Weak discrete maximum principle of finite element methods in convex polyhedra

Abstract

We prove that the Galerkin finite element solution u h u_h of the Laplace equation in a convex polyhedron Ω \varOmega , with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree r ⩾ 1 r\geqslant 1 , satisfies the following weak maximum principle: ‖ u h ‖ L ∞ ( Ω ) ⩽ C ‖ u h ‖ L ∞ ( ∂ Ω ) , \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 C independent of the mesh size h h . By using this result, we show that the Ritz projection operator R h R_h is stable in L ∞ L^\infty norm uniformly in h h for r ≥ 2 r\geq 2 , i.e., ‖ R h u ‖ L ∞ ( Ω ) ⩽ C ‖ u ‖ L ∞ ( Ω ) . \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.