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

Let $Ω$ 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 $u \in H^{1}(Ω) \cap C(\overlineΩ)$ (with $R_{h} u$ interpolating $u$ at the boundary nodes) satisfies \begin{align*} \Vert R_{h} u\Vert_{L^{\infty}(Ω)} \leq C \vert \log h \vert \Vert u\Vert_{L^{\infty}(Ω)}. \end{align*} If we further assume $u \in W^{1,\infty}(Ω)$, then \begin{align*} \Vert R_{h} u\Vert_{W^{1, \infty}(Ω)} \leq C \vert \log h \vert \Vert u\Vert_{W^{1, \infty}(Ω)}. \end{align*}

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