多孔介质中流体问题广泛存在于石油、天然气等地下流体资源的开发中，其数学模型是依赖于时间的强耦合非线性偏微分方程组。流体问题的数值模拟涉及长时间域、大范围的计算，这导致计算量比较大，计算时间长，收敛速度较慢，因此对大规模、长时间流体问题的数值模拟设计高效快速算法是十分必要的。耦合模型是一类鞍点问题，同时涉及复杂界面情形，因此对其设计快速算法不是一件容易的事情。本项目以二维非线性Stokes/Darcy-Forchheimer耦合模型为切入点，对鞍点问题，设计合理的解耦算法，构造健壮的非线性几何多重网格方法，进行非线性多重网格方法收敛性的理论分析和最优阶误差估计。我们将采用混合元方法离散，并通过构造合适的光滑子，网格间的插值算子以及限制、提升算子，给出算法设计。最后，将结果推广到三维情形，模拟实际问题，实现相应的计算程序。本项目旨在为非线性多重网格方法在不同领域的应用和发展提供理论与技术支持。

Mathematical and physical model for fluid flow in porous media leads to highly coupled time-dependent nonlinear partial differential equations, which are widely used in the exploration and production of underground fluid resources such as oil, natural gas in petroleum reservoir. Numerical simulation of porous flow not only covers wide areas also long duration of time, this results in high computation complexity, slow convergence and long computation time. Therefore, it is urgent to design fast solvers for the coupled nonlinear models, and further research is needed. Notice that it is not easy to construct fast algorithms for coupled nonlinear models because they are very complicated with the involved complex interface conditions and usually appear as saddle point problems. Inspired by this, the project mainly focus on fast algorithms for two dimensional coupled Stokes/Darcy-Forchheimer model. The coupled nonlinear models, which are decoupled by the appropriate algorithms, are discretized by mixed finite element methods. Robust and efficient nonlinear geometry multigrid methods are designed with suitable smoothers and lifting operators for the considered nonlinear models, and rigorous theoretical analysis and numerical experiments are given. Moreover, efficient nonlinear multigrid methods are expanded to three dimensional problems. The project is a further exploration and supplement to applications and theoretical research on fast algorithms for coupled nonlinear models for fluid flow in porous media. In addition, the completion of this project will provide theoretical and technical support to the various applications of nonlinear multigrid methods.