  It is shown that discretizations based on variational or weak formulations of
the plate bending problem with simple support boundary conditions do not lead
to failure of convergence when polygonal domain approximations are used and the
imposed boundary conditions are compatible with the nodal interpolation of the
restriction of certain regular functions to approximating domains. It is
further shown that this is optimal in the sense that a full realization of the
boundary conditions leads to failure of convergence for conforming methods. The
abstract conditions imply that standard nonconforming and discontinuous
Galerkin methods converge correctly while conforming methods require a suitable
relaxation of the boundary condition. The results are confirmed by numerical
experiments.
