WELL-POSEDNESS AND FINITE ELEMENT APPROXIMATION FOR THE CONVECTION MODEL IN SUPERPOSED FLUID AND POROUS LAYERS

Those studying complex domains containing superposed fluid and porous layers will frequently encounter the phenomenon of natural convection. The mathematical model of this problem can be described using a Navier-Stokes/Darcy system coupled with heat equations. The interface conditions for Navier-Stokes/Darcy systems are the Beavers-Joseph-Saffman conditions; the heat equations in two subdomains are coupled by the continuity of temperature and heat flux. By constructing a perturbation system of the original model, we prove its well-posedness and propose an effective finite element algorithm. The algorithm separates the model into two subproblems, which can improve the computing efficiency. By using the energy estimate method and the inductive method, we prove the stability of the algorithm, and the complete error analysis is also derived afterwards. Finally, numerical tests are implemented to verify our theoretical results.