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 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:with a constant C independent of the mesh size h. By using this result, we show that Ritz projection operatorThus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.