  In this article we consider a priori error and pointwise estimates for finite
element approximations of solutions to semilinear elliptic boundary value
problems in d>=2 space dimensions, with nonlinearities satisfying critical
growth conditions. It is well-understood how mesh geometry impacts finite
element interpolant quality, and leads to the reasonable notion of shape
regular simplex meshes. It is also well-known how to perform both mesh
generation and simplex subdivision, in arbitrary space dimension, so as to
guarantee the entire hierarchy of nested simplex meshes produced through
subdivision continue to satisfy shape regularity. However, much more
restrictive angle conditions are needed for basic a priori quasi-optimal error
estimates, as well as for a priori pointwise estimates. These angle conditions,
which are particularly difficult to satisfy in three dimensions in any type of
unstructured or adaptive setting, are needed to gain pointwise control of the
nonlinearity through discrete maximum principles. This represents a major gap
in finite element approximation theory for nonlinear problems on unstructured
meshes, and in particular for adaptive methods. In this article, we close this
gap in the case of semilinear problems with critical or sub-critical nonlinear
growth, by deriving a priori estimates directly, without requiring the discrete
maximum principle, and hence eliminating the need for restrictive angle
conditions. Our main result is a type of local Lipschitz property that relies
only on the continuous maximum principle, together with the growth condition.
We also show that under some additional smoothness assumptions, the a priori
error estimate itself is enough to give pointwise control the discrete
solution, without the need for restrictive angle conditions. Numerical
experiments confirm our theoretical conclusions.
