degree
$k$ for the velocity and backward Euler time discretization. We provide several
numerical results to confirm the theoretical rates of convergence and
illustrate the performance and flexibility of the method for a range of model
parameters.
