Stokes - Darcy Coupling System With Interfacial Boundary Conditions Based On Finite Element Method

Stokes-Darcy model arises in many applications such as groundwater system in karst aquifers, interaction between surface and subsurface flows and petroleum extraction. It describes a free flow of a liquid by the Stokes equation and the confined flow in a porous media by Darcy equation; the two flows are coupled through interface conditions. The difference of local models and existence of the interface condition lead to various mathematical and numerical difficulties. They make the decoupling methods for solving Stokes-Darcy coupling problem becomes one of the hot topics in computational mathematics and computational fluid dynamics.This thesis investigates, from both the mathematical and numerical viewpoint, the coupling of surface and porous media flows. The behavior of the fluid can be described by the Stokes-Darcy equations, or the Stokes equations in the surface sub-region, and Darcy’s law in the porous media sub-region. On the interface, conditions of conservation of mass, balance of forces and the Beavers-Joseph conditions(or its simplified case: Beavers-Joseph-Saffman conditions) are well accepted.First, we introduce the Stokes-Darcy coupling problems and build the model with the Beavers-Joseph or Beavers-Joseph-Saffman interface conditions, respectively. The decoupling method enables us to solve the coupling Stokes-Darcy flow problem in parallel by simply lagging the interfacial coupling terms onto the each region by using the uniform mesh, and then we can use the interface condition to solve these problems. We can make use of the weak formulation of Stokes-Darcy coupling system to obtain the finite element approximation, so we can transfer the Stokes-Darcy coupling system to the simplified linear system.Second, we investigate the well-posedness of a coupled Stokes-Darcy model with Beavers-Joseph interface boundary conditions. And then we obtain the convergence rate and error esitimate to guarantee the computition of a coupled Stokes-Darcy model.At last, numerical experiments illustrate the validity of the theoretical results.