  This paper analyses discontinuous Galerkin finite element methods (DGFEM) to
approximate a regular solution to the von K\'arm\'an equations defined on a
polygonal domain. A discrete inf-sup condition sufficient for the stability of
the discontinuous Galerkin discretization of a well-posed linear problem is
established and this allows the proof of local existence and uniqueness of a
discrete solution to the non-linear problem with a Banach fixed point theorem.
The Newton scheme is locally second-order convergent and appears to be a robust
solution strategy up to machine precision. A comprehensive a priori and a
posteriori energy-norm error analysis relies on one sufficiently large
stabilization parameter and sufficiently fine triangulations. In case the other
stabilization parameter degenerates towards infinity, the DGFEM reduces to a
novel $C^0$ interior penalty method (IPDG). Moreover, a reliable and efficient
a posteriori error analysis immediately follows for the DGFEM of this paper,
while the different norms in the known $C^0$-IPDG lead to complications with
some non-residual type remaining terms. Numerical experiments confirm the
best-approximation results and the equivalence of the error and the error
estimators. A related adaptive mesh-refining algorithm leads to optimal
empirical convergence rates for a non convex domain.
