定常Navier-Stokes方程基于速度投影的等阶元稳定化方法

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

摘要:

针对定常的Navier-Stokes方程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，增加了压力投影稳定项。基于速度投影的稳定化方法增强了$L^2$ 范数的稳定性，而不同以往增强的是$H^1$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不需要求解速度梯度投影。假设连续的Navier-Stokes方程存在唯一一支非奇解的情况下，证明了该离散格式是稳定的，且存在唯一的一支非奇解。此外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。

Abstract:

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.