  A new weak Galerkin (WG) finite element method is introduced and analyzed in
this paper for the biharmonic equation in its primary form. This method is
highly robust and flexible in the element construction by using discontinuous
piecewise polynomials on general finite element partitions consisting of
polygons or polyhedra of arbitrary shape. The resulting WG finite element
formulation is symmetric, positive definite, and parameter-free. Optimal order
error estimates in a discrete $H^2$ norm is established for the corresponding
WG finite element solutions. Error estimates in the usual $L^2$ norm are also
derived, yielding a sub-optimal order of convergence for the lowest order
element and an optimal order of convergence for all high order of elements.
Numerical results are presented to confirm the theory of convergence under
suitable regularity assumptions.
