A Banach space mixed formulation for the unsteady Brinkman-Forchheimer equations
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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 a semidiscrete continuous-in-time finite element approximation 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. We provide several numerical results for a fully discrete scheme employing a Backward Euler time discretization to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.
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