A Banach space mixed formulation for the unsteady Brinkman-Forchheimer equations

TL;DR

This paper introduces a novel mixed formulation for unsteady Brinkman-Forchheimer equations using a pseudostress tensor, providing theoretical analysis and numerical validation of the method's accuracy and stability.

Contribution

It develops a new mixed formulation involving a pseudostress tensor for unsteady flows, with rigorous existence, uniqueness, and error analysis in a Banach space framework.

Findings

Proved well-posedness of the mixed formulation.

Established convergence rates for finite element approximations.

Numerical results confirm theoretical error estimates.

Abstract

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.