本项目主要研究间断Galerkin（DG）方法数值求解Navier-Stokes（NS）方程支配的变分和半变分不等式。很多流体动力学问题有滑移和渗透现象，需要由带有摩擦型非线性滑移或渗透边界条件的NS方程描述。这类问题的边界条件具有次可微性，可以被变分或半变分不等式很好的刻画。与连续有限元相比，DG方法求解NS方程更具稳定性、准确性，且能保持物理量的局部守恒性，捕捉到物理量的间断信息，可以轻松的实现自适应算法，且非常适合并行运算。通过本项目的研究，我们将推导设计这类变分和半变分不等式的DG格式，建立先验误差分析；然后研究数值格式的后验误差估计，为实现自适应DG算法提供可靠且有效的误差估计子。本项目的目标是建立起一套完整的体系，为高效计算这类问题提供理论基础，也为相关实际问题的计算模拟、算法设计提供理论指导，达到在现有硬件资源条件下，扩大计算规模和提高计算精度的目的。

The project studies discontinuous Galerkin (DG) methods for numerically solving variational and hemivariational inequalities controlled by Navier-Stokes (NS) equations. Slip and leak phenomena appear in many fluid dynamics problems, which can be described by NS equations with nonlinear slip or leak boundary conditions of frictional type. The boundary conditions of these problems are subdifferentiable, so these problems can be modeled by variational or hemivariational inequalities. To solve the NS equations, compared with continuous finite element method, DG methods are more stable and accurate. DG methods are local conservative by design, and can capture discontinuous physical quantities. In addition, they are ideally suited for implementing adaptive algorithm and parallel computing. Through the research of this project, we will derive and design stable and efficient DG schemes for these variational and hemivariational inequalities, establish a priori error analysis, and study a posteriori error estimates to derive reliable and efficient error estimators for implementing adaptive DG algorithm. The main goal of this project is to establish a theoretical foundation for efficiently solving variational and hemivariational inequalities controlled by NS equations; furthermore, it will serve as a theorectical guideline of numerical simulation and algorithms design for related practical problems, so that we can solve large scale problems with high precision under existing hardware conditions.