Weak discrete maximum principle of finite element methods in convex polyhedra

Abstract: We prove that the Galerkin finite element solution $u_h$ of the Laplace equation in a convex polyhedron $\varOmega$, with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree $r\ge 1$, satisfies the following weak maximum principle: \begin{align*} \left|u_{h}\right|{L^{\infty}(\varOmega)} \le C\left|u{h}\right|{L^{\infty}(\partial \varOmega)} , \end{align*} with a constant $C$ independent of the mesh size $h$. By using this result, we show that the Ritz projection operator $R_h$ is stable in $L^\infty$ norm uniformly in $h$ for $r\geq 2$, i.e. \begin{align*} |R_hu|{L^{\infty}(\varOmega)} \le C|u|_{L^{\infty}(\varOmega)}. \end{align*} Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.