标题： Navier-Stokes方程的时间序列展开和稳定化有限元法的数值积分 显示英文标题

标题： Numerical Integration of Navier-Stokes Equations by Time Series Expansion and Stabilized FEM

摘要： 本文档介绍了一种用于不可压缩Navier-Stokes（NS）方程的高级数值方法，该方法在有限元法（FEM）框架内使用时间序列展开（TSE）技术。该技术通过引入一种新的稳定策略得到增强，该策略采用发散级数重求和（DSR）技术，显著提高了算法的计算效率。稳定机制经过精心设计，以提高计算级数项的稳定性和有效性，从而能够应用阶乘级数（FS）算法进行级数重求和。这种方法对于解决与NS方程精确且稳定的数值解相关的问题至关重要，这些问题是计算流体力学（CFD）应用中的关键问题。 本文档详细阐述了Stokes问题的变分公式，并使用Ladyzhenskaya-Babuska-Brezzi（LBB）条件进行了收敛性分析。随后介绍了NS方程以及稳定技术的实现细节，并通过层流绕过圆柱体的数值测试展示了该方法的有效性和在流体动力学模拟中的广泛应用潜力。稳定性的结果显示，计算稳定性和精度有了显著提升，为该领域的未来研究提供了有前景的方向。 显示英文摘要

摘要： This manuscript introduces an advanced numerical approach for the integration of incompressible Navier-Stokes (NS) equations using a Time Series Expansion (TSE) method within a Finite Element Method (FEM) framework. The technique is enhanced by a novel stabilization strategy, incorporating a Divergent Series Resummation (DSR) technique, which significantly augments the computational efficiency of the algorithm. The stabilization mechanism is meticulously designed to improve the stability and validity of computed series terms, enabling the application of the Factorial Series (FS) algorithm for series resummation. This approach is pivotal in addressing the challenges associated with the accurate and stable numerical solution of NS equations, which are critical in Computational Fluid Dynamics (CFD) applications. The manuscript elaborates on the variational formulation of Stokes problem and present convergence analysis of the method using the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. It is followed by the NS equations and the implementation details of the stabilization technique, underscored by numerical tests on laminar flow past a cylinder, showcasing the method's efficacy and potential for broad applicability in fluid dynamics simulations. The results of the stabilization indicate a substantial enhancement in computational stability and accuracy, offering a promising avenue for future research in the field.