Abstract A two-level variational multiscale meshless local Petrov-Galerkin (VMS-MLPG) method is presented for incompressible Navier-Stokes equations based on two local Gauss integrations which effectively replace a unit operator (first level) and an orthogonal project operator (second level). The present VMS-MLPG method allows arbitrary combinations of interpolation functions for the velocity and pressure fields, specifically the equal-order interpolations that are easy to implement and satisfy the Babuska-Breezi (B-B) condition. The prediction accuracy and the numerical stability of the proposed method for the lid-driven cavity flow and the backward facing step flow problems are analyzed and validated by comparing with the SUMLPG method and the benchmark solutions. It is shown that the present VMS-MLPG method can guarantee the numerical stability and obtain the reasonable solutions for incompressible Navier-Stokes equation.

中文翻译：

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不可压缩 Navier-Stokes 方程的两级变分多尺度无网格局部 Petrov-Galerkin (VMS-MLPG) 方法

摘要 针对不可压缩 Navier-Stokes 方程，提出了一种基于两个局部高斯积分的两级变分多尺度无网格局部 Petrov-Galerkin (VMS-MLPG) 方法，该方法有效地替代了单位算子（第一级）和正交投影算子（第二级）。 ）。目前的 VMS-MLPG 方法允许速度场和压力场的插值函数的任意组合，特别是易于实现并满足 Babuska-Breezi (BB) 条件的等阶插值。通过与 SUMLPG 方法和基准解决方案的比较，分析和验证了所提出的盖子驱动腔流和后向阶梯流问题的预测精度和数值稳定性。