An Lρ spaces-based mixed virtual element method for the steady ρ-type Brinkman–Forchheimer problem based on the velocity–stress–vorticity formulation

Abstract In this paper we devise and analyze a Banach-spaced mixed virtual element scheme for the steady motion of $\rho $-type Brinkman–Forchheimer equation with strongly symmetric stress. Our approach introduces stress and vorticity as additional variables, enabling the elimination of pressure from the original unknowns, which can later be recovered using a postprocessing formula based solely on the stress. Consequently, a mixed variational formulation of the velocity and these new unknowns has been obtained within a Banach space framework. We then propose the $\mathbb{H}({\mathbf{div}}_\varrho ;\varOmega )$-conforming virtual element method, where $\varrho $ is the conjugate of $\rho $, to discretize this formulation and establish the existence and uniqueness of the discrete solution, along with stability bounds, using the Browder–Minty theorem without imposing any assumptions on the data. Furthermore, convergence analysis for all variables in their natural norms is conducted, demonstrating an optimal rate of convergence. Finally, several numerical experiments are presented to illustrate the efficiency and validity of the proposed method.