  This paper derives some discrete maximum principles for $P1$-conforming
finite element approximations for quasi-linear second order elliptic equations.
The results are extensions of the classical maximum principles in the theory of
partial differential equations to finite element methods. The mathematical
tools are based on the variational approach that was commonly used in the
classical PDE theory. The discrete maximum principles are established by
assuming a property on the discrete variational form that is of global nature.
In particular, the assumption on the variational form is verified when the
finite element partition satisfies some angle conditions. For the general
quasi-linear elliptic equation, these angle conditions indicate that each
triangle or tetrahedron needs to be $\mathcal{O}(h^\alpha)$-acute in the sense
that each angle $\alpha_{ij}$ (for triangle) or interior dihedral angle
$\alpha_{ij}$ (for tetrahedron) must satisfy $\alpha_{ij}\le \pi/2-\gamma
h^\alpha$ for some $\alpha\ge 0$ and $\gamma>0$. For the Poisson problem where
the differential operator is given by Laplacian, the angle requirement is the
same as the existing ones: either all the triangles are non-obtuse or each
interior edge is non-negative. It should be pointed out that the analytical
tools used in this paper are based on the powerful De Giorgi's iterative method
that has played important roles in the theory of partial differential
equations. The mathematical analysis itself is of independent interest in the
finite element analysis.
