  In this note we examine the a priori and a posteriori analysis of
discontinuous Galerkin finite element discretisations of semilinear elliptic
PDEs with polynomial nonlinearity. We show that optimal a priori error bounds
in the energy norm are only possible for low order elements using classical a
priori error analysis techniques. We make use of appropriate quasinorms that
results in optimal energy norm error control.
  We show that, contrary to the a priori case, a standard a posteriori analysis
yields optimal upper bounds and does not require the introduction of
quasinorms. We also summarise extensive numerical experiments verifying the
analysis presented and examining the appearance of layers in the solution.
