The pointwise stabilities of piecewise linear finite element method on non-obtuse tetrahedral meshes of nonconvex polyhedra

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{align*} \Vert R_{h} u\Vert_{L^{\infty}(\Omega)} \leq C \vert \log h \vert \Vert u\Vert_{L^{\infty}(\Omega)}. \end{align*} If we further assume $u \in W^{1,\infty}(\Omega)$, then \begin{align*} \Vert R_{h} u\Vert_{W^{1, \infty}(\Omega)} \leq C \vert \log h \vert \Vert u\Vert_{W^{1, \infty}(\Omega)}. \end{align*}