论文标题
在弗洛伊达尔网上的三个维度上的斯托克斯复合物上的两个猜想
Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes
论文作者
论文摘要
近年来，对不可压缩的Stokes方程式的离散化引起了极大的关注，这些方程式确切地保留了不可压缩的限制。这些具有很大的兴趣，因为这些离散化是压力稳定的，即速度的误差估计不取决于压力中的误差。在几乎不可压缩的线性弹性固体中也会产生类似的考虑。现在，与该特性的离散化符合离散化，在两个维度上已经充分理解，但在三个维度中仍然了解得很糟糕。在这项工作中，我们陈述了有关此主题的两个猜想。首先是Scott-Vogelius元素对在均匀网格上具有稳定的速度度稳定，$ k \ ge 4 $;文献中可用的最佳结果是$ k \ ge 6 $。第二个是存在$ k \ ge 5 $的差异内核的稳定空间分解。我们提供了支持我们猜想的数值证据。
In recent years a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust, i.e. the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions, but remain poorly understood in three dimensions. In this work we state two conjectures on this subject. The first is that the Scott-Vogelius element pair is inf-sup stable on uniform meshes for velocity degree $k \ge 4$; the best result available in the literature is for $k \ge 6$. The second is that there exists a stable space decomposition of the kernel of the divergence for $k \ge 5$. We present numerical evidence supporting our conjectures.
