定常Navier-Stokes方程基于速度投影的等阶元稳定化方法

针对定常的Navier-Stokes方程,本文给出并分析了基于速度场L~2投影的新型稳定化有限元方法.速度-压力逼近采用了P_1/P_1元.为了克服等阶元不满足inf-sup条件的问题,本文增加了压力投影稳定项.基于速度场L~2投影的稳定化方法,本文增强了L~2范数的稳定性.该稳定化格式的优点是所有的计算都在同一套网格上执行,不需要嵌套网格且只涉及速度场投影而不需要求解速度梯度投影.在连续的Navier-Stokes方程存在唯一一支非奇解的情况下,本文证明了该离散格式是稳定的.此外,本文还得出了离散解的误差估计.数值实验证实该方法是有效的.

Velocity Projection-Based Stabilized Finite Element Method for Steady Navier-Stokes Equations using equal order interpolation

A new type of velocity $L^2$ projection-based stabilized finite element method for steady Navier-Stokes equations is proposed and analyzed. Velocity and pressure are approximated equal order element $P_1/P_1$. To overcome the violation of discrete inf-sup condition when equal order elements are used, pressure projection stabilized term is added. Velocity projection-based stabilized method directly increases the $L^2$-stability instead of $H^2$-stability. The main advantages of the proposed methods lies in that, all the computations are performed at the same element level, without the need of nested meshes and the projection of the gradient of velocity. It is showed that this discrete model is stable and has a unique branch of nonsingular solutions , given the continuous Navier-Stokes equations has an unique branch of nonsingular solutions . Moreover, error estimates are derived, and numerical experiments show that the method is valid.