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A multiscale finite element method for the incompressible Navier-Stokes equations

机译：不可压缩Navier-Stokes方程的多尺度有限元方法

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This paper presents a new multiscale finite element method for the incompressible Navier-Stokes equations. The proposed method arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. Modeling of the unresolved scales corrects the lack of stability of the standard Galerkin formulation and yields a method that possesses superior properties like that of the streamline upwind/Petrov-Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The multiscale method allows arbitrary combinations of interpolation functions for the velocity and the pressure fields, specifically the equal order interpolations that are easy to implement but violate the celebrated Babuska-Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor τ appears naturally via the solution of the fine-scale problem. A family of 2-D elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed. Convergence studies for the method on uniform, skewed as well as composite meshes are presented. Numerical simulations of the nonlinear steady and transient flow problems are shown that exhibit the good stability and accuracy properties of the method.
机译：本文为不可压缩的Navier-Stokes方程提出了一种新的多尺度有限元方法。所提出的方法是由速度场分解为粗/分辨标度和细/未分辨标度而产生的。对未解决的比例进行建模可纠正标准Galerkin配方缺乏稳定性的问题，并产生一种具有优异特性的方法，如流线上风/ Petrov-Galerkin（SUPG）方法和Galerkin /最小二乘（GLS）方法。多尺度方法允许对速度场和压力场进行插值函数的任意组合，特别是容易实现但违反著名的Babuska-Brezzi条件的等阶插值。本方法的显着特征是稳定张量τ的结构通过精细尺度问题的解决而自然地出现。已经开发了包括3和6个节点三角形以及4和9个节点四边形的2-D元素族。提出了关于均匀，偏斜以及复合网格的方法的收敛性研究。结果表明，非线性稳态和瞬态流动问题的数值模拟显示了该方法的良好稳定性和准确性。

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