基于间断有限元方法的 Navier-Stokes 变分不等问题高效算法研究

Navier-Stokes variational inequality problem is of great importance in fluid mechanics, because the fluid processes can be accurately described by the variational inequality when the motion of fluid has nonlinear slip boundary conditions. Based on the merit of allowing the discontinuous solution in domain and dangling nodes in mesh of discontinuous finite element method, and particularity of the nonlinear slip boundary conditions, this project will investigate high accuracy and high efficiency numerical methods for Navier-Stokes variational inequality problem. This project is composed of the following parts: establishing discontinuous finite element methods and building adaptive discontinuous finite element methods for 2D Navier-Stokes variational inequality problem; setting up dimensional splitting algorithms under discontinuous finite element formulation for 3D Navier-Stokes variational inequality problem, and providing new numerical simulation for investigating blood flow in thoracic aorta. This project will promote the cross development of computational mathematics and fluid mechanics, improve the numerical theory and numerical methods for Navier-Stokes variational inequality problem.

Navier-Stokes 变分不等问题在流体力学中占有十分重要的地位, 特别是当流体的流动边界是非线性滑移边界时, 流体的流动过程则需要通过建立变分不等式进行研究。 鉴于间断有限元允许解在区域内部间断以及网格剖分中出现悬点的优势和非线性滑移边界条件的特殊性, 本项目将研究 Navier-Stokes 变分不等问题的高精度、高效率的数值方法。 具体研究内容包括: 就二维 Navier-Stokes 变分不等问题建立间断有限元方法和自适应间断有限元方法; 在间断有限元的框架下, 就三维 Navier-Stokes 变分不等问题构造维数分裂算法, 并实际应用于动脉硬化患者血液流动的数值模拟中。 本项目的研究成果将推进计算数学、流体力学的交叉发展, 进一步完善 Navier-Stokes 变分不等问题的数值理论和数值方法。

本项目综合运用了计算数学、变分不等式和流体力学的知识，研究了Navier-Stokes 变分不等式问题的高效算法。具体研究内容包括：针对二维 Navier-Stokes 变分不等问题建立了间断有限元方法和自适应间断有限元方法；针对三维 Navier-Stokes 变分不等问题构造了维数分裂算法。进一步，研究了算法的收敛性和稳定性，编写了新的程序并通过数值算例验证了理论分析结果。本项目已发表8篇SCI论文，相关研究成果在理论方面能够丰富变分不等问题的数值算法，在应用方面能够指导动脉硬化患者血液流动中数值模拟的相关医学问题，并且推动了计算数学、变分不等式与流体力学的交叉发展。