Weak discrete maximum principle of finite element methods in convex polyhedra

TL;DR

This paper establishes a weak maximum principle for finite element solutions of the Laplace equation in convex polyhedra, demonstrating stability of the Ritz projection operator in the supremum norm and removing previous logarithmic factors.

Contribution

It proves a weak maximum principle for finite element solutions in convex polyhedra and shows uniform $L^$ stability of the Ritz projection operator for higher-degree elements.

Findings

Weak maximum principle holds with mesh-independent constant.

Ritz projection operator is stable in $L^$ norm for $r  2$.

Removes logarithmic factor in previous convex polyhedral domain results.

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.