  We present a mixed finite element method with triangular and parallelogram
meshes for the Kirchhoff-Love plate bending model. Critical ingredient is the
construction of low-dimensional local spaces and appropriate degrees of freedom
that provide conformity in terms of a sufficiently large tensor space and that
allow for any kind of physically relevant Dirichlet and Neumann boundary
conditions. For Dirichlet boundary conditions and polygonal plates, we prove
quasi-optimal convergence of the mixed scheme. An a posteriori error estimator
is derived for the special case of the biharmonic problem. Numerical results
for regular and singular examples illustrate our findings. They confirm
expected convergence rates and exemplify the performance of an adaptive
algorithm steered by our error estimator.
