耦合问题基于高斯积分的稳定化有限元方法研究

In this project, we will study the new stabilized finite element method for the Navier-Stokes(Stokes)-Darcy coupled problem. This problem comes from the model of the procedure of pollutants discharged into aquifers. we use P1/P1 element or nonconforming P1/P1 element for the incompressible flow and P1/P1 element for the porous media problem. For the low and equal order finite elements don't satisfy the Inf-Sup stability condition, a new stabilized method based on local Gauss integration is introduced. In computation, moreover, because the scale and the coefficients in two sub-domains are different, we allow the meshes in different sub-domains to be nonmatching at the interface so that one can adjust the mesh sizes to deal with problems caused by heterogeneities in scales or anisotropies. Firstly, the new stabilized method is used to solve the stationary or nonstationary coupled problem. The stability and convergence of this method are proved, and the numerical experiments are presented to show the efficiency of this method, and the superconvergence of this method is discussed. Then the stabilized combined with two-level method is used to solve this coupled problem under the appropriate mesh size. The study of the project have more important value in the mathematical theory and numerical method, and the related subjects are also the frontier and popular topics in the field of partial differential equations about fluid mechanics.

本项目拟对不可压缩流Navier-Stokes/Stokes与渗流Darcy耦合问题的稳定化有限元方法进行研究。该问题主要源于对实际生活中地表污水渗透到地下水模型的研究。在不可压缩流体区域和渗流区域分别对速度和压力采用P1-P1或NCP1-P1元和P1-P1元，由于低次等阶有限元不满足Inf-Sup条件，从而引入基于局部高斯积分的稳定化方法。另由于两个子区域上的尺度和系数均不同，所以允许在两个子区域上的网格在界面处不匹配，这样便于调整网格尺度以处理不统一性所造成的问题。本项目首先用此稳定化方法研究定常和非定常状态的耦合问题，讨论算法的稳定性和收敛性，给出数值试验说明其优越性，并讨论该算法的超收敛性。然后在合适的网格尺度下，把该方法和两层网格方法结合起来解决这类问题。项目研究在数学理论和数值方法上均有较高的价值，相关问题也是目前关于流体力学的偏微分方程领域前沿和热门的研究课题。

本项目主要围绕不可压缩流NS方程，NS-Darcy耦合问题，牛顿流和非牛顿流方程展开研究工作。这些非线性问题都是来源于生活中的实际问题，例如耦合问题是模拟实际生活中地表污水渗透到地下水的过程，是否满足牛顿粘性定律也是自然界常见的流体形态等。我们采用两层稳定化非协调有限元方法对定常NS方程进行稳定性和收敛性分析并进行数值实验，对带有非线性滑动边界条件的NS方程利用非协调有限元和弱解的最大值原理讨论了速度和压力的先验误差估计，并进行收敛性分析；对耦合NS-Darcy问题，利用稳定化非协调有限元方法进行稳定性和收敛性分析；对牛顿流与非牛顿流问题，我们采用新的数学理论和方法，研究和分析了全局解和奇异解及其渐近行为。项目共发表高水平SCI论文8篇，录用1篇，发表国际重要期刊论文2篇，招收和指导硕士研究生1名。