  The generalized weak Galerkin (gWG) finite element method is proposed and
analyzed for the biharmonic equation. A new generalized discrete weak second
order partial derivative is introduced in the gWG scheme to allow arbitrary
combinations of piecewise polynomial functions defined in the interior and on
the boundary of general polygonal or polyhedral elements. The error estimates
are established for the numerical approximation in a discrete H^2 norm and a
L^2 norm. The numerical results are reported to demonstrate the accuracy and
flexibility of our proposed gWG method for the biharmonic equation.
