Preprint 2023-30
Jessika Camaño, Ricardo Oyarzúa, Miguel Serón, Manuel Solano:
A mass conservative finite element method for a nonisothermal Navier-Stokes/Darcy coupled system
Abstract:
We propose and analyze an H(div)-conforming and mass conservative finite element method for the coupling of nonisothermal fluid flow with nonisothermal porous media flow. The governing equations are the Navier-Stokes/heat system, commonly known as the Boussinesq system, in the free-fluid region, and the Darcy-heat coupled model in the membrane. These systems are coupled through buoyancy terms and a set of transmission conditions on the fluid-membrane interface, including mass conservation, balance of normal forces, the Beavers-Joseph-Saffman law, and continuity of heat flux and fluid temperature. We consider a velocity-pressure-temperature variational scheme for the Boussinesq system in the free-fluid region whereas in the membrane region we consider a dual-mixed formulation for the Darcy system coupled with a primal equation for the temperature model. In this way, the unknowns of the resulting formulation are given by the velocity, pressure and temperature in both domains. For the associated Galerkin scheme, we combine an H(div)-conforming scheme for the fluid variables and a conforming Galerkin discretization for the heat equation. Therefore, the resulting numerical scheme produces exactly divergence-free velocities and also allows preserve the law of conservation of mass at a discrete level. The analysis of the continuous and discrete problems is carried out by means of a fixed-point strategy under a sufficiently small data assumption. We derive optimal error estimates under an additional assumption over the data and present numerical results illustrating the performance of the method.
