A strongly mass-conservative method for the coupled Navier-Stokes and Darcy-Forchheimer equations

Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong SAR, P.R. China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 18 March 2025
Accepted: 19 February 2026

Abstract

In this paper, we propose and analyze a strongly mass-conservative numerical scheme for the coupled Navier-Stokes and Darcy-Forchheimer system in both two and three spatial dimensions. The two subproblems are coupled through physically relevant interface conditions, including mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman condition. We employ a staggered discontinuous Galerkin method for the Navier-Stokes equations and use standard mixed finite elements for the Darcy-Forchheimer problem. The proposed formulation incorporates the interface conditions directly, without introducing Lagrange multipliers on the interface or artificial numerical fluxes on the mesh skeleton. As a consequence, although discontinuous Galerkin elements are used in the free-flow region, the resulting discrete velocity field is globally H(div)-conforming across the entire domain. In particular, the incompressibility constraint is satisfied exactly in the free-flow region, thereby yielding strong mass conservation over the entire computational domain. Under a suitable small-data assumption, we establish the well-posedness of the resulting nonlinear discrete system. Owing to the exact preservation of mass conservation, the proposed scheme exhibits a pressure-robust behavior, in the sense that the velocity approximation is insensitive to pressure effects. Numerical experiments are presented to illustrate the stability and robustness of the method, including its performance in regimes involving small viscosity, large pressure, and limited solution regularity.