A three-field Banach spaces-based mixed formulation for the unsteady Brinkman–Forchheimer equations

Abstract

We propose and analyze a new mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Besides the velocity, our approach introduces the velocity gradient and a pseudostress tensor as further unknowns. As a consequence, we obtain a three-field Banach spaces-based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation, and derive the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then propose a semidiscrete continuous-in-time approximation on simplicial grids based on the Raviart–Thomas elements of degree k≥0 for the pseudostress tensor and discontinuous piecewise polynomials of degree k for the velocity and the velocity gradient. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness and derive the stability bounds for both schemes, and under a quasi-uniformity assumption on the mesh, we establish the corresponding error estimates. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.