  The Morley finite element method (FEM) is attractive for semilinear problems
with the biharmonic operator as a leading term in the stream function vorticity
formulation of 2D Navier-Stokes problem and in the von K\'{a}rm\'{a}n
equations. This paper establishes a best-approximation a~priori error analysis
and an a~posteriori error analysis of discrete solutions close to an arbitrary
regular solution on the continuous level to semilinear problems with a
trilinear nonlinearity. The analysis avoids any smallness assumptions on the
data and so has to provide discrete stability by a perturbation analysis before
the Newton-Kantorovic theorem can provide the existence of discrete solutions.
An abstract framework for the stability analysis in terms of discrete operators
from the medius analysis leads to new results on the nonconforming
Crouzeix-Raviart FEM for second-order linear non-selfadjoint and indefinite
elliptic problems with $L^\infty$ coefficients. The paper identifies six
parameters and sufficient conditions for the local a~priori and a~posteriori
error control of conforming and nonconforming discretisations of a class of
semilinear elliptic problems first in an abstract framework and then in the two
semilinear applications. This leads to new best-approximation error estimates
and to a~posteriori error estimates in terms of explicit residual-based error
control for the conforming and Morley FEM.
