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

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

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

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