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 $r\geqslant 1$, satisfies the following weak maximum principle: \begin{align*} \left \|u_{h}\right \|_{L^{\infty }(\varOmega )} \leqslant 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 )} \leqslant C\|u\|_{L^{\infty }(\varOmega )} . \end{align*} Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.