A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations. (19th August 2020)

• Record Type: Journal Article
• Authors: Caucao, Sergio; Yotov, Ivan
• Is Part Of: IMA journal of numerical analysis. Volume 41:Number 4(2021)
• Page Start: 2708
• Page End: 2743
• Publication Date: 2020-08-19
• Subjects: unsteady Brinkman–Forchheimer equations -- pseudostress-velocity formulation -- mixed finite element methods; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/draa035
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: We propose and analyse 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 and pressure, 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.