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.

翻译：近些年来，不可压缩Stokes方程精确保持不可压缩约束的离散化方法受到了广泛关注。这些方法具有显著意义，因为此类离散格式是压力鲁棒的，即速度误差估计不依赖于压力误差。类似的考量也出现在几乎不可压缩线弹性固体问题中。具有该性质的协调离散格式在二维情形下已得到充分理解，但在三维情形下仍认识有限。本文就这一主题提出两个猜想：其一，Scott-Vogelius单元对在速度次数$k \ge 4$的一致网格上满足inf-sup稳定性，而现有文献中的最佳结果为$k \ge 6$；其二，对于$k \ge 5$，存在散度核的稳定空间分解。我们给出了支持这些猜想的数值证据。