Let \(\Omega \) be a Lipschitz polyhedral (can be nonconvex) domain in \({\mathbb {R}}^{3}\), and \(V_{h}\) denotes the finite element space of continuous piecewise linear polynomials. On non-obtuse quasi-uniform tetrahedral meshes, we prove that the finite element projection \(R_{h}u\) of \(u \in H^{1}(\Omega ) \cap C({\overline{\Omega }})\) (with \(R_{h} u\) interpolating u at the boundary nodes) satisfies

$$\begin{aligned} \Vert R_{h} u\Vert _{L^{\infty }(\Omega )} \le C \vert \log h \vert \Vert u\Vert _{L^{\infty }(\Omega )}. \end{aligned}$$

If we further assume \(u \in W^{1,\infty }(\Omega )\), then

$$\begin{aligned} \Vert R_{h} u\Vert _{W^{1, \infty }(\Omega )} \le C \vert \log h \vert \Vert u\Vert _{W^{1, \infty }(\Omega )}. \end{aligned}$$

中文翻译：

---

非凸多面体的非钝四面体网格上分段线性有限元方法的点稳定性

令\（\ Omega \）为\（{\ mathbb {R}} ^ {3} \）中的Lipschitz多面域（可以为非凸），而\（V_ {h} \）表示连续的有限元空间分段线性多项式。在非钝角准均匀四面体网格中，我们证明了有限元投影\（R_ {H} U \）的\（U \在H ^ {1}（\欧米茄）\帽C（{\划线{\ Omega}}）\）（其中\（R_ {h} u \）在边界节点处插值u）满足

$$ \ begin {aligned} \ Vert R_ {h} u \ Vert _ {L ^ {\ infty}（\ Omega}} \ le C \ vert \ log h \ vert \ Vert u \ Vert _ {L ^ {\狡猾的}（\ Omega}}。\ end {aligned} $$

如果我们进一步假设\（u \ in W ^ {1，\ infty}（\ Omega）\），则

$$ \ begin {aligned} \ Vert R_ {h} u \ Vert _ {W ^ {1，\ infty}（\ Omega}} \ le C \ vert \ log h \ vert \ Vert u \ Vert _ {W ^ {1，\ infty}（\ Omega}}。\ end {aligned} $$