Abstract

Based on two-grid discretizations and quadratic equal-order finite elements for the velocity and pressure approximations, we develop a three-step defect-correction stabilized algorithm for the incompressible Navier-Stokes equations, where non-homogeneous Dirichlet boundary conditions are considered and high Reynolds numbers are allowed. In this developed algorithm, we first solve an artificial viscosity stabilized nonlinear problem on a coarse grid in a defect step and then correct the resulting residual by solving two stabilized and linearized problems on a fine grid in correction steps. While the fine grid correction problems have the same stiffness matrices with only different right-hand sides. We use a variational multiscale method to stabilize the system, making the algorithm has a broad range of potential applications in the simulation of high Reynolds number flows. Under the weak uniqueness condition, we give a stability analysis of the present algorithm, analyze the error bounds of the approximate solutions, and derive the algorithmic parameter scalings. Finally, we perform a series of numerical examples to demonstrate the promise of the proposed algorithm.

中文翻译：

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非齐次狄利克雷边界条件不可压缩流的三步缺陷校正稳定算法

摘要

基于速度和压力近似的双网格离散化和二次等阶有限元，我们开发了一种用于不可压缩纳维-斯托克斯方程的三步缺陷校正稳定算法，其中考虑了非齐次狄利克雷边界条件，并且高雷诺数是允许的。在这个开发的算法中，我们首先在缺陷步骤中解决粗网格上的人工粘度稳定非线性问题，然后通过在校正步骤中解决细网格上的两个稳定和线性化问题来校正所得残差。而细网格校正问题具有相同的刚度矩阵，只是右侧不同。我们采用变分多尺度方法来稳定系统，使得该算法在高雷诺数流的模拟中具有广泛的潜在应用。在弱唯一性条件下，我们对算法进行了稳定性分析，分析了近似解的误差界，并推导了算法的参数标度。最后，我们通过一系列数值示例来证明所提出算法的前景。