本项目计划研究不可压缩流体和多孔介质流耦合问题的mortar元算法。该耦合问题由不同的偏微分方程构成，Navier-Stokes方程用来描述不可压缩流体的运动规律，而多孔介质流满足Darcy法则。交界面条件为：质量守恒定律、力的平衡性条件和Beavers-Joseph-Saffman法则。准备开展以下三部分工作：.第一、研究带传质方程耦合问题的mortar元算法，这是一个有挑战性的课题，其中有三个未知量。在此类耦合系统中，流体的粘性依赖于某种化学物质的浓度。第二、开展对Navier-Stokes方程与Darcy-Forchheimer流耦合问题mortar元算法的研究工作。对于高速流体，多孔介质中将会出现湍流，Darcy-Forchheimer方程可以用来描述多孔介质中高速流体的运动规律。第三、针对上述耦合系统，建立mortar元算法的后验误差估计，并设计出自适应算法。

In this project, I plan to investigate mortar element method for the coupling of incompressible flow and porous media flow. The model is relevant to a variety of physical processes. The coupled system consists of different partial differential equations where the free fluid is governed by Navier-Stokes equations and Darcy law can simulate the porous media flow. The fluid and porous media flow are coupled by such interface conditions: mass conservation (or called as flux continuity), the balance of the normal forces and the Beavers-Joseph-Saffman law which is the most accepted condition. The project consists of the following parts..Firstly, study the coupled problem with transport. The model is closer to the reality when Navier-Stokes equations and Darcy flow are coupled with a transport equation where the viscosity depends on the concentration of a chemical. It is a challenging problem. Secondly, investigate mortar element method for the coupling of Navier-Stokes equations and Darcy-Forchheimer flow. When the velocity of flow in the porous media is high, the nonlinear relationship between velocity and pressure can be described by Darcy-Forchheimer equation. Thirdly, analyze a posteriori error estimates of mortar element method for the above coupling systems. And based on residual-type a posteriori error estimator, adaptive algorithm will be developed.