Uniform lower and upper bounds for positive finite-element approximations
to semilinear elliptic equations in several space dimensions subject to
mixed Dirichlet-Neumann boundary conditions are derived. The main feature is
that the non-linearity may be non-monotone and unbounded. The discrete minimum
principle provides a positivity-preserving approximation if the discretization
parameter is small enough and if some structure conditions on the non-linearity
and the triangulation are assumed. The discrete maximum principle also holds
for degenerate diffusion coefficients. The proofs are based on Stampacchia's truncation
technique and on a variational formulation. Both methods are settled on
careful estimates on the truncation operator.