Discrete maximum principles are essential measures of the qualitative reliability of the given numerical method, therefore they have been in the focus of intense research, including nonlinear elliptic boundary value problems describing stationary states in many nonlinear processes. In this paper we consider a general class of nonlinear elliptic problems which covers various special cases and applications. We provide exactly computable conditions on the geometric characteristics of widely studied finite element shapes: triangles, tetrahedra, prisms and rectangles, and guarantee the validity of discrete maximum priciples under these conditions.