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.