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

Let $\mathrm{\Omega }$ 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 $u\in W^{1,\mathrm{\infty }}(\mathrm{\Omega })$, 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*}