非定常Navier-Stokes方程的并行两水平稳定有限元算法

Parallel Two-level Stabilized Finite Element Algorithms for Unsteady Navier-Stokes Equations

使用标准的有限元方法求解非定常Navier-Stokes方程所得速度误差常受压力误差影响,且误差随粘性系数的减少而增大。为了增强压力的鲁棒性,本文引入grad-div稳定项,以提高近似解的精度,提出数值求解非定常Navier-Stokes方程的并行两水平grad-div稳定有限元算法,其时间和空间离散分别采用隐式Euler格式和Galerkin有限元方法。首先在全局粗网格上求解非线性grad-div稳定问题,然后在相互重叠的细网格子区域上并行求解grad-div稳定问题,以校正粗网格解。最后给出数值实验验证理论分析的正确性和算法的有效性。

In numerical solution of unsteady Navier-Stokes equations with standard finite element method,errors of computed velocity are usually affected by pressure errors,where smaller viscosity coefficients lead to greater velocity errors.To improve pressurerobustness,in this paper,we introduce a grad-div stabilization term to improve accuracy of approximate solutions.We present parallel two-level grad-div stabilized finite element algorithms for unsteady Navier-Stokes equations,where implicit Euler scheme and Galerkin finite element methods are used for temporal and spatial discretizations,respectively.At each time step,firstly we solve a nonlinear Navier-Stokes problem with grad-div stabilization on a coarse grid,and then linearized and grad-div stabilized problems are solved with Stokes,Oseen and Newton iterations on overlapping fine grid subdomains in a parallel manner to correct the coarse grid solution.Finally,numerical experiments are given to verify correctness of theoretical predictions and demonstrate effectiveness of the algorithms.