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

Abstract

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

5 pages