A High-Order, Analytically Divergence-Free Approximation Method for the Time-Dependent Stokes Problem
A High-Order, Analytically Divergence-Free Approximation Method for the Time-Dependent Stokes Problem
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DOI：
10.1137/151006196
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发表时间：
2016-04
期刊：
影响因子：
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通讯作者：
C. Keim;H. Wendland
中科院分区：
文献类型：
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作者：
C. Keim;H. Wendland
我们发展并分析了一种新的含时Stokes方程的高阶近似方法。我们的方法基于Leray投影，它将Stokes方程转化为向量值的热方程。然后，我们使用无网格近似空间在空间中进行离散。这些离散近似空间使用一对精心选择的矩阵值核函数，其中，速度由解析上无发散的函数来近似。此外，解的压力部分与速度是同时计算的，因此不需要解决额外的辅助问题，也不需要满足inf-sup条件。最后，由于我们在空间中使用配置，所以不需要空间数值积分。我们将在周期背景下对该方法进行严格的分析，但也指出该方法如何在更一般的情况下使用。
We develop and analyze a new high-order approximation method for the time-dependent Stokes equation. Our method is based upon a Leray projection, which converts the Stokes equation into a vector-valued heat equation. We then employ meshfree approximation spaces to discretize in space. These discrete approximation spaces use a well-chosen pair of matrix-valued kernels such that, among other things, the velocity is approximated by an analytically divergence-free function. Moreover, the pressure part of the solution is computed simultaneously with the velocity, so that no additional auxiliary problem has to be solved and so that no inf-sup conditions have to be satisfied. Finally, since we use collocation in space, no spatial numerical integration is required. We will give a rigorous analysis of the method in the periodic setting but also point out how the method can be used in more general situations.