Robust Numerical Schemes for the Coupled Navier-Stokes and Darcy Equations

The coupled Navier-Stokes and Darcy equations play an important role in modeling many realistic applications and typical examples can be found in the groundwater system in karst aquifers, and the interaction between surface and subsurface. Designing efficient and accurate numerical schemes to solve this model can lead to a better understanding of the physical phenomena. As such, great effort has been devoted to designing and analyzing robust and efficient numerical schemes for such coupled models. The coupled models rely on suitable interface conditions, and the Beavers-Joseph interface condition is shown to be practically valid. The major mathematical difficulty in using the Beavers-Joseph interface condition lies in the introduction of the indefinite interface term. To alleviate this difficulty, most existing works focus on a simplified interface condition, which is not accurate and desirable in many applications. In this project, our objective is to devise and analyze robust and accurate schemes for the coupled Navier-Stokes and Darcy equations with Beavers-Joseph interface condition. The project will develop (1) a robust scheme for the coupled model without resorting to a La-grange multiplier; (2) an unfitted scheme for the coupled model robust with respect to small cuts; (3) efficient and robust partitioned methods. To this end, we will first combine the pressure-robust discretizations and standard mixed finite element methods to solve the coupled model. The resulting scheme will deliver divergence-free fluid velocity approximations. We will develop new mathematical tools to tackle the indefinite interface term arising from the Beavers-Joseph interface condition. Then, we will adapt the spatial discretizations to allow unfitted meshes, which is attractive when solving problems in a multi-domain setting since one can alleviate the cumbersome meshing process caused by the presence of complicated geometries and curved interfaces. We will develop suitable stabilization techniques to ensure the robustness of the scheme with respect to small cuts. Finally, we will develop and analyze partitioned methods via a suitable decoupling of the interface conditions, which enables us to solve the Navier-Stokes and Darcy equations independently and efficiently. The accuracy, efficiency and capabilities of the scheme will be verified by rigorous mathematical analysis and extensive numerical simulations for benchmark problems. The developed methodologies and mathematical tools can be applied to various multiphysics problems. Moreover, we expect that the proposed project will provide a new framework and large advances for realistic applications involving the coupled fluid flow and porous media flow.