三维定常Navier-Stokes方程有限元/有限体积方法非奇异解束L~∞优化阶分析研究

Optimal L~∞ analysis of non-singular finite element methods/finite volume methods for the stationary 3D Navier-Stokes equations

本文主要对三维定常Navier-Stokes方程有限元/有限体积方法非奇异解束L~∞优化阶分析进行研究,利用低阶宏元逼近、精细的三线性项估计技巧及Green函数和加权技巧,得到相应的有限元方法关于速度梯度和压力变量L~∞的优化阶分析;以有限元解为插值,利用有限元与有限体积方法之间等价性,突破有限体积体系试验函数与检验函数不在同一空间且仅有O(h)阶误差的限制,得到有限体积方法与有限元方法解之间有趣的结果:速度梯度和压力变量L^2模具有O(h^(3/2))阶的超逼近结果,且L~∞模具有O(h)阶的优化收敛结果.进一步得到相应的有限体积方法非奇异解束L~∞模的优化阶收敛分析.

In this paper, we develop and analyze the L∞ stability and convergence analysis for non-singular finite element and finite volume solutions for the stationary 3D Navier-Stokes equations. We obtain optimal esti- mates for the gradient of velocity and the pressure in the L∞-norm by applying the stabilization of a macro-element and technical lemmas including weighted L2-norm estimates for the regularized Green＇s functions associated with the Stokes problem. Moreover, using the finite element solutions as interpolations, the relationship between the finite element method and the finite volume method is used to obtain the interesting super-close convergence rate with O（h3/2） in the L2-norm and the optimal rate with O（h） in the L∞-norm between the finite element method and the finite volume method for the velocity gradient and the pressure. Furthermore, optimal error estimates in the L∞-norm are derived for the first time for the velocity gradient and pressure without a logarithmic factor O（｜ log h｜） for the stationary 3D Naiver-Stokes equations.