Weak discrete maximum principle of finite element methods in convex polyhedra.

We prove that the Galerkin finite element solution uh of the Laplace equation in a convex polyhedron Ω, with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree r ≥ 1, satisfies the following weak maximum principle: ||uh||L∞(Ω) ≤ C||uh||L∞(∂Ω), with a constant C independent of the mesh size h. By using this result, we show that the Ritz projection operator Rh is stable in L∞ norm uniformly in h for r ≥ 2, i.e., ||Rhu||L∞(Ω) ≤ C||u||L∞(Ω). Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains. [ABSTRACT FROM AUTHOR]